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Tensor product categorifications, Verma modules and the blob 2-category

September 2021
Quantum Topology 12(4)

DOI:10.4171/QT/156

Authors:

Grégoire Naisse

None

Pedro Vaz

Catholic University of Louvain

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal \mathfrak{sl}_2 Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of \mathfrak{sl}_2 . From the above we derive a categorification of the blob algebra.

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Quantum Topol. 12 (2021), 705–812

DOI 10.4171/QT/156

Published by EMS Press

This work is licensed under a CC BY 4.0 license.

Tensor product categoriﬁcations,

Verma modules and the blob 2-category

Abel Lacabanne,1Grégoire Naisse,2and Pedro Vaz3

Abstract. We construct a dg-enhancement of KLRW algebras that categoriﬁes the tensor

product of a universal sl2Verma module and several integrable irreducible modules. When

the integrable modules are two-dimensional, we construct a categorical action of the blob

algebra on derived categories of these dg-algebras which intertwines the categorical action

of sl2. From the above we derive a categoriﬁcation of the blob algebra.

Mathematics Subject Classiﬁcation (2020). Primary: 18N25; Secondary: 20G42.

Keywords. Higher representation theory, 2-Verma, blob algebra, dg-enhanced KLRW

algebras.

Contents

1 Introduction................................ 706

2 Quantum sl2and the blob algebra . . . . . . . . . . . . . . . . . . . . 716

3 Dg-enhanced KLRW algebras . . . . . . . . . . . . . . . . . . . . . . 724

4 A categoriﬁcation of M./ ˝V . x

N / ................... 736

5 Cups, caps and double braiding functors . . . . . . . . . . . . . . . . . 742

6 A categoriﬁcation of the blob algebra . . . . . . . . . . . . . . . . . . 751

7 Variants and generalizations . . . . . . . . . . . . . . . . . . . . . . . 764

Appendices .................................. 776

A Detailed proofs and computations . . . . . . . . . . . . . . . . . . . . 776

B Homological toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . 800

References................................... 809

1Fonds de la Recherche Scientiﬁque - FNRS Grant no. MIS-F.4536.19.

2Fonds de la Recherche Scientiﬁque - FNRS, under Grant no. 1.A310.16. Grégoire Naisse is

also grateful to the Max Planck Institute for Mathematics in Bonn for its hospitality and ﬁnancial

support.

3Fonds de la Recherche Scientiﬁque - FNRS Grant no. MIS-F.4536.19.

706 A. Lacabanne, G. Naisse, and P. Vaz

1. Introduction

Dualities are fundamental tools in mathematics in general and in higher repre-

sentation theory in particular. For example, Stroppel’s version of Khovanov ho-

mology [41,42] and Khovanov’s HOMFLY–PT homology [19] can be seen as

instances of higher Schur–Weyl duality (see also [43] for further explanations). In

this paper we construct an instance of higher Schur–Weyl duality between Uq.sl2/

and the blob algebra of Martin and Saleur [30] by using a categoriﬁcation of the

tensor product of a Verma module and several two-dimensional irreducibles.

1.1. State of the art

1.1.1. Schur–Weyl duality, Uq.sl2/and the Temperley–Lieb algebra. Schur–

Weyl duality connects ﬁnite-dimensional modules of the general linear and sym-

metric groups. In particular, it states that over an algebraically closed ﬁeld the

actions of GLmand Sron the r-folded tensor power of the natural module Vof

GLmcommute and are the centralizers of each other. In the quantum version, GLm

and Srare replaced respectively by the quantum general linear algebra Uq.glm/

and the Hecke algebra Hr.q/. We note that these consequences of (quantum)

Schur–Weyl duality remain true if one replaces the general linear with the special

linear group. For example, in the case of mD2, the centralizer of the action

of Uq.sl2/on V˝ris the Temperley–Lieb algebra TLr, a well-known quotient of

the Hecke algebra. One of the applications of this connection is the construction

of the Jones–Witten–Reshetikhin–Turaev Uq.sl2/-tangle invariant as a state-sum

model (a linear combination of elements of TLr) called Kauﬀman bracket, which

was the version categoriﬁed by Khovanov [18] in the particular case of links.

1.1.2. The blob algebra. It was shown in [14] that, for a projective Uq.sl2/

Verma module Mwith highest weight (in the sense that is the eigenvalue), the

endomorphism algebra of M˝V˝ris the blob algebra BrDBr.q; / of Martin

and Saleur [30]. This algebra Br, which was unfortunately called Temperley–Lieb

algebra of type B in [14], is in fact a quotient of the Temperley–Lieb algebra of

type B [10,11]. Note that the parameters and qin [14] are not algebraically

independent but can be easily made independent by working with a universal

Verma module as in [26].

The blob algebra Brcan be given a diagrammatic presentation in terms of

Q.q; /-linear combinations of ﬂat tangle diagrams [14] on rC1strands, with

Tensor product categoriﬁcations and the blob 2-category 707

generators

uiWD : : : : : :

for iD1; : : : ; r 1, and

WD : : :

taken up to planar isotopy ﬁxing the endpoints, and subject to the usual Temperley–

Lieb relation of type A:

D .q Cq1/;

and the blob relations:

D .q C1q1/ ;

q1D.q C1q1/q :

Note that this generators-relations deﬁnition of the blob algebra makes also sense

over ZŒq˙1; ˙1.

Remark 1.1. In [30], the blob algebra is given a diﬀerent presentation, where

the generator of type B is pictured as a dot on the left-most strand, and is an

idempotent. We use the presentation given in [14], which is isomorphic to the one

in [30] over Z.q; / (but not over ZŒq˙1; ˙1). This presentation is closer to the

representation theory of Uq.sl2/and is the one that arises from our categoriﬁcation

construction.

More generally, we consider the category Bwith objects given by M˝V˝r

for various r2N, and hom-spaces given by Uq.sl2/-intertwiners. This category,

that we call the blob category, has a very similar diagrammatic description as the

708 A. Lacabanne, G. Naisse, and P. Vaz

blob algebra, where objects are collections of rC1points on the horizontal line.

The hom-spaces are presented by ﬂat tangles connecting these points, with the

left-most point of the source always connected to the left-most point of the target,

allowing 4-valent intersections between the ﬁrst two strands. These diagrams are

subject to the same relations as the blob algebra. We stress that, in contrast to

the Temperley–Lieb category of type A, the blob category is not monoidal with

respect to juxtaposition of diagrams since the blue strand in the pictures above

needs to be on the left-hand side of any diagram.

1.1.3. Webster categoriﬁcation. In a seminal paper [49], Webster has con-

structed categoriﬁcations of tensor products of integrable modules for symmetriz-

able Kac–Moody algebras, generalizing Lauda’s [27], Khovanov and Lauda [21,

22], and Chuang and Rouquier [6], and Rouquier’s [40] categoriﬁcation of quan-

tum groups, and their integrable modules. Webster further used his categoriﬁ-

cations to give a link homology theory categorifying the Witten–Reshetikhin–

Turaev invariant of tangles. The construction in [49] involves algebras, called

KLRW algebras (or tensor product algebras), that are ﬁnite-dimensional algebras

presented diagrammatically, generalizing cyclotomic KLR algebras. Categories

of ﬁnitely generated modules over KLRW algebras come equipped with an action

of Khovanov–Lauda–Rouquier’s 2-Kac–Moody category, and their Grothendieck

groups are isomorphic to tensor products of integrable modules. Link invariants

and categoriﬁcations of intertwiners are constructed using functors given by the

derived tensor product with certain bimodules over KLRW algebras.

1.1.4. Verma categoriﬁcation: dg-enhancements. In [36,37,34], the second

and third authors have given a categoriﬁcation of (universal, parabolic) Verma

modules for (quantized) symmetrizable Kac–Moody algebras. In its more general

form [34], the categoriﬁcation is given as a derived category of dg-modules over

a certain dg-algebra, similar to a KLR algebra but containing an extra generator

in homological degree 1. This dg-algebra can also be endowed with a collection

of diﬀerent diﬀerentials, each of them turning it into a dg-algebra whose homol-

ogy is isomorphic to a cyclotomic KLR algebra. This can be interpreted as a

categoriﬁcation of the projection of a universal Verma module onto an integrable

module. Categoriﬁcation of Verma modules was used by the second and third

authors in [35] to give a quantum group higher representation theory construction

of Khovanov–Rozansky’s HOMFLY–PT link homology.

1.2. The work in this paper. For a formal parameter, let M./ be the universal

Uq.sl2/-Verma module with highest weight , and V . x

N / WD V .N1/˝   ˝ V .Nr/,

Tensor product categoriﬁcations and the blob 2-category 709

where V .Nj/is the irreducible of highest weight qNj,Nj2N. In this paper

we combine Webster’s categoriﬁcation with the Verma categoriﬁcation to give a

categoriﬁcation of M./˝V . x

N /. Then we construct a categoriﬁcation of the blob

algebra by categorifying the intertwiners of M./ ˝V . x

N / where all the Njare 1.

1.2.1. Dg-enhanced KLRW algebras and categoriﬁcation of tensor products

(§§ 3and 4). Fix a commutative unital ring k. The KLRW algebra is the

k-algebra spanned by planar isotopy classes of braid-like diagrams whose strands

are of two types: there are black strands labeled by simple roots of a symmetrizable

Kac–Moody algebra gand carrying dots, and there are red strands labeled by

dominant integral weights. KLRW algebras are cyclotomic algebras in the sense

that they generalize cyclotomic KLR algebras to a string of dominant integral

weights, where the “violating condition” [49, Deﬁnition 4.3] plays the role of the

cyclotomic condition. KLRW algebras were also deﬁned without the violating

condition, in which case we call them non-cyclotomic or aﬃne KLRW algebras.

In the case of sl2, for b2Nand x

N2Nr, we denote by Tx

b(resp. z

b) the (resp.

aﬃne) KLRW algebra spanned by bblack strands (all labeled by the simple root

of sl2) and rred strands, labeled in order N1; : : : ; Nrfrom left to right.

Following a procedure analogous to [37,34], we construct in §3an algebra

T; x

b, with a formal parameter, that contains the aﬃne KLRW algebra z

bas a

subalgebra. In a nutshell, T;x

bis deﬁned by putting a vertical blue strand labeled

by on the left of the diagrams of z

b, and adding a new generator that we call

a nail (this corresponds with the “tight ﬂoating dots” of [37,34]). We draw this

new generator as





Note that a nail can only be placed on the left-most strand, which is always blue.

The nails are subject to the following local relations:



;



D 



;



D0:

When x

ND¿is the empty sequence, we recover the dg-enhanced nilHecke

algebra from [37]. The subalgebra spanned by all diagrams without a nail is

isomorphic to the aﬃne KLRW algebra z

710 A. Lacabanne, G. Naisse, and P. Vaz

As we will see, the algebra T; x

bcan be equipped with three Z-gradings: two

internal gradings, one as in Webster’s original deﬁnition and an additional grading

(see Deﬁnition 3.2), as well as a homological grading. The ﬁrst two of these

gradings categorify the parameters qand respectively, and we call them q- and

-gradings. As usual, the homological grading allows us to categorify relations

involving minus signs. We write qk(resp. k) for a grading shift up by kin the q-

(resp. -)grading, and Œk for a grading shift up by kin the homological grading,

for k2Z.

We let the nail be in homological degree 1, while diagrams without a nail

are in homological degree 0. As in the categoriﬁcation of Verma modules, if

we endow the algebra T;x

bwith a trivial diﬀerential, then it becomes a dg-

algebra categorifying M./ ˝V . x

N / (see below). We can also equip T;x

bwith a

diﬀerential dN, for N0, which acts trivially on diagrams without a nail, while

dN0

@

WD N



and extending using the graded Leibniz rule. The dg-algebra .T ; x

b; dN/is formal

with homology isomorphic to the KLRW algebra T.N;x

N /

b(see Theorem 3.13).

The usual framework using the algebra map T;x

b!T; x

bC1that adds a black

strand at the right of a diagram gives rise to induction and restriction dg-functors

Eband Fbbetween the derived dg-categories Ddg.T  ;x

b; 0/ and Ddg.T ; x

bC1; 0/. The

following describes the categorical Uq.sl2/-action:

Theorem 4.1.There is a quasi-isomorphism

Cone.Fb1Eb1! EbFb/Š

! M

Œˇ Cjx

Nj2bq

Idb;

of dg-functors.

As usual in the context of categoriﬁcation, the notation LŒˇ Cj x

Nj2bqon the right-

hand side is an inﬁnite coproduct categorifying multiplication by the rational

fraction .qjx

Nj2b 1qj x

NjC2b /=.q q1/interpreted as a Laurent series.

Turning on the diﬀerential dNgives endofunctors EN

b,FN

bon the derived

category DdgLb0T; x

b; dN. In this case, the right-hand side in Theorem 4.1

becomes quasi-isomorphic to a ﬁnite sum and we recover the usual action on

categories of modules over KLRW algebras (see Proposition 4.3).

Tensor product categoriﬁcations and the blob 2-category 711

In [33], the second author introduced the notion of an asymptotic Grothendieck

group, which is a notion of a Grothendieck group for (multi)graded categories

of objects admitting inﬁnite iterated extensions (like inﬁnite composition se-

ries or inﬁnite resolutions) whose gradings satisfy some mild conditions. De-

note by QK

0./the asymptotic Grothendieck group (tensored over Z..q; //

with Q..q; //). The categorical Uq.sl2/-actions on the derived categories

DdgLb0T; x

b; 0and DdgLb0T ;x

b; dNdescend to the asymptotic

Grothendieck group and we have the main result of §4, which reads as follow-

ing:

Theorem 4.7.There are isomorphisms of Uq.sl2/-modules

QK

0.T ; x

N; 0/ ŠM ./ ˝V . x

N /;

and

QK

0.T ; x

N; dN/ŠV .N / ˝V . x

N /;

for all N2N.

In §7.1 we prove that in the case of bD1,x

ND1; : : : ; 1 and ND1, the

dg-algebra .T ;1;:::;1

1; d1/is isomorphic to a dg-enhanced zigzag algebra, general-

izing [45, §4].

1.2.2. The blob 2-category (§§ 5and 6). We study the case of x

ND1; : : : ; 1

in more detail. We deﬁne several functors on Ddg.T ; x

N; 0/ commuting with

the categorical action of Uq.sl2/. As in [49], these are deﬁned as a ﬁrst step

via (dg-)bimodules over the above mentioned dg-enhancements of KLRW-like

algebras. To simplify matters, let T;r be the dg-enhanced KLRW algebra with

rstrands labeled 1 and a blue strand labeled . The categorical Temperley–Lieb

action is realized by a pair of biadjoint functors, constructed in the same way as

in [49]. They are given by derived tensoring with the .T  ;r ; T ;r˙2/-bimodules

Biand x

Bigenerated respectively by the diagram

1

:::

and its mirror along a horizontal axis. We stress again that the blue strand is on

the left. Moreover, these diagrams are subjected to some local relations (see §5.1).

712 A. Lacabanne, G. Naisse, and P. Vaz

Taking the derived tensor product with these bimodules deﬁnes the coevaluation

and evaluation dg-functors as

BiWD Bi˝L

TW Ddg.T  ;r ; 0/ ! Ddg.T ;r C2; 0/;

BiWD x

Bi˝L

TW Ddg.T ;rC2; 0/ ! Ddg.T ;r ; 0/:

In §6.1 we extend [49] and prove that these functors satisfy the relations of the

Temperley–Lieb algebra:

Corollaries 6.3 and 6.5.There are natural isomorphisms

Bi˙1ıBiŠId;x

BiıBiŠqIdŒ1 ˚q1IdŒ1:

We deﬁne the double braiding functor in the same vein, using the .T ;r ; T  ;r /-

bimodule Xgenerated by the diagram

1

:::

modulo the deﬁning relations of T ;r , and the extra local relations



;



The double braiding functor is then deﬁned as the derived tensor product

„WD X˝L

TW Ddg.T  ;r ; 0/ ! Ddg.T ;r ; 0/:

The functors Bi,x

Biand „intertwine the categorical Uq.sl2/-action on

Ddg.T  ;r ; 0/:

Proposition 6.1.We have natural isomorphisms Eı„Š„ıEand Fı„Š„ıF,

and also EıBiŠBiıE,FıBiŠBiıF, and similarly for x

Bi.

The ﬁrst main result of §6is that the blob algebra acts on Ddg.T ;r ; 0/. This fol-

lows from the Temperley–Lieb action in §6.1 and Corollary 6.11, Proposition 6.14,

and Corollary 6.17, summarized below.

Tensor product categoriﬁcations and the blob 2-category 713

Corollary 6.11, Proposition 6.14, and Corollary 6.17.The functor

„WDdg.T  ;r ; 0/ ! Ddg.T ;r ; 0/

is an autoequivalence, with inverse given by

„1WD RHOMT.X; /WDdg.T  ;r ; 0/ ! Ddg.T ;r ; 0/:

There are quasi-isomorphisms

Cone.q2„Œ1 ! q2IdŒ1/Œ1 '

! Cone.„ ı„! 1„/;

and

q.Id/Œ1 ˚1q1.Id/Œ1 '

! N

B1ı„ıB1;

of dg-functors

One of the main diﬃculties in establishing the results above is that, in order

to compute derived tensor products, we have to take left (resp. right) coﬁbrant

replacements of several dg-bimodules. As observed in [29, §2.3], while the left

(resp. right) module structure remains unchanged when passing to the left (resp.

right) coﬁbrant replacement, the right (resp. left) module structure is preserved

only in the A1sense. As a consequence, constructing natural transformations

between compositions of derived tensor product functors often requires to use

A1-bimodules maps. We have tried to avoid as much as possible to end up in this

situation, replacing the potentially unwieldy A1-bimodules by quasi-isomorphic

dg-bimodules.

Let Brbe a certain subcategory (see §6.3) of the derived dg-category of

.T ;r ; 0/-.T ;r ; 0/-bimodules generated by the dg-bimodules corresponding with

the dg-functors identity, „˙1and Biıx

Bi. Given two dg-bimodules in Br,

we can compose them in the derived sense by replacing both of them with a

bimodule coﬁbrant replacement (i.e. a coﬁbrant replacement as dg-bimodule, and

not only left or right dg-module), and taking the usual tensor product. This gives

a dg-bimodule, isomorphic to the derived tensor product of the two initial dg-

bimodules. In particular, it equips QK

0.Br/with a ring structure. We show

that Bris a categoriﬁcation of the blob algebra Brwith ground ring extended to

Q..q; //:

Corollary 6.19.There is an isomorphism of Q..q; //-algebras

QK

0.Br/ŠBr.q; /:

714 A. Lacabanne, G. Naisse, and P. Vaz

This result generalizes to the blob category. However, a technical issue we

ﬁnd here is that dg-categories up to quasi-equivalence do not form a 2-nbdash

category, but rather an .1; 2/-category [9]. Concretely in our case, we consider a

sub-.1; 2/-category of this .1; 2/-category, where the objects are the derived

dg-categories Ddg.T ;r ; 0/ for various r2N, and the 1-hom are generated

by the dg-functors identity, „˙1,x

Biand Bi. Moreover, these 1-hom are stable

.1; 1/-categories, and thus their homotopy categories are triangulated (see [28]).

In particular, we write QK

0.B/for the category with the same objects as Band

with hom-spaces given by the asymptotic Grothendieck groups of the homotopy

category of the 1-hom of B. By [9] and [46], we can compute these hom-

spaces by considering usual derived categories of dg-bimodules, and we obtain

the following, again after extending the ground rings to Q..q; //:

Corollary 6.21.There is an equivalence of categories

QK

0.B/ŠB:

1.2.3. The general case: symmetrizable g.The deﬁnition of dg-enhanced

KLRW algebras in §3generalizes immediately to any symmetrizable g. We

indicate this generalization in §7.2. We expect that the results of §3and §4extend

to this case without diﬃculty.

1.2.4. Quiver Schur algebras. Quiver Schur algebras were introduced geomet-

rically by Stroppel and Webster in [44] to give a graded version of the cyclo-

tomic q-Schur algebras of Dipper, James and Mathas [7]. Independently, Hu and

Mathas [13] constructed a graded Morita equivalent variant of the quiver Schur

algebras in [44] as graded quasi-hereditary covers of cyclotomic KLR algebras

for linear quivers. While the construction in [44] is geometric, the construction

in [13] is combinatorial/algebraic.

More recently, Khovanov, Qi and Sussan [23] gave a variant of the quiver Schur

algebras in [13] for the case of cyclotomic nilHecke algebras, and showed that

Grothendieck groups of their algebras can be identiﬁed with tensor products of

integrable modules of Uq.sl2/. Following similar ideas, in §7.3 we construct a

dg-algebra, which we conjecture to be the quiver Schur variant of the dg-enhanced

KLRW algebra of §3(Conjectures 7.15 and 7.17).

1.2.5. Appendix. We have moved the most computational proofs to §A, leaving

only a sketch of some of the proofs in the main text. The reader can also ﬁnd

in §Bsome explanations and results about homological algebra, A1-structures

and asymptotic Grothendieck groups.

Tensor product categoriﬁcations and the blob 2-category 715

1.3. Possible future directions and applications

1.3.1. Khovanov homology for tangles of type B. The topological interpreta-

tion of the blob algebra in [14, §3.4] gives rise to a Jones polynomial for tangles

of type B (i.e. tangles in the annulus). We expect that by introducing braiding

functors as in [49], we obtain a link homology of type B, yielding invariants of

links in the annulus akin to ones introduced by Asaeda, Przytycki, and Sikora [3]

(see also [4,12,38]).

Given a link in the annulus, the invariant obtained from our construction

would be a dg-endofunctor of the derived dg-category of dg-modules over the

dg-enhanced KLRW algebra .T ;;; 0/. This means that the empty link is sent

to the dg-endomorphism space of the identity functor, which coincides with the

Hochschild cohomology of T;;, and is inﬁnite-dimensional (the center of T;;

is already inﬁnite-dimensional). By restricting to the subcategory of dg-modules

over .T ;;

0; 0/, it becomes 1-dimensional since T;;

0Šk. With this restriction,

we conjecture that our “would-be” invariant coincides with the usual annular

Khovanov homology.

The following is a work in progress with A. Wilbert. As it is the case of using

Webster’s machinery [49], computing the tangle invariant of type B using our

framework could be unwieldy. A more computation-friendly alternative could be

to use dg-bimodules overannular arc algebras constructed using the annular TQFT

of [3], as done in [1, §5.3] (see also [8, §5]). Furthermore, evidences show there

is a (at least weak) categorical action of the blob algebra on the derived category

of dg-modules over these annular arc algebras.

In a diﬀerent direction, one could try to extend our results to construct a

Khovanov invariant for links in handlebodies, in the spirit of the handlebody

HOMFLY–PT-link homology of Rose and Tubbenhauer in [39].

1.3.2. Constructions using homotopy categories. KLRW algebras are given

diagrammatically, which is the often an appropriate framework for constructions

with an additive ﬂavor. Nevertheless, the various functors realizing the various

intertwiners and the braiding need to pass to derived categories of modules. This

makes it harder to describe explicitly the 2-categories realizing these symmetries

since a bimodule for two of those algebras induces an A1-bimodule on the level

of derived categories. This was pointed out by Mackaay and Webster in [29], who

gave explicit constructions of categoriﬁed intertwiners in order to prove the equiv-

alence between the several existing gln-link homologies. One of the things [29]

tells us is how to construct homotopy versions of Webster’s categoriﬁcations.

716 A. Lacabanne, G. Naisse, and P. Vaz

A construction using homotopy categories for the results in this paper seems

desirable from our point of view. We hope it can be done either by mimicking [29],

which can turn out to be a technically challenging problem, or alternatively, by

a construction of dg-enhancements for redotted Webster algebras, as considered

in [25] and [20] to give a homotopical version of some of the above, but whose

low-tech presentation might hide diﬃculties.

1.3.3. Generalized blob algebras and variants. The results of [14] were ex-

tended in [26], where the ﬁrst and third authors have computed the endomorphism

algebra of the Uq.glm/-module Mp.ƒ/ ˝V˝nfor Mp.ƒ/ a parabolic universal

Verma modules and Vthe natural module of Uq.glm/, which is always a quotient

of an Ariki–Koike algebra. As particular cases (depending on pand the relation

between nand m) we obtain Hecke algebras of type Bwith two parameters, the

generalized blob algebra of Martin and Woodcock [31] or the Ariki–Koike alge-

bra itself. With this result in mind it is tantalizing to ask for an extension to glm

of the work in this paper. Modulo technical diﬃculties the methods in this paper

could work for glmin the case of a parabolic Verma module for a 2-block par-

abolic subalgebra, which is the case where the generators of the endomorphism

algebra satisfy a quadratic relation. Constructing a categoriﬁcation of the Ariki–

Koike algebra or the generalized blob algebra as the blob 2-category in §6looks

quite challenging at the moment, in particular for a functor-realization of the cy-

clotomic relation and the relation D0(for the generalized blob algebra in the

presentation given in [26, Theorem 2.24]).

Acknowledgements. The authors thank Catharina Stroppel for interesting dis-

cussions, and for pointing us [30], helping to clarify the confusion with the ter-

minology of “blob algebra” and “Temperley–Lieb algebra of type B.” The authors

would also like to thank the referee for his/her numerous, detailed and helpful

comments.

2. Quantum sl2and the blob algebra

2.1. Quantum sl2.Recall that quantum sl2can be deﬁned as the Q.q/-algebra

Uq.sl2/, with generic q, generated by K; K1; E and Fwith relations

KE Dq2EK; KK1D1DK1K;

KF Dq2FK; EF FE DKK1

qq1:

Tensor product categoriﬁcations and the blob 2-category 717

Quantum sl2becomes a bialgebra when endowed with comultiplication

.E/ WD E˝1CK1˝E; .F / WD F˝KC1˝F ;

.K˙1/WD K˙1˝K˙1;

and with counit ".K˙1/WD 1,".E/ WD ".F / WD 0.

There is a Q.q/-linear anti-involution Nof Uq.sl2/deﬁned on the generators

N .E/ WD q1K1F; N.F / WD q1EK; N.K / WD K: (1)

It is easily checked that

ı ND.N˝ N / ı: (2)

2.1.1. Integrable modules. For each N2N, there is a ﬁnite-dimensional

irreducible Uq.sl2/-module V .N /, called integrable module, with basis elements

¹vN;0; vN;1 ; : : : ; vN;N ºand action

KvN;i WD qN2i vN;i ;

FvN;i WD vN;i C1;

EvN;i WD ŒiqŒN iC1qvN;i 1;

where Œnqis the n-th quantum integer

ŒnqWD qnqn

qq1Dqn1Cqn12C    C q1n:

In particular, let VWD V .1/ be the fundamental Uq.sl2/-module.

The module V .N / can be equipped with the Shapovalov form

.;/NWV .N / V .N / ! Q.q/;

which is a non-degenerate bilinear form such that .vN;0; vN;0 /ND1and which

is N-Hermitian: for any v; v02V .N / and u2Uq.sl2/, we have .u v; v0/ND

.v; N .u/ v0/N. A computation shows that

.vN;i ; vN;j /NDıi;j qi .N i / ŒiqŠŒN qŠ

ŒN iqŠ;

where Œ0qŠWD 1and ŒnqŠWD ŒnqŒn 1q: : : Œ2qŒ1q.

718 A. Lacabanne, G. Naisse, and P. Vaz

2.1.2. Verma modules. Let ˇbe a formal parameter and write WD qˇas a

formal variable. Let bbe the standard upper Borel subalgebra of sl2and Uq.b/

be its quantum version. It is the Uq.sl2/-subalgebra generated by K; K1and E.

Let Kbe a 1-dimensional Q.; q /-vector space, with ﬁxed basis element v. We

endow Kwith an Uq.b/-action by declaring that

K˙1vWD ˙1v; EvWD 0;

extending linearly through the obvious inclusion Q.q/ ,!Q.q; /. The universal

Verma module M./ is the induced module

M./ WD Uq.sl2/˝Uq.b/K:

It is irreducible and inﬁnite-dimensional with Q.q; /-basis elements

¹v;0 WD v; v;1 ; : : : ; v;i ; : : : º

and action

Kv;i WD q2i v ;i ;

Fv;i WD v;i C1;

Ev;i WD Œi qŒˇ iC1qv ;i 1;

where

Œˇ CkqWD qk1qk

qq1:

The Verma module M./ can also be equipped with a Shapovalov form .;/,

which is again a non-degenerate bilinear form such that .v; v/D1and which

is N-Hermitian: for any v; v02M ./ and u2Uq.sl2/, we have .u v; v0/D

.v; N .u/ v0/. One easily calculates that

.v;i ; v ;j /Dıi;j iqi2ŒiqŠŒˇqŒˇ 1qŒˇ iC1q:

2.1.3. Tensor products. Given Wand W0two Uq.sl2/-modules, their tensor

product W˝W0is again a Uq.sl2/-module with the action induced by . Explic-

itly, we have

K˙1.w ˝w0/WD K˙1w˝K˙1w0;

F.w ˝w0/WD F w ˝K w0Cw˝F w0;

E.w ˝w0/WD Ew ˝w0CK1w˝E w0;

for all w2Wand w02W0.

Tensor product categoriﬁcations and the blob 2-category 719

For x

ND.N1; : : : ; Nr/2Nrwe write

V .x

N / WD V .N1/˝    ˝ V .Nr/

and

M˝V .x

N / WD M./ ˝V .N1/˝    ˝ V .Nr/:

In the particular case N1D    D NrD1, we write Vrfor the r-th folded tensor

product V˝V˝    ˝ V.

2.1.4. Weight spaces. The module M˝V .x

N / decomposes into weight spaces

M˝V .x

N /qkWD ¹v2M˝V . x

N /WKv Dqkvº:

Note that we have M˝V .x

N / ŠL`0M˝V . x

N /qj

Nj2` , where jx

Nj WD PiNi.

2.1.5. Basis. Let Pr

bbe the set of weak compositions of binto rC1parts, that

bWD °.b0; b1; : : : ; br/2NrC1W

iD0

biDb±:

Consider also

Pr;x

bWD ¹.b0; b1; : : : ; br/2Pr

bWbiNifor 1irº  Pr

In addition to the induced basis by the tensor product, the space M˝V .x

N /

admits a basis that will be particularly useful for categoriﬁcation. For

D.b0; : : : ; br/2Pr

we write

vWD Fbr.Fb1.F b0.v/˝vN1;0/   ˝ vNr;0 /:

Then, M˝V . x

N / has a basis given by

¹vW2Pr;x

b; b 0º:

In particular, we have that M˝V .x

N /qj

Nj2b has a basis given by ¹vº2Pr;

One can describe inductively the change of basis from ¹vº2Pr;

to the

induced basis as follows:

v.b0;:::;br/D

min.br;Nr/

kD0

q.1k/.brk/ br

kv.b0;:::;br1Cbrk/ ˝vNr;k ;

720 A. Lacabanne, G. Naisse, and P. Vaz

for any .b0; : : : ; br/2Pr

band

v.b0;:::;br1/˝vNr;n D

kD0

.1/nkq.nk/.n2/n

kv.b0;:::;br1Cnk;k/ ;

for any .b0; : : : ; br1/2Pr1

band 0nNr, with n

kqWD ŒnqŠ

ŒkqŠŒnkqŠ.

We can also use these formulas to inductively rewrite a vector vwith 2Pr

in terms of various vfor 2Pr;x

b. Indeed, we have

v.b0;:::;br/D

min.br;Nr/

kD0

q.1Nr/.brk/

jD1;j ¤k

Œbrj q

jD1;j ¤k

Œk j q

v.b0;:::;br1Cbrk;k/ ;

for any .b0; : : : ; br/2Pr

2.1.6. Shapovalov forms for tensor products. Following [49, §4.7], we con-

sider a family of bilinear forms .;/; x

Non tensor products of the form

M./ ˝V . x

N /

satisfying the following properties:

(1) each form .;/;x

Nis non-degenerate;

(2) for any v; v02M./ ˝V . x

N / and u2Uq.sl2/,

.u v; v0/; x

ND.v; N .u/ v0/;x

(3) for any f2Q.q; / and v; v02M./ ˝V . x

N /,

.f v; v0/;x

ND.v; f v 0/;x

NDf .v; v 0/;x

(4) if v ; v02M./ ˝V . x

N /, then

.v; v0/; x

ND.v ˝vN;0; v 0˝vN;0/;N 0;

where N0D.N1; : : : ; Nr; N /.

Similarly to [49, Proposition 4.33] we have:

Proposition 2.1. There exists a unique system of such bilinear forms which are

given by

.v; v0/; x

ND.v; v0/…

; x

for every v; v02M./ ˝V . x

N / where .;/…

; x

Nis the product of the universal

Shapovalov form on M./ and of the Shapovalov forms on the various V .Ni/.

Tensor product categoriﬁcations and the blob 2-category 721

2.2. The blob algebra. Recall that the blob algebra Bris the Q.; q/-algebra

with generators u1; : : : ; ur1and , and subject to the relations of type A:

uiujDujui;for jijj> 1, (3)

uiuiC1uiDui;for 1ir2, (4)

uiui1uiDui;for 2ir1, (5)

iD .q Cq1/; for 1ir1, (6)

and to the blob relations:

uiDui; for 2ir, (7)

u1u1D .q C1q1/u1;(8)

q12D.q C1q1/ q: (9)

Note that is invertible, with inverse given by 1DCq21q2, and that

the relations (3)–(6) imply that the generators u1; : : : ; ur1generate a subalgebra

isomorphic to the Temperley–Lieb algebra of type A.

The blob algebra has several well-known diagrammatic presentations. The

most classical one already appeared in [30], but (a slight modiﬁcation of ) the one

in [14] is more convenient for our purposes. This presentation is given by setting

uiD: : : : : :

;

D: : : ;

where diagrams are taken up to planar isotopy and read from bottom to top, and

with local relations

D .q Cq1/; (6)

D .q C1q1/ ; (8)

q1D.q C1q1/q ; (9)

722 A. Lacabanne, G. Naisse, and P. Vaz

corresponding to (6), (8), and (9) (explaining why we kept the same numbering).

Note that the relations (3)–(5) and (7) are encoded by the planar isotopies.

Remark 2.2. In the graphical description of Brgiven in [14] the generator 

is presented as a double braiding (see [14, Figure 1]). We don’t follow that

interpretation in our diagrammatics in order to simplify pictures, but we keep the

terminology (see §5.3 ahead).

Remark 2.3. With respect to [14] our conventions switch .; q/ and .1; q 1/,

which can be interpreted as exchanging the double braiding by the double inverse

braiding.

There is an action of Bron M˝Vrthat intertwines with the quantum

sl2-action. This action can be described locally, identifying the ﬁrst vertical strand

in Brwith the identity on M./, and the ith vertical strand with the identity on

the i-th copy of Vin M˝Vr. Then the action is given using the following maps

WV˝V! Q.q; /; 8

v1;0 ˝v1;0 7! 0;

v1;0 ˝v1;1 7! 1;

v1;1 ˝v1;0 7! q1;

v1;1 ˝v1;1 7! 0;

WQ.q; / ! V˝V; 1 7! qv1;0 ˝v1;1 Cv1;1 ˝v1;0;

WM˝V! M˝V;

v;k ˝v1;0 7! 1q2k v;k ˝v1;0

q.q q1/ŒkqŒˇ kC1qv;k 1˝v1;1;

v;k ˝v1;1 7! .1Cq21q2.kC1/ /v ;k ˝v1;1

1q2.kC1/ .q q1/v;kC1˝v1;0;

where the formula for is obtained by acting twice with an R-matrix. In our

conventions, we have Dfı‚21 ıfı‚where ‚is given by the action of

nD0

.1/nqn.n1/=2 .q q1/n

ŒnqŠFn˝En;

‚21 by the action of

nD0

.1/nqn.n1/=2 .q q1/n

ŒnqŠEn˝Fn;

Tensor product categoriﬁcations and the blob 2-category 723

and, for any, k2N,

f .v;k ˝v1;0/WD 1=2qkv ;k ˝v1;0

and

f .v;k ˝v1;1/WD 1=2qkv ;k ˝v1;1:

The following will be useful later:

Lemma 2.4. The action of Brtranslates in terms of v-vectors of M˝Vras

: : : : : :

v.:::;bi1;bi;biC1;biC2;::: / 7! q1Œbiqv.:::;bi1CbiCbiC11;biC2;::: / ;(10)

: : : : : :

v7! qŒ2qv.:::;bi1;1;0;bi;::: / qv.:::;bi1C1;0;0;bi;::: / qv.:::;bi1;0;1;bi;::: / ;

(11)

: : : W

v.b0;b1;::: / 7! .1qb0qŒb0q/v.0;b0Cb1;::: / Cq2Œb0qv.1;b0Cb11;::: / :(12)

Proof. A computational proof is given in §A.

As a matter of fact, this completely determines EndUq.sl2/.M ˝Vr/:

Theorem 2.5 ([14, Theorem 4.9]). There is an isomorphism

BrŠEndUq.sl2/.M ˝Vr/: (13)

The blob category Bis the Q.; q/-linear category given by

objects are non-negative integers r2N;

HomB.r; r0/is given by Q.; q/-linear combinations of string diagrams con-

necting rC1points on the bottom to r0C1points on the top, with the ﬁrst

strand always connecting the left-most point to the left-most point, where the

strings cannot intersect each other except for diagrams like . Diagrams are

considered up to planar isotopy and subject to the relations (6), (8), and (9).

724 A. Lacabanne, G. Naisse, and P. Vaz

Let TL be the Temperley–Lieb category of type A, deﬁned diagrammatically. It

is a Q.q/-linear monoidal category equivalent to Fund.sl2/, the full monoidal

subcategory of Uq.sl2/-mod generated by V. Note that Bcan be endowed with a

structure of module category over TL, by gluing diagrams on the right.

Also consider the full subcategory MV Uq.sl2/-mod given by the modules

M./ ˝V˝rfor all r2N. It is a module category over Fund.sl2/by acting on

the right with tensor product of Uq.sl2/-modules.

Theorem 2.6 ([14, Theorem 4.9]). There are equivalences of categories such that

B TL

MV Fund.sl2/

˝acts

commutes.

Remark 2.7. Note that [14] considers projective Verma modules with integral

highest weight. The case of universal Verma modules was studied in [26], albeit

not in the categorical setup.

3. Dg-enhanced KLRW algebras

In [37] and [34] it was explained how to construct a “dg-enhancement” of cyclo-

tomic nilHecke algebras to pass from a categoriﬁcation of the integrable module

V .N / to a categoriﬁcation of the Verma module M./. This suggests that one

might try to go from a categoriﬁcation of V .N / ˝V . x

N / to a categoriﬁcation of

M./ ˝V . x

N / by constructing a dg-enhancement of KLRW algebras [49, §4],

which we do next.

3.1. Preliminaries and conventions. Before deﬁning the various algebras, we

ﬁx some conventions, and we recall some common facts about dg-structures

(a reference for this is [15]). First, let kbe a commutative unital ring for the

remaining of the paper.

3.1.1. Dg-algebras. AZn-graded dg-(k-)algebra .A; dA/is a unital ZZn-

graded (k-)algebra ADL.h;g/2ZZnAh

g, where we refer to the Z-grading as

homological (or h-degree) and the Zn-grading as g-degree, with a diﬀerential

dWA!Asuch that

Tensor product categoriﬁcations and the blob 2-category 725

dA.Ah

g/Ah1

gfor all g2Zn; h 2Z;

dA.xy/ DdA.x/y C.1/degh.x/ xdA.y/;

d2

AD0.

The homology of .A; dA/is H.A; dA/WD ker.d /= im.d /, which is a ZZn-graded

algebra that decomposes as Lh2Z;g2ZnHh

g.A; dA/WD Hh.Ag; dA/. A morphism

of dg-algebras fW.A; dA/!.A0; dA0/is a morphism of algebras that preserves the

ZZn-grading and commutes with the diﬀerentials. Such a morphism induces

a morphism fWH.A; dA/!H.A0; dA0/. We say that fis a quasi-isomorphism

whenever fis an isomorphism. Also, we say that .A; dA/is formal if there is a

quasi-isomorphism .A; dA/'

!.H.A; dA/; 0/.

Remark 3.1. Note that, in contrast to [15], the diﬀerential decreases the homo-

logical degree instead of increasing it.

Similarly, a Zn-graded left dg-module is a ZZn-graded module Mwith a

diﬀerential dMsuch that

dM.M h

g/Mh1

gfor all g2Zn; h 2Z;

dM.x m/ DdA.x/ yC.1/degh.x/xdM.y/;

d2

MD0.

Homology, maps between dg-modules and quasi-isomorphisms are deﬁned as

above, and there are similar notions of Zn-graded right dg-modules and dg-bi-

modules.

In our convention, a Zm-graded category is a category with a collection of m

autoequivalences, strictly commuting with each others. The category .A; dA/-mod

of (left) Zn-graded dg-modules over a dg-algebra .A; dA/is a ZZn-graded

abelian category, with kernels and cokernels deﬁned as usual. The action of Z

is given by the homological shift functor

Œ1W.A; dA/-mod ! .A; dA/-mod

acting by

increasing the degree of all elements in a module Mup by 1, i.e. degh.mŒ1/ D

degh.m/ C1;

switching the sign of the diﬀerential dM Œ1 WD dM;

introducing a sign in the left-action r.mŒ1/ WD .1/degh.r /.r m/Œ1.

726 A. Lacabanne, G. Naisse, and P. Vaz

The action of g2Znis given by increasing the Zn-degree of elements up by g,

in the sense that

.gM /g0CgWD .M /g0;

or in other terms, an element x2Mwith degree g0becomes of degree g0Cg

in gM. There are similar deﬁnitions for categories of right dg-modules and dg-

bimodules, with the subtlety that the homological shift functor does not twist the

right-action:

.mŒ1/ rWD .m r/Œ1:

As usual, a short exact sequence of dg-(bi)modules induces a long exact sequence

in homology.

Let fW.M; dM/!.N; dN/be a morphism of dg-(bi)modules. Then, one

constructs the mapping cone of fas

Cone.f / WD .M Œ1 ˚N; dC/; dCWD dM0

f dN:(14)

The mapping cone is a dg-(bi)module, and it ﬁts in a short exact sequence:

0! N{N

! Cone.f / M Œ1

! M Œ1 ! 0;

where {Nand M Œ1 are the inclusion and projection N{N

!M Œ1 ˚NM Œ1

! M Œ1.

3.1.2. Hom and tensor functors. Given a left dg-module .M; dM/and a right

dg-module .N; dN/, one constructs the tensor product

.N; dN/˝.A;dA/.M; dM/WD ..M ˝AN /; dM˝N/;

dM˝N.m ˝n/ WD dM.m/ ˝nC.1/degh.m/m˝dN.n/: (15)

If .N; dN/(resp. .M; dM/) has the structure of a dg-bimodule, then the tensor

product inherits a left (resp. right) dg-module structure.

Given a pair of left dg-modules .M; dM/and .N; dN/, one constructs the

dg-hom space

HOM.A;dA/..M; dM/; .N; dN// WD .HOMA.M ; N /; dHOM.M;N //;

dHOM.M;N / .f / WD dNıf.1/degh.f /fıdM;(16)

where HOMAis the ZZn-graded hom space of maps between ZZn-graded

A-modules. Again, if .M; dM/(resp. .N; dN/) has the structure of a dg-bimodule,

then it inherits a left (resp. right) dg-module structure.

Tensor product categoriﬁcations and the blob 2-category 727

In particular, given a dg-bimodule .B; dB/over a pair of dg-algebras .S ; dS/-

.R; dR/, we obtain tensor and hom functors

B˝.R;dR/./W.R; dR/-mod ! .S; dS/-mod;

HOM.S;dS/.B ; /W.S; dS/-mod ! .R; dR/-mod;

which form a adjoint pair .B ˝.R;dR//`HOM.S;dS/.B; /. Explicitly, the

natural bijection

ˆB

M;N WHom.S;dS/.B ˝.R ;dR/M; N / '

! Hom.R;dR/.M; HOM.S ;dS/.B; N //;

(17)

is given by

.f WB˝.R;dR/M! N / 7! .m 7! .b 7! f .b ˝m///:

3.1.3. Diagrammatic algebras. We always read diagram from bottom to top.

We say that a diagram is braid-like when it is given by strands connecting a

collection of points on the bottom to a collection of points on the top, without

being able the turnback. Suppose these diagrams can have singularities (like dots,

4-valent crossings, or other similar decorations).

Abraid-like planar isotopy is an isotopy ﬁxing the endpoints and that does not

create any critical point, in particular it means we can exchange distant singulari-

ties fand g:



Suppose that the diagrams carry a homological degree (associated to singulari-

ties), and consider linear combination of such diagrams. Then, a graded braid-like

planar isotopy is an isotopy ﬁxing the endpoints, that does not create any critical

point and such that we get a sign whenever we exchange two distant singularities

fand g:



D.1/jfjjgj



where jfj(resp. jgj) is the homological degree of f(resp. g).

728 A. Lacabanne, G. Naisse, and P. Vaz

3.2. Dg-enhanced KLRW algebras. Let x

ND.N1; : : : ; Nr/. Recall the KLRW

algebra [49, §4] (also called tensor product algebra) on bstrands Tx

bis the

diagrammatic k-algebra generated by braid-like diagrams on bblack strands and

rred strands. Red strands are labeled from left to right by N1; : : : ; Nrand cannot

intersect each other, while black strands can intersect red strands transversely, they

can intersect transversely among themselves and can carry dots. Diagrams are

taken up to braid-like planar isotopy, and satisfy local relations (18)–(23) which

are given below, together with the violating condition that a black strand in the

leftmost region is 0:

   D 0:

We write z

bfor the same construction but without the violating condition.

The following are the deﬁning (local) relations of Tx

The nilHecke relations:

D0; D;(18)

D C ;D C :(19)

The black/red relations:

;

;(20)

Ni;

DNi

;(21)

;

(22)

kC`D

Ni1

k `

:(23)

Tensor product categoriﬁcations and the blob 2-category 729

Multiplication is given by vertical concatenation of diagrams if the labels and

colors of the strands agree, and is zero otherwise. As explained in [49, §4],

the algebra Tx

bis ﬁnite-dimensional and Z-graded (we refer to this grading as

q-grading), with

degq !D 2; degq !D2; (24)

degq0

@Ni

ADdegq0

@Ni

ADNi:(25)

In the case of x

ND.N / the algebra T.N /

bcontains a single red strand labeled

N, and is isomorphic to the cyclotomic nilHecke algebra NHN

Deﬁnition 3.2. The dg-enhanced KLRW algebra T;x

bis the diagrammatic k-al-

gebra carrying an homological degree generated by braid-like diagrams on bblack

strands, rred strands and a blue strand on the left. Red strands are labeled from

left to right by N1; : : : ; Nrand the blue strand is labeled . Black strands can carry

dots and intersect transversely with black and red strands. Moreover, the left-most

black strand can be nailed on the blue strand, giving a 4-valent vertex as follows:



We put the crossings and the dot in homological degree 0, while the nail is in

homological degree 1. These diagrams are taken modulo graded braid-like planar

isotopy, and subject to the local relations (18)–(23) of Tx

b, together with the local

relations:



;



D 



;



D0: (26)

Remark 3.3. Note that there can be no black or red strand at the left of the blue

strand.

730 A. Lacabanne, G. Naisse, and P. Vaz

Remark 3.4. Note that since nails are stuck on the left, we can not exchange

them using a graded braid-like planar isotopy. Thus, because nails are the only

generators carrying a non-zero homological degree, we could consider diagrams

up to usual braid-like planar isotopy. However, the homological degree of the

nail will play an important role in the categoriﬁcation of the structure constant

Œˇ Ckqappearing in M./ ˝V .x

N /, and graded braid-like planar isotopy will

play an important role in §5.

Clearly, there is an injection of algebra z

b,!T;x

bgiven by adding a vertical

blue strand at the left of a diagram in z

We endow T;x

bwith an extra Z2-grading, the ﬁrst one being inherited from

band denoted q, the second is written . We declare that

degq; 0

@

WD .0; 2/;

and the elements without a nail are all in degree zero and have the same q-degree

as in (24) and (25), so that the inclusion z

b,!T;x

bpreserves the q-grading.

One easily checks that it is well deﬁned.

In the case of x

ND¿, the algebra T;¿

bcontains only a blue strand labeled

and no red strands, and is isomorphic to the dg-enhanced nilHecke algebra

introduced in [37, Deﬁnition 2.3]. To match with the notation from [37], we write

AbWD T;¿

We will often endow T; x

bwith a trivial diﬀerential, turning it into a Z2-graded

dg-algebra .T ;x

b; 0/.

3.3. Basis theorem. For any D.b0; b1; : : : ; br/2Pr

b, deﬁne the idempotent

1WD



:::

b0N1

:::

b1N2

...

:::

of T; x

b. Note that T; x

bŠL;2Pr

b1T; x

b1as ZZ2-graded k-module.

3.3.1. Polynomial action. We now deﬁne an action of the dg-algebra T; x

bon

Polr

bWD M

2Pr

Polb";

Tensor product categoriﬁcations and the blob 2-category 731

the free module over the ring

PolbWD ZŒx1; : : : ; xb˝V.!1; : : : ; !b/

generated by "for each 2Pr

We recall the action of the symmetric group Sbon Polbused in [37, §2.2]. We

view Sbas a Coxeter group with generators iD.i i C1/. The generator iacts

on Polbas follows,

i.xj/WD xi.j /;

i.!j/WD !jCıi;j .xixiC1/!iC1:

For ;  2Pr

b, an element of 1T; x

b1acts by zero on any Polb"0for

0¤and sends Polb"to Polb". It remains to describe the action of the

local generators of T; x

bon a polynomial f2Polb. First, similarly as in [49,

Lemma 4.12], we put

::: ::: fWD xif;

: : : : : : fWD fi.f /

xixiC1

;

: : :

: : : fWD f;

: : :

: : : fWD xN

if;

where we identify xi2Polb"with xi2Polb", and where we only have

drawn the i-th or the i-th and .i C1/-th black strands, counting from left to right.

Furthermore, we put



::: fWD !1f:

Lemma 3.5. The rules above deﬁne an action of T ;x

bon Polr

Proof. We easily check that the relations (18)–(23) and (26) are satisﬁed. 

732 A. Lacabanne, G. Naisse, and P. Vaz

Fix D.b0; : : : ; br/2Pr

b. Let NHnbe the nilHecke algebra on n-strands (it

is described as a diagrammatic algebra with only black strands having dots and

relations (18) and (19)). There is a map

WAb0˝NHb1˝    ˝ NHbr! T;x

diagrammatically given by



: : :

Ab0˝

: : :

NHb1˝    ˝

: : :

NHbr



7!



:::

Ab0NHb1NHbr

(27)

where we recall that Ab0is isomorphic to the dg-enhanced nilHecke algebra

of [37], identifying the nilHecke generators with each other and the nail with the

“leftmost ﬂoating dot.” The tensor product Ab0˝NHb1˝    NHbracts on Polr

through . This action is only non-zero on Polb"and it is readily checked that

this action coincides with the tensor product of the polynomial actions of Ab0on

ZŒx1; : : : ; xb0˝V.!1; : : : ; !b0/Polbfrom [37, §2.2], and of the usual action

of the nilHecke algebra NHbion ZŒxb0CCbi1C1; : : : ; xb0CCbiPolb(see for

example [21, §2.3]).

Lemma 3.6. The map is injective.

Proof. It follows immediately from the faithfulness of the polynomial actions

of Ab0(see [37, Corollary 3.9]) and of NHbi(see [21, Corollary 2.6]). 

3.3.2. Left-adjusted expressions. We recall the notion of a left-adjusted expres-

sion as in [37, §2.2.1]: a reduced expression i1ikof an element w2SrCbis

said to be left-adjusted if i1C    C ikis minimal. One can obtain a left-adjusted

expression of any element of SrCbby taking recursively its representative in the

left coset decomposition

SnD

tD1

Sn1n1t:

As one easily conﬁrms, if we think of permutations in terms of string diagrams,

then a left-reduced expression is obtained by pulling every strand as far as possible

to the left.

Tensor product categoriﬁcations and the blob 2-category 733

3.3.3. A basis of T;x

b.We now turn to the diagrammatic description of a basis

of T; x

bsimilar to [34, §3.2.3]. For an element 2Pr

band 1kb, we deﬁne

the tightened nail k21T; x

b1as the following element:

kWD

:::

NiC1

:::



where the nailed strand is the k-th black strand counting from left to right. This

element has degree degh;q; .k/D.1; 4.k 1/ C2.N1C    C Ni/; 2/.

Lemma 3.7. Tightened nails anticommute with each other, up to terms with a

smaller number of crossings:

k`D `kCR;  2

kD0CR0;

where R(resp. R0) is a sum of diagrams with strictly fewer crossings than k`

(resp. 2

k), for all 1k; ` b.

Proof. Similar to [34, Lemma 3.12], and omitted. 

Remark 3.8. If k; ` b0, then we have k`D `k. Moreover, if k62 ¹b0C1;

b0Cb1C1; : : : ; b0C    C brC1º, then we have 2

kD0.

Now ﬁx ;  2Pr

band consider the subset of permutations SSrCb, viewed

as diagrams with a blue strand, bblack strands and rred strands, such that

the blue strand is always on the left of the diagram;

the strands are ordered at the bottom by 1and at the top by 1;

for any reduced expression of w2S, there are no red/red crossings.

Example 3.9. If DD.0; 1; 1/, then the set Shas two elements, namely

and :

Note that the second element is not left-adjusted.

For each w2S;N

lD.l1; : : : ; lb/2 ¹0; 1ºband N

aD.a1; : : : ; ab/2Nbwe

deﬁne an element bw ;N

l;N

a21T; x

m1as follows:

734 A. Lacabanne, G. Naisse, and P. Vaz

(1) we choose a left-reduced expression of win terms of diagrams as above;

(2) for each 1ib, if liD1, then we nail the i-th black strand at the

top, counting from the left, on the blue strand by pulling it from its leftmost

position;

(3) ﬁnally, for each 1ib, we add aidots on the i-th black strand at the top.

Let BWD ¹bw;N

l;N

aWw2S;N

l2 ¹0; 1ºb;N

a2Nbº.

Example 3.10. We continue the example of DD.0; 1; 1/. If we choose

lD.1; 0/ and N

aD.0; 1/ for wthe permutation with a black/black crossing, after

left-adjusting it, then we obtain

bw;N

l;N

aD:

Theorem 3.11. The set Bis a basis of 1T; x

b1as a ZZ2-graded k-module.

Proof. By Lemma 3.7, with arguments similar to [34, Proposition 3.13], one

shows that this set generates 1T;x

m1as a k-module. The proof consists in an

induction on the number of crossings, allowing to apply braid-moves in order to

reduce diagrams. In order to show that this set is linearly independent over k, we

apply Lemma 3.6.

In the following, we draw T;x

b1with D.b0; : : : ; br/as a box diagram



: : :

b0N1

:::

Nr1

: : :

br1Nr

: : :

T; x

Moreover, when we draw something like



: : :

b0N1

:::

: : :

NiC1

:::

: : :

T; x

b1

with p0and 0t < bi, it means we consider the subset of T; x

b1given

replacing the box labeled T; x

b1with any diagram of T; x

b1in the diagram above,

and consider it as a diagram of T ;x

b1.

Tensor product categoriﬁcations and the blob 2-category 735

Corollary 3.12. As a ZZ2-graded k-module, T; x

b1decomposes as a direct

sum



: : :

b0N1

:::

Nr1

: : :

br1Nr

: : :

T; x

Š: : :

b0N1

:::

Nr1

: : :

br1

: : :



T;N 0

iD0M

0t<bi

p0

: : :

b0N1

:::

: : :

NiC1

:::

: : :

T; x

b1

iD0M

0t<bi

p0

: : :

::: : : :

: : :

NiC1

:::

: : :



T; x

b1

where N0D.N1; : : : ; Nr1/, and the isomorphism is given by inclusion.

Proof. The claim follows immediately from Theorem 3.11.

3.4. Dg-enhancement. For each N2N, we want to deﬁne a non-trivial dif-

ferential dNon T;x

b. First, we collapse the Z2-grading into a single Z-grading,

which we also call q-degree, through the map Z2!Z; .a; b/ 7! aCbN (i.e.

specializing DqN). Then, we put

dN0

@

WD N



736 A. Lacabanne, G. Naisse, and P. Vaz

and dN.t/ WD 0for all element tof z

bT; x

b, and extending by the graded

Leibniz rule with respect to the homological grading. A straightforward compu-

tation shows that dNrespects all the deﬁning relations of T;x

b, and therefore is

well deﬁned.

Theorem 3.13. The Z-graded dg-algebra .T ;x

b; dN/is formal with

H.T ; x

b; dN/ŠT.N;x

N /

were .N; x

N / WD .N; N1; : : : ; Nr/2NrC1.

Proof. The proof follows by similar arguments as in [34, Theorem 4.4], by using

Corollary 3.12. We leave the details to the reader. 

4. A categoriﬁcation of M./ ˝V . x

N /

In this section we explain how derived categories of .T ; x

b; 0/-dg-modules cate-

gorify the Uq.sl2/-module M./ ˝V . x

N /. Since the construction is very similar

to the one in [37] and [34], we will assume some familiarity with [37] and [34],

and we will refer to these papers for several details.

We introduce the notations

Œkq

./WD

k1

pD0

qk12p ./;

Œˇ Ckq

./WD M

p0

q1C2pCk./Œ1 ˚1q1C2pk./;

where we recall that qab./is a shift up by .a; b/ in the Z2-grading, and ./Œ1

is a shift up by 1in the homological grading. We write ˝for ˝k, and ˝bfor

˝.T ;

b;0/. We also write Ddg.T ; x

b; 0/ for the dg-enhanced derived category of

Z2-graded dg-modules over .T ;x

b; 0/ (see §B.2 for a precise deﬁnition).

4.1. Categorical action. Let 1b ;1 2T; x

bC1be the idempotent given by

1b;1 WD X

2Pr

b

:::

b0N1

:::

b1N2

...

:::

Tensor product categoriﬁcations and the blob 2-category 737

There is a (non-unital) map of algebras T; x

b!T; x

bC1given by adding a vertical

black strand to the right of a diagram from T;x



: : :

:::

Nr1

:::

D7!



:::

Nr1

:::

D(28)

sending the unit 12T; x

Nto the idempotent 1b;1. This map gives rise to derived

induction and restriction dg-functors

IndbC1

bWDdg.T ;x

b; 0/ ! Ddg .T ; x

bC1; 0/;

IndbC1

b./WD .T ; x

bC1; 0/1b;1 ˝L

b./;

ResbC1

bWDdg.T ;x

bC1; 0/ ! Ddg .T ; x

b; 0/;

ResbC1

b./WD RHOMbC1..T ; x

bC1; 0/1b;1;/;

which are adjoint (see §B.3). By Corollary 3.12, we know that .T ; x

bC1; 0/ is a

coﬁbrant dg-module over .T ;x

b; 0/, so that we can replace derived tensor products

(resp. derived homs) by usual tensor products

IndbC1

b./Š.T ; x

bC1; 0/1b;1 ˝b./;

ResbC1

b./Š1b;1.T  ;x

bC1; 0/ ˝bC1./:

Then, we deﬁne

FbWD IndbC1

EbWD 1q1C2bj x

NjResbC1

and Idbis the identity dg-functor on Ddg.T ; x

b; 0/.

Theorem 4.1. There is a quasi-isomorphism

Cone.Fb1Eb1! EbFb/Š

! M

Œˇ Cjx

Nj2bq

Idb;

of dg-functors.

Proof. Consider the map

Wq2.T ; x

b1b1;1 ˝b11b1;1T;x

b/! 1b;1T ; x

bC11b;1;

738 A. Lacabanne, G. Naisse, and P. Vaz

given by

x˝b1y7! xby;

where bis a crossing between the b-th and .b C1/-th black strands. Diagram-

matically, one can picture it as

:::

7!

: : :

where the bent black strands informally depict the induction/restriction functors.

Then, as in [34, Theorem 5.1], we obtain an exact sequence of .T ; x

b; T ; x

b/-bi-

modules

0! q2.T ;x

b1b1;1 ˝b11b1;1T;x

! 1b;1T ; x

bC11b;1



! M

p0

q2p.T ; x

b/˚2q2pC2jx

Nj4b .T ; x

b/! 0;

where is the projection onto the following summands

p0:::

b0N1

:::

br1



T; x

b1

:::

b0N1

:::

br1



T; x

b1

of Corollary 3.12 (i.e. when iDrand tDbr1). Note that, a priori, this

only deﬁnes a map of left modules. Fortunately, by applying similar arguments

as in [34, Lemma 5.4], it is possible to show that it deﬁnes a map of bimodules.

Exactness follows from a dimensional argument using Corollary 3.12.

4.1.1. Recovering V.N / ˝V. x

N /.Introducing the diﬀerential dNfrom §3.4 in

the picture, the map (28) lifts to a map of dg-algebras .T ; x

b; dN/!.T ; x

bC1; dN/.

Then we deﬁne dg-functors

b./WD .T ; x

bC1; dN/1b;1 ˝b./;

b./WD q2bj x

NjN1b;1.T ; x

bC1; dN/˝bC1./:

These deﬁnitions corresponds with derived induction and (shifted) derived restric-

tion dg-functors along (28), by Corollary 3.12 again.

Recall the notion of a strongly projective dg-module from [32] (or see §B.1.2).

Tensor product categoriﬁcations and the blob 2-category 739

Proposition 4.2. As .T ; x

b; dN/-module, .T ; x

bC1; dN/is strongly projective.

Proof. As in [34, Proposition 5.15], and omitted. 

By Proposition B.2, Theorem 4.1 can be seen as a quasi-isomorphism of

mapping cones

Cone.FN

b1EN

b1! EN

bFN

! Cone M

p0

q1C2pCNCjx

Nj2b Idb

! M

p0

q1C2pNjx

NjC2b Idb;

where hNis given by multiplication by the element

:::



Proposition 4.3. There is a quasi-isomorphism

Cone M

p0

q1C2pCNCjx

Nj2b Idb

! M

p0

q1C2pNjx

NjC2b IdbŠ

! M

ŒN Cjx

Nj2bq

Idb;

where LŒkqMWD LŒkqM Œ1.

Proof. As in [34, Proposition 5.9], and omitted. 

4.1.2. Induction along red strands. Take x

ND.N1; : : : ; Nr/and N0D.x

N; NrC1/.

Consider the (non-unital) map of algebras T;x

b!T;N 0

bthat consists in adding

a vertical red strand labeled NrC1at the right a diagram:



:::

Nr1

:::

: : :

D7!



:::

Nr1

:::

NrC1

Let IWDdg.T ;x

b; 0/ !Ddg .T ;x

b; 0/ be the corresponding induction dg-functor,

and let N

IWDdg.T ; x

N0; 0/ !Ddg .T ;x

b; 0/ be the restriction dg-functor.

Proposition 4.4. There is an isomorphism N

IıIŠId.

Proof. The statement follows from Corollary 3.12.

740 A. Lacabanne, G. Naisse, and P. Vaz

4.2. Categoriﬁcation theorem. In this section we suppose that kis a ﬁeld. Re-

call the notion of an asymptotic Grothendieck group K

0from [33, §8] (or §B.4).

Since .T ; x

b; 0/ is a positive c.b.l.f. dimensional Z2-graded dg-algebra (see Deﬁ-

nition B.5), we have by Theorem B.6 that K

0.T ; x

b; 0/ is a free Z..q; //-module

generated by the classes of projective T;x

b-modules with a trivial diﬀerential. Let

QK

0./WD K

0./˝Z..q;// Q..q; //:

For each 2Pr

b, there is a projective T; x

b-module given by

PWD T; x

b1:

Recall the inclusion WAb0˝NHb1˝    ˝ NHbr,!T; x

bdeﬁned in (27). It

is well known (see for example [21, §2.2.3]) that NHnadmits a unique primitive

idempotent up to equivalence given by

enWD #nxn1

1xn2

2xn12NHn;

where #n2Snis the longest element, w1w2wkWD w1w2wk, with ibeing

a crossing between the i-th and .i C1/-th strands, and xiis a dot on the i-th

strand. There is a similar result for NHb0Ab0(see [37, §2.5.1]). Moreover, for

degree reasons, any primitive idempotent of T; x

bis the image of a collection of

idempotents under the inclusion for some , and thus is of the form

eWD .eb1˝    ˝ ebn/:

It is also well known (see for example [21, §2.2.3]) that there is a decomposition

NHnŠqn.n1/=2 M

ŒnqŠ

NHnen;

as left NHn-modules. For the same reasons, we obtain

PŠqPr

iD0bi.bi1/=2 M

iD0ŒbiqŠ

T; x

be:

4.2.1. Categorifed Shapovalov form. Let T; x

NWD Lb0T;x

b. As in [21,

§2.5], let WT; x

N!.T ; x

N/op be the map that takes the mirror image of diagrams

along the horizontal axis. Given a left .T ; x

N; 0/-module M, we obtain a right

.T ; x

N; 0/-module M with action given by m rWD .1/degh.r/ degh.m/ .r/ m

for m2Mand r2T;x

N. Then we deﬁne the dg-bifunctor

.;/WDdg.T ; x

N; 0/ Ddg.T ; x

N; 0/ ! Ddg .k; 0/;

.W; W 0/WD W ˝L

.T ;

N;0/ W0:

Tensor product categoriﬁcations and the blob 2-category 741

Proposition 4.5. The dg-bifunctor deﬁned above satisﬁes:

..T ; x

0; dN/; .T ; x

0; dN// Š.k; 0/;

.IndbC1

bM; M 0/Š.M; ResbC1

bM0/for all M; M 02Ddg.T ; x

N; 0/;

.LfM; M 0/Š.M; LfM0/ŠLf.M; M 0/for all f2Z..q; //;

.M; M 0/Š.I.M /; I.M 0//.

Proof. Straightforward, except for the last point which follows from Proposi-

tion 4.4, together with the adjunction I`N

I.

Comparing Proposition 4.5 to §2.1.6, we deduce that .;/has the same

properties on the asymptotic Grothendieck group of .T ;r ; 0/ as the Shapovalov

form on M˝Vr.

4.2.2. The categoriﬁcation theorem. Let EWD Lb0Eband FWD Lb0Fb. By

Theorem 4.1 and Proposition B.7, we know that QK

0.T ; x

N; 0/ is an Uq.sl2/-mod-

ule, with action given by the pair ŒF; ŒE.

Lemma 4.6. We have

dimQ..q;//.QK

0.T ; x

b; 0// dimQ..q;//.M./ ˝V . x

N /qr2b /:

Moreover, QK

0.T ; x

N; 0/ is spanned by the classes ¹ŒPº2Pr; x

Proof. It is well known (see for example [36, Lemma 7.2]) that whenever k > n,

then the unit element in NHkcan be rewritten as a combination of elements having

nconsecutive dots somewhere on the left-most strand. Thus, for any 02Pr

b, we

obtain that 10can be rewritten as a combination of elements factorizing through

elements in ¹1º2Pr; x

.

We consider M./ ˝V . x

N / over the ground ring Q..q; // instead of Q.q; /.

Theorem 4.7. There are isomorphisms of Uq.sl2/-modules

QK

0.T ; x

N; 0/ ŠM ./ ˝V . x

N /;

and

QK

0.T ; x

N; dN/ŠV .N / ˝V . x

N /;

for all N2N.

742 A. Lacabanne, G. Naisse, and P. Vaz

Proof. We have a Q..q; //-linear map

M./ ˝V . x

N / ! QK

0.T ; x

N; 0/; v7! ŒP:

By Lemma 4.6, this map is surjective. It commutes with the action of K˙1

and Ebecause of Corollary 3.12. By Proposition 4.5, the map intertwines

the Shapovalov form with the bilinear form induced by the bifunctor .;/on

QK

0.T ; x

N; 0/. Thus, it is a Q..q; //-linear isomorphism. Since the map inter-

twines the Shapovalov form with the bifunctor .;/, and commutes with the

action of Eand K˙1, we deduce by non-degeneracy of the Shapovalov form that

it also commutes with the action of F. Thus, it is a map of Uq.sl2/-modules.

The case QK

0.T ; x

N; dN/follows from Theorem 3.13 together with [49, The-

orem 4.38]. 

5. Cups, caps and double braiding functors

Throughout this section, we ﬁx x

ND.1; 1; : : : ; 1/ 2Nrand write T;r WD T; x

and ˝TWD ˝.Lr0T;r /. Also, when we will talk about (bi)modules, we will

generally mean Z2-graded dg-(bi)module, assuming it is clear from the context.

5.1. Cup and cap functors. Following [49, §7] (see also [48, §4.3]), we deﬁne

the cup bimodule Bifor 1irC1as the .T ;r C2; 0/-.T ;r ; 0/-bimodule

generated by the diagrams



: : :

b01

: : :

b11

...

: : :

bi1

: : :

i11

...

: : :

(29)

for all .b0; : : : ; bi2; bi1; b0

i1; bi; : : : ; br/2NrC2. Here, generated means that

elements of Biare given by taking the diagram above and gluing any diagram of

T;r C2on the top, and any diagram of T;r on the bottom. The diagrams in Biare

considered up to graded braid-like planar isotopy, with the cup being in homolog-

ical degree 0, and subject to the same local relations as the dg-enhanced KLRW

algebra (20)–(23) and (26), together with the following extra local relations:

D0; D0; (30)

D;D  :(31)

Tensor product categoriﬁcations and the blob 2-category 743

We set the Z2-degree of the generator in (29) as

degq; WD .0; 0/:

Similarly, we deﬁne the cap bimodule x

Biby taking the mirror along the hori-

zontal axis of Bi. However, we declare that the cap is in homological degree 1,

and with Z2-degree given by

degq; WD .1; 0/:

Note that since the red cap has a 1homological degree, it anticommutes with

the nails when applying a graded planar isotopy.

From this, one deﬁnes the coevaluation and evaluation dg-functors as

BiWD Bi˝L

TW Ddg.T  ;r ; 0/ ! Ddg.T ;r C2; 0/;

BiWD x

Bi˝L

TW Ddg.T ;rC2; 0/ ! Ddg.T ;r ; 0/:

5.1.1. Biadjointness. Note that

BiŠqRHOMT.Bi;/Œ1;

by Proposition 5.1 below. Thus, q1x

BiŒ1 is right adjoint to Bi. Similarly, we

obtain that qx

BiŒ1 is left-adjoint to Bi.

The unit and counit of Biaq1x

BiŒ1 gives a pair of maps of bimodules

iWq.T  ;r /Œ1 ! x

Bi˝L

TBi; "iWBi˝Lx

Bi! q.T  ;r /Œ1;

and similarly qx

BiŒ1 aBigives

NiWq1.T ; r /Œ1 ! Bi˝Lx

Bi;N"iWx

Bi˝L

TBi! q1.T ;r /Œ1:

5.1.2. Tightened basis. Take D.b0; : : :; brC2/2PrC2

band 2Pr

b. Let Nibe

given by .b0; b1; : : :; bi2; bi1Cbi1CbiC1;O

bi;O

biC1; biC2; : : : br/2Pr

b. For

each 1`bi, consider the map

g`WqbiC12`.1NiT ;r 1/! 1Bi1;

744 A. Lacabanne, G. Naisse, and P. Vaz

given by gluing on the top the following element:



: : :

:::

: : :

bi1

: : :

`1

: : :

bi`

: : :

biC1

:::

: : :

Recall the basis Bof Theorem 3.11. We claim that

`D1

g`.NiB/; (32)

is a basis for 1Bi1. We postpone the proof of this for later.

5.2. Coﬁbrant replacement of Bi.As explained in [48, §4.3], Biadmits an

easily describable coﬁbrant replacement as a left module. But before describing

it, let us introduce some extra notations. Let Ti ; be the left .T ;r ; 0/-module

generated by the elements



: : :

b01

: : :

b11

...

1 1 11

: : :

bi11

: : :

bi1

...

: : :

for all .b0; b1; : : :; br/. We deﬁne similarly Ti; and Ti; .

Let pBibe the left .T ;r C2; 0/-module given by the dg-module

pBiWD

q.Ti ;

q2.Ti; ˚Ti ;

q.Ti ;

where the diﬀerential is given by the arrows, which are the maps given by adding

the term in the label at the bottom of , or . Similarly, we deﬁne a right

coﬁbrant replacement x

Biq'

x

Biby taking the symmetric along the horizontal

line and shifting everything by q1./Œ1.

Proposition 5.1. There is a surjective quasi-isomorphism of left Z2-graded

.T ;r C2; 0/-modules

pBi

Bi:

Tensor product categoriﬁcations and the blob 2-category 745

Proof. Consider the surjective map Ti; Bithat closes the elements at

the bottom by a cup:

7! :

This map is indeed surjective since any black strand going to the left of the cap

factors through a black strand going to the right, using (31). Then the claim follows

by observing that

qTi ;

2Ti; ˚Ti; Bi

qTi ;

is an exact sequence.

Indeed, by Theorem 3.11, we know that adding a black/red crossing is an

injective operation, and thus the sequence is exact on q2Ti; . For the same

reason we also have that

ker

qTi ;

˚Ti;

qTi ;

ŠTi; \Ti;

By Theorem 3.11, we know that if an element can be written as a diagram with a

black strand crossing a red strand on the left, and as a diﬀerent diagram with the

same black strand crossing a red strand on the right, then it can be rewritten as a

diagram with the same strand going straight, but carrying a dot. These elements

correspond exactly with the image of the preceding map in the complex, which is

thus exact at the second position. Finally, we observe that

BiŠTi; = .Ti; CTi; /;

by constructing an inverse of the map that adds a cup on the bottom, by pulling

the cup to the bottom. It is not hard, but a bit lengthy, to check that it respects the

deﬁning relations of Biin the quotient Ti; =.Ti; CTi; /.

746 A. Lacabanne, G. Naisse, and P. Vaz

Corollary 5.2. The elements in (32)form a ZZ2-graded k-basis for 1Bi1.

Proof. As in Theorem 3.11, one can show that the elements in (32) span the space

1Bi1, mainly using (31) and (23). Linear independence follows from a dimen-

sional argument, using Proposition 5.1 and Theorem 3.11. The computation of the

dimensions can be done at the non-categoriﬁed level, and thus is a consequence

of (10) of Lemma 2.4.

Therefore, the map Pg`WLbi

`D1qbiC12`.1 NiT ;r /'

!1Biof right modules

is an isomorphism, where Niand g`are as in §5.1.2. In particular, Biis a coﬁbrant

right dg-module.

With Theorem 4.7 in mind, this means that x

Biacts on QK

0.T ; x

N; 0/ as the

cap of Bon M˝Vr(see (10)), and Proposition 5.1 means that Biacts as the cup

(see (11)).

5.3. Double braiding functor. Inspired by the deﬁnition of the braiding functor

in [49, §6] (see also [48, §4.1]), we introduce a double braiding functor that will

play the role of a categoriﬁcation of the action of on M˝Vr.

Deﬁnition 5.3. The double braiding bimodule X(see Remark 2.2 for an explana-

tion about the terminology) is the .T ;r ; 0/-.T ;r ; 0/-bimodule generated by the

diagrams



: : :

b01

: : :

b11

...

: : :

for all .b0; : : : ; br/2NrC1. We consider diagrams in Xup to graded braid-like

planar isotopy with the generators being in homological degree 0, and subject to

the relations(20)-(23) and (26), and the extra local relations



;



:(33)

We set the Z2-degree of the generator as

degq; 0



AWD .0; 1/:

Tensor product categoriﬁcations and the blob 2-category 747

We deﬁne the double braiding functor as

„WD X˝L

TW Ddg.T  ;r ; 0/ ! Ddg.T ;r ; 0/:

5.3.1. Tightened basis. Let us now describe a basis of the bimodule X, similar

to the basis of T;r

bgiven in Theorem 3.11. We ﬁx and two elements of Pr

band

recall the set Sdeﬁned in §3.3.3. For each w2S;N

lD.l1; : : : ; lb/2 ¹0; 1ºb

and N

aD.a1; : : : ; ab/2Nbwe deﬁne an element xw;N

l;N

a21X1as follows:

(1) choose a left-reduced expression of win terms of diagrams as above,

(2) for each 1ib, if liD1, nail the i-th black strand (counting on the top

from the left) on the blue strand by pulling it from its leftmost position,

(3) for each 1ib, add aidots on the i-th black strand at the top,

(4) ﬁnally, attach the ﬁrst red strand to the blue strand by pulling it from its

leftmost position.

Deﬁnition 5.4. Deﬁne the unbraiding map

uWX ! T ;r ;

as the map given by removing the double braiding



7!



Note that the unbraiding map is a map of .T ;r

b; 0/-.T ;r

b; 0/-bimodules.

Theorem 5.5. The set ¹xw ;N

l;N

aWw2S;N

l2 ¹0; 1ºb;N

a2Nbºis a ZZ2-graded

k-basis of 1X1.

Proof. Showing that this set generates 1X1is similar to [34, Proposition 3.13]

and we leave the details to the reader.

To show that the elements .xw;N

l;N

a/w;N

l;N

aare linearly independent we consider

a linear combination Pw;N

l;N

a˛w;N

l;N

axw;N

l;N

aD0and apply the unbraiding map u.

We now pull the ﬁrst red strand to its original position before the last step of the

construction of xw;N

l;N

a. This has the eﬀect of adding dots on some black strands

because of (21).

We now rewrite u.Pw;N

l;N

a˛w;N

l;N

axw;N

a/D0in terms of the tightened basis

of T;r

b. We carefully look at the terms with the highest number of crossings:

748 A. Lacabanne, G. Naisse, and P. Vaz

by pulling the dots at the top, we obtain diﬀerent elements of the tightened basis

of T;r

bplus terms with a lower number of crossings. From the freeness of the

tightened basis of T;r

b, we deduce that the coeﬃcient of the terms with the highest

number of crossings must be zero and we can proceed by a descending induction

on the number of crossings. 

Corollary 5.6. The unbraiding map uWX !T;r is injective.

Proof. The matrix of uin terms of tightened bases can be made in column echelon

form with pivots being 1.

5.4. Coﬁbrant replacement of X.We now want to construct a left coﬁbrant

replacement for X. Take D.b2; : : : ; br/2Pr2

band consider the idempotent

1k;`; WD 1k;` ;b2;:::;br. We also write

1`; WD : : :

: : :

b21

...

: : :

so that for example

10;kC`; D

1

: : :

˝N

1`; :

For k0; ` 0and 2Pr2

b, we deﬁne

k;`; WD

k1

tD0

Y1;t

k;`; WD qk2t C1.T  ;r

b11;kC`1; /Œ1;

k;`; WD Y00

k;`; ˚

k1

tD0

Y0;t

k;`; ;

Y00

k;`; WD 1qk.T ;r

b10;kC`; /;

Y0;t

k;`; WD qk2t .T  ;r

b10;kC`; /Œ1:

Note that Y1

0D0and Y0

0D1.T ;r

b10;`; /.

Tensor product categoriﬁcations and the blob 2-category 749

We write

XkWD M

`0;2Pr2

X1k;`;;

kWD M

`0;2Pr2

k;`; ;

kWD M

`0;2Pr2

k;`; ;

and similarly for Y1;t

k,Y00

kand Y0;t

Deﬁne the coﬁbrant .T ;r

b; 0/-module pXkgiven by the mapping cone

pXkWD Cone.Y 1

! Y0

k/;

where {kWD Pk1

tD0{t

kfor

kWY1;t

k! Y00

k˚Y0;t

1

˝N

1kC`1; 7! 0





:::

˝N

1`Ck1t; ;

1

˝N

1`Ck1;1

Note that each {t

kis injective, and therefore so is {k. Then, consider the left module

map

kWpXk! Xk;

given by kWD 0

kCPk1

tD0t

kwhere

0

kWY00

k! Xk;

1

:::

˝N

1`; 7! : : :



˝N

1`; ;

and

t

kWY0;t

k! Xk;

1

:::

˝N

1`; 7!

: : : : : :



˝N

1`; ;

for all 0tk1.

750 A. Lacabanne, G. Naisse, and P. Vaz

Lemma 5.7. The map kWpXk!Xkis surjective.

Proof. The statement can be proved by observing that Xkis generated as a left

.T ;r

b; 0/-module by the elements

:::



˝N

1`; ;

: : : : : :



˝N

1`; ;

for all 0tk1. The details can be found in §A.2.

Lemma 5.8. The sequence

0! Y1

! Y0

k

! Xk! 0;

is a short exact sequence of left Z2-graded .T ;r ; 0/-modules.

Proof. Since we already have a complex with an injection and a surjection, it is

enough to show that

gdim XkDgdim Y0

kgdim Y1

where gdim is the graded dimension in the form of a Laurent series belonging

to NJh˙1; ˙1; q˙1K. This can be shown by induction on k, and the details are

in §A.2.

From that, we induce a right .T  ;r ; 0/-A1-action on pXWD Lk0pXk(see

§B.1.3), turning it into a Z2-graded .T  ;r ; 0/-.T ;r ; 0/-A1-bimodule, and we

obtain:

Proposition 5.9. The map WD Pk0kWpXXis a quasi-isomorphism of

Z2-graded .T ;r ; 0/-.T ;r ; 0/-A1-bimodules.

Proof. It is an immediate consequence of Lemma 5.8.

Again, having Theorem 4.7 in mind, it means „acts on QK

0.T ; x

N; 0/ as the

element of Bon M˝Vr(see (12)).

Tensor product categoriﬁcations and the blob 2-category 751

6. A categoriﬁcation of the blob algebra

As in [49, §7], the cup and cap functors respect a categorical instance of the

Temperley–Lieb algebra relations (3)–(6). We additionally show that the double

braiding functor respects a categorical version of the blob relations (8) and (9).

Note that Webster also proves that the cup and cap functors intertwine the cate-

gorical Uq.sl2/-action, which categoriﬁes the fact that the Temperley–Lieb algebra

describes morphisms of Uq.sl2/- modules. We start by proving the same for these

functors in the dg-setting as well as for the double braiding functors:

Proposition 6.1. We have natural isomorphisms Eı„Š„ıEand Fı„Š„ıF,

and also EıBiŠBiıE,FıBiŠBiıF, and similarly for x

Bi.

Proof. Since Eand Fare given by derived tensor product with a dg-bimodule that

is coﬁbrant both as left and as right module, all compositions are given by usual

tensor product of dg-bimodules. Then, the ﬁrst isomorphism is equivalent to

1b;1.T  ;r

bC1/˝bC1XbC1ŠXb˝b1b;1.T  ;r

bC1/;

which in turn follows from Theorem 5.5 and Corollary 3.12. The case with Fis

identical, and so is the proof for Biusing Corollary 5.2.

Then, we use all this to show that compositions of the functors Bi;N

Biand „

realize a categoriﬁcation of B.

6.1. Temperley–Lieb relations. This section is an extension of Webster’s re-

sults [49, §7] for the dg-enhanced KLRW algebra T;r .

Proposition 6.2. There is an isomorphism

Bi˙1˝L

TBiŠT;r ;

of Z2-graded .T ;r ; 0/-.T ;r ; 0/-A1-bimodules.

Proof. We prove N

Bi1˝L

TBiŠT;r , the other case follows similarly. Using

Proposition 6.1 and the fact that BiıIŠIıBifor i < r 1(where we recall Iis

the induction along a red strand deﬁned in §4.1.2), we can work locally, supposing

752 A. Lacabanne, G. Naisse, and P. Vaz

that iDr1and biD0. Then, we have that N

Bi1˝TpBilooks like

q20

A˚

which is isomorphic to

T;r

0˚0

because of (30). Note that it is an isomorphism of dg-bimodules, since all the

higher composition maps of the A1-structure must be zero by degree reasons,

concluding the proof. 

Corollary 6.3. There is a natural isomorphism N

Bi˙1ıBiŠId.

Proposition 6.4. There is a distinguished triangle

q.T  ;r /Œ1 i

! x

Bi˝L

TBi

N"i

! q1.T ;r /Œ1 0

!

of Z2-graded .T ;r ; 0/-.T ;r ; 0/-A1-bimodules.

Proof. We have

Bi˝L

TBiŠN

Biq˝TBi;

which looks like

q.Bi/Œ1 ˚q1.Bi/Œ1

Tensor product categoriﬁcations and the blob 2-category 753

Thus, since BiŠT;r , we have that

H. N

Bi˝L

TBi/Šq.T  ;r /Œ1 ˚q1.T ;r /Œ1:

In order to compute i, recall (or see §B.3.1) that the unit of the adjunction

.Bi˝L

T/`.RHOMT.Bi;// is given by

0

iWT;r ! RHOMT.Bi; Bi˝L

TT;r /ŠHOMT.pBi; Bi/;

t7! Œx 7! Nxt ;

where Nxis the image of xunder the map pBiBi. Moreover, HOMT.pBi; Bi/

is given by

q2HOMT.Ti; ; Bi/Œ2 ˚HOM.Ti ; ; Bi/

and then, 0

iis the map T;r '

!HOM.Ti; ; Bi/ŠBithat adds a cup on the

top. Thus, iidentiﬁes q.T  ;r /Œ1 with q.Bi/Œ1 H. N

Bi˝L

TBi/in homology.

Similarly, the counit of the adjunction .x

Bi˝L/`.RHOMT.x

Bi;// is

N"0Wx

Bi˝L

TRHOMT.x

Bi; T ;r /Šx

Biq˝THOMT.x

Bi; T ;r /! T;r ;

t˝f7! f .N

t/:

Then, we obtain that x

Biq˝THOMT.x

Bi; T ;r /is isomorphic to

q2.Bi/Œ2 ˚Bi

and thus, N"0is the isomorphism

! T;r :

Therefore, N"identiﬁes q1.T  ;r /Œ1 with q1.Bi/Œ1 H. N

Bi˝L

TBi/in

homology. 

754 A. Lacabanne, G. Naisse, and P. Vaz

Because the connecting morphism in Proposition 6.4 is zero, the triangle splits

and we have

Bi˝L

TBiŠq.T  ;r /Œ1 ˚q1.T ;r /Œ1:

Corollary 6.5. There is a natural isomorphism

BiıBiŠqIdŒ1 ˚q1IdŒ1:

6.2. Blob relations. Proving the blob relations requires some preparation.

6.2.1. Quadratic relation. We deﬁne recursively the following element by set-

ting z0WD 0,

z1WD z1WD ;(34)

ztC2WD

:::

ztC2

:::

ztC1

:::

ztC1

(35)

for all t0. Note that z2is given by a single crossing

z2D

since the second term is zero in this case. One easily sees that degq.zt/D22t .

Deﬁne a map of left modules

kWq2.Xk/Œ1 ! X˝TY1

as '1

kWD Pk1

tD0'1;t

k, where each

'1;t

kWq2.Xk/Œ1 ! X˝TY1;t

kŠM

`;

qk2t C1.X11;kC`1;/Œ1;

is given by multiplication on the bottom by

::: :::



zkt

˝N

1`; :

Tensor product categoriﬁcations and the blob 2-category 755

Also deﬁne a map of left modules

kWM

`;

q2.T ;r

b1k;`; /Œ1 ! X˝TY0

as '0

kWD '0

0CPk1

tD0'0;t

k, where each

0WM

`;

q2.T ;r

b1k;`; /Œ1 ! X˝TY00

kŠM

`;

1qk.X10;kC` ;/;



:::

˝N

1`; 7! 

k1

tD0

::: :::

zkt

t

˝N

1`; ;

and where

'0;t

kWM

`;

q2.T ;r

b1k;`; /Œ1 ! X˝TY0;t

kŠM

`;

qk2t .X10;k C`; /Œ1;



:::

˝N

1`; 7!

::: :::

zkt

t

˝N

1`; :

Recall that the unbraiding map (Deﬁnition 5.4)

uWX ,!T ;r

is given by



7!



Lemma 6.6. The diagram

X˝TY1

kX˝TY0

k1X

q2.Xk/Œ1 L`; q2.T ;r

b1k;`; /Œ1 0

1˝{k



u˝k

commutes.

756 A. Lacabanne, G. Naisse, and P. Vaz

Proof. The proof is a straightforward computation using (20) and (22) together

with (33). We leave the details to the reader. 

Thus, there is an induced map

'kWCone.q2.Xk/Œ1 u

! M

`;

q2.T ;r

b1k;`; /Œ1/Œ1

! Cone.X ˝L

TXk

1˝u

! 1Xk/;

as left modules.

Theorem 6.7. The map

'WD

kD0

.1/k'kWCone.q2XŒ1 u

! q2T;r

bŒ1/Œ1

! Cone.X ˝L

TX1˝u

! 1X/;

is a quasi-isomorphism.

Proof. The statement can be proven by showing that Cone.'/ has a trivial homol-

ogy, and thus is acyclic. This is done in details in §A.3.1.

The next step is to prove that 'deﬁnes a map of A1-bimodules. Luckily, by

the following proposition, we do not need to use any A1-structure here.

Proposition 6.8. The map

X˝TpX1˝

! X˝TX;

is a quasi-isomorphism of A1-bimodules.

Proof. Tensoring to the left is a right-exact functor, thus Lemma 5.8 gives us an

exact sequence

X˝TY1

1˝{k

! X˝TY0

1˝k

! Xk! 0:

It is not hard to see that 1˝{kis injective, and thus we have a short exact sequence

0! X˝TY1

1˝{

! X˝TY0

1˝k

! Xk! 0;

so that 1˝kis a quasi-isomorphism. 

Tensor product categoriﬁcations and the blob 2-category 757

Taking a mapping cone preserves quasi-isomorphisms. Thus, we have a quasi-

isomorphism

Cone.X ˝L

TX1˝u

! 1X/ '

! Cone.X ˝TX1˝u

! 1X/: (36)

Let

Q'WCone.q2X Œ1 u

! q2T;r

bŒ1/Œ1 ! Cone.X ˝TX1˝u

! 1X/

be the map given by composing 'with the quasi-isomorphism in (36). We also

write Q'0WD .1 ˝/ ı'0. Therefore, by Lemma B.3, proving that 'is a map

of A1-bimodules ends up being the same as proving that Q'0is a map of dg-

bimodules.

Theorem 6.9. The map 'is a map of Z2-graded .T ;r ; 0/-.T  ;r ; 0/-A1-bimod-

ules.

Proof. The statement follows by proving that Q'0is a map of dg-bimodules, which

is done in details in §A.3.2.

Corollary 6.10. There is an exact sequence

0! q2.X/Œ1 u

! q2.T ;r

b/Œ1 Q'0

! X˝TX1˝u

! 1X! 0;

of dg-bimodules.

Corollary 6.11. There is a quasi-isomorphism

Cone.q2„Œ1 ! q2IdŒ1/Œ1 '

! Cone.„ ı„! 1„/;

of dg-functors.

6.2.2. Inverse of „.Recall the notations from §5.4.

Lemma 6.12. As a right .T  ;r ; 0/-module, 11;kC`1;Xis generated by the ele-

ments



:::

k1

˝N

1`; and



˝N

1`Ck1;:(37)

Proof. The statement can be proven using an induction on k, as done in details

in §A.3.

758 A. Lacabanne, G. Naisse, and P. Vaz

Lemma 6.13. The map

. ı {k/WHOMT.Y 0

k; X/ HOMT.Y 1

k; X/;

is surjective.

Proof. We have

HOMT.Y 0

k; X/ ŠM

`; qk.10;kC`;X/ ˚

k1

tD0

1q.k2t / .10;kC` ;X/Œ1;

HOMT.Y 1

k; X/ ŠM

`; k1

tD0

1q.k2t C1/ .11;kC`1; X /Œ1:

Then, the map

. ı {k/Wqk.10;kC`; X / ˚1q.k2t/ .10;kC` ;X/Œ1

! 1q.k2t C1/ .11;kC`1; X /Œ1

is given by gluing





:::

˝N

1`Ck1t; ;

1

˝N

1`Ck1;1

on the top of diagrams, for all 0tk1.

Then we observe that the map

. ı {k/Wqk.10;kC`; X / ! 1q.k2tC1/.11;k C`1;X/Œ1

sends

:::

s

˝N

1`; ;7!

0if s < t ;



:::

k1

˝N

1`; if sDt;



:::

s1

˝N

1`; if s > t ;

Tensor product categoriﬁcations and the blob 2-category 759

for all 0sk1. Thus, . ı {k/WHOMT.Y 0

k; X/ HOMT.Y 1

k; X/

has a triangular form when applied to the elements above, and is surjective by

Lemma 6.13.

Proposition 6.14. The functor

„WDdg.T  ;r ; 0/ ! Ddg.T ;r ; 0/

is an autoequivalence, with inverse given by

„1WD RHOMT.X; /WDdg.T  ;r ; 0/ ! Ddg.T ;r ; 0/:

Proof. By Lemma 6.13 and Proposition 5.9, we have

RHOMT.X1; X10/ŠHOMT.X1; X10/:

Then, we compute

gdim HOMT.X1; X10/Dgdim HOMT.P;P0/;

using the fact that „decategoriﬁes to the action of . More precisely, as in [49,

§4.7], the bifunctor RHOMT.;/decategoriﬁes to a sesquilinear version of the

Shapovalov form when restricted to a particular subcategory of Ddg.T ;r ; 0/, and

this sesquilinear form respects . w; w 0/D.w; w0/. Finally, we observe that the

map

HOMT.P;P0/IdX˝./

,! HOMT.X1; X10/;

is injective, since the map P!X1given by gluing

: : :



:::

on the top of diagrams is injective. This can be seen by composing the above

map P!X1with the injection uWX1!P, and observing it yields

an injective map. Therefore, RHOMT.X1; X10/Š1T;r 10, and „is an

autoequivalence. 

6.2.3. Categoriﬁcation of relation (8)

Lemma 6.15. There is a quasi-isomorphism

X˝L

TB1ŠX˝TpB1

! X˝TB1;

of A1-bimodules.

760 A. Lacabanne, G. Naisse, and P. Vaz

Proof. Let us write XWD X˝TT1; . Then we have

X˝TpB1Š

q.X

q2.X ˚X

q.X

The statement follows by observing that the ﬁrst map is injective, and its image

coincides with the kernel of the second one. 

Our goal will be to show the following:

Proposition 6.16. There is a quasi-isomorphism

q.T ;r /Œ1 ˚1q1.T ;r /Œ1 '

! N

B1˝L

TX˝L

TB1;

of A1-bimodules.

For this, we will need to understand the left A1-action on B1q:

B1qWD

q.T ˚q1.T /Œ

We start by constructing a composition map T˝B1q!B1q, by deﬁning it on

each generator of T. We extend it by ﬁrst rewriting elements in Tas basis elements

and then applying recursively the deﬁnition in terms of generating elements (so

that it is well deﬁned). Dots and crossings act on each of the summand by simply

adding the three missing vertical strands between the -strand and the remaining

of the diagram, and gluing on top. For example in q1.T /Œ1, we have



:::

D7!

1 1

:::

The action of the nail is a bit trickier. On q.T /Œ1 and on q1.T /Œ1 it

acts by gluing



7!



Tensor product categoriﬁcations and the blob 2-category 761

on the top of the diagrams. On Tit acts by



7! 0

@





;



2T˚T ;

and on Tby



7! 0

@

;



2T˚T :

One can easily verify that this respects the diﬀerential in B1q. The higher multi-

plication maps T˝B1q˝T!B1qand T˝T˝B1qcompute the defect of

the map T˝B1q!B1qfor being a left T-action. Concretely, it means that we

can compute these higher multiplication maps by looking how both side of each

deﬁning relation of Tact on B1q.

For example, the relation



is respected on q1.T /Œ1 up to adding the elements appearing in the right

of the following equation:







so that the higher multiplication map T˝T˝q1.T /Œ1 !B1qgives



˝17! 0

@

;



2T˚T :

762 A. Lacabanne, G. Naisse, and P. Vaz

Note that it means the higher maps only involve elements coming from (26). Also,

one can easily verify that the other two relations in (26) are already respected for

the multiplication map T˝q1.T /Œ1 !B1q, so that our computation

above completely determine T˝T˝q1.T /Œ1 !B1q. There is a similar

higher multiplication map T˝q1.T /Œ1 ˝T!B1q, which is non-trivial

in the case



˝1˝



for similar reasons. We will not need to compute the other higher composition

maps.

Proof of Proposition 6.16.Tensoring B1qwith X˝TB1gives a complex where

the elements are locally of the form

;

which, after eliminating the acyclic subcomplex, yields



˚q10

@

AŒ1 :

All higher multiplications maps vanish: except for T˝T˝.B1q˝TX˝B1/!

.B1q˝TX˝B1/and T˝.B1q˝TX˝B1/˝T!.B1q˝TX˝B1/, all of these are

zero for degree reasons, and the remaining two are zero by the calculations above.

Therefore, what remains is isomorphic to q.T  ;r /Œ1 ˚1q1.T ;r /Œ1, as

dg-bimodules. We conclude by applying Lemma 6.15.

Tensor product categoriﬁcations and the blob 2-category 763

Corollary 6.17. There is a quasi-isomorphism

q.Id/Œ1 ˚1q1.Id/Œ1 '

! N

B1ı„ıB1;

of dg-functors.

6.3. The blob 2-category. In this section, we suppose kis a ﬁeld. Let B.r; r 0/

be the subcategory of dg-functors Ddg .T ;r ; 0/ !Ddg.T ;r 0; 0/ c.b.l.f. generated

by all compositions of „; Biand N

Bi, and identity functor whenever rDr0, where

c.b.l.f. generated means it is given by certain (potentially inﬁnite) iterated exten-

sions of these objects (see Deﬁnition B.9 for a precise deﬁnition). As explained

in §B.4.4, there is an induced morphism

QK

0.B.r; r0// (66)

! HomQ..q;//.QK

0.T ;r ; 0/; QK

0.T ;r 0; 0//;

sending the equivalence class of an exact dg-functor to its induced map on the

asymptotic Grothendieck groups of its source and target (this is similar to the

fact that an exact functor between triangulated categories induces a map on their

triangulated Grothendieck groups).

Recall the blob category B, but consider it as deﬁned over Q..q; // instead of

Q.q; /.

Theorem 6.18. There is an isomorphism

HomB.r; r0/ŠQK

0.B.r; r0//:

Proof. Comparing the action of Bon M˝Vrfrom §2.2 with the coﬁbrant

replacement pXfrom §5.4, and pBiand pN

Bifrom §5.2, we deduce there is a

commutative diagram

HomB.r; r0/HomQ..q;//.M ˝Vr; M ˝Vr0/

QK

0.B.r; r0// HomQ..q;//.QK

0.T ;r ; 0/; QK

0.T ;r 0; 0//

(13)



(66)

where the arrow fis the obvious surjective one, sending to Œ„, and cup/caps to

ŒBi/ŒN

Bi. Because the diagram commutes and using Theorem 2.5, we deduce that

fis injective, and thus it is an isomorphism. 

764 A. Lacabanne, G. Naisse, and P. Vaz

In particular, if we write BrWD B.r; r/, then we have:

Corollary 6.19. There is an isomorphism of Q..q; //-algebras

QK

0.Br/ŠBr:

By Faonte [9], we know that A1-categories form an .1; 2/-category, where

the hom-spaces are given by Lurie’s dg-nerves [28] of the dg-categories of

A1-functors (or equivalently quasi-functors, see §B.2.1). Thus, we can deﬁne

the following:

Deﬁnition 6.20. Let Bbe the .1; 2/-category deﬁned by

objects are non-negative integers r2N(corresponding to Ddg.T ;r ; 0/);

HomB.r; r0/is Lurie’s dg-nerve of the dg-category B.r; r0/.

We refer to Bas the blob 2-category.

We deﬁne QK

0.B/to be the category with objects being non-negative integers

r2Nand homs are given by asymptotic Grothendieck groups of the homotopy

categories of HomB.r; r 0/. These homs are equivalent to QK

0.B.r; r0//.

Corollary 6.21. There is an equivalence of categories

QK

0.B/ŠB:

7. Variants and generalizations

7.1. Zigzag algebras. In [45, §4] it was proven that for gDsl2the KLRW

algebra T1;:::;1

1with rred strands and only one black strand is isomorphic to a

preprojective algebra AŠ

rof type A. It is a Koszul algebra, whose quadratic dual

was used by Khovanov and Seidel in [24] to construct a categorical braid group

action.

Let kbe a ﬁeld of any characteristic and let Qrbe the following quiver

0 1 2  r

0j1



1j0

1j2

2j1

and kQrits path algebra. We endow kQrwith a ZZ2-grading by declaring that

deg.iji˙1/ WD .0; 1; 0/; deg. / WD .1; 0; 2/:

Tensor product categoriﬁcations and the blob 2-category 765

We consider the ﬁrst grading as homological, and the second and third gradings

are called the q-grading and the -grading respectively. We denote the straight

path that starts on i1and ends at inby .i1ji2j:::jin1jin/and the constant path

on iby .i/. The set ¹.0/;  ; .r /ºforms a complete set of primitive orthogonal

idempotents in kQr.

Deﬁnition 7.1. Let AŠ

rbe algebra given by the quotient of the path algebra kQr

by the relations

.iji1ji / D.i jiC1ji/; for i > 0,

.0j1j0/ D.0j1j0/;

2D0:

We usually consider AŠ

ras a dg-algebra .AŠ

r; 0/ with zero diﬀerential. We can

also consider a version of AŠ

rwith a non-trivial diﬀerential dgiven by

d.X / WD ´.0j1j0/ if XD;

0otherwise;

of which one easily checks that it is well deﬁned.

Proposition 7.2. The ZZ2algebra AŠ

ris isomorphic to the ZZ2algebra T;r

in rred strands and 1black strand by the map sending

.i/ 7!



: : : : : :

;

where the black strand comes right after the ith red, and

.i 1ji/ 7!



: : : : : :

;

.i C1ji/ 7!



: : : : : :

;

7!



: : : : : : :

766 A. Lacabanne, G. Naisse, and P. Vaz

Furthermore, the isomorphism upgrades to isomorphisms of dg-algebras

.AŠ

r; 0/ Š.T  ;r

1; 0/ and .AŠ

r; d / Š.T ;r

1; d1/:

Proof. First, one can show by a straightforward computation that the map deﬁned

above respects all deﬁning relations of AŠ

r. Moreover, by turning any dot in T;r

to a double crossings using (21), it is not hard to construct an inverse of the map

deﬁned above. We leave the details to the reader. 

Corollary 7.3. The homology of .AŠ

r; d / is concentrated in homological degree 0

and is isomorphic to the preprojective algebra AŠ

Moreover, by Proposition 7.2, the results in §6can be pulled to the derived

category of Z2-graded .AŠ

r; 0/-modules, endowing Ddg.AŠ

r; 0/ ŠDdg .T ;r

1; 0/

with a categorical action of Br.

7.2. Dg-enhanced KLRW algebras: the general case. Fix a symmetrizable

Kac–Moody algebra gwith set of simple roots Iand dominant integral weights

WD .1; : : : ; d/.

7.2.1. Dg-enhanced KLRW algebras: gsymmetrizable. We recall that the

KLRW algebra [49, §4] TN



b.g/on bstrands is the diagrammatic k-algebra gen-

erated by braid-like diagrams on bblack strands and rred strands. Red strands

are labeled from left to right by 1; : : : ; rand cannot intersect each other, while

black strands are labeled by simple roots and can intersect red strands transversally,

they can intersect transversally among themselves and can carry dots. Diagrams

are taken up to braid-like planar isotopy and satisfy the following local relations:

the KLR local relations (2.5a)–(2.5g) in [49, Deﬁnition 2.4];

the local black/red relations (38)–(41) for all 2

and for all ˛j,˛k2I,

given below;

a black strand in the leftmost region is 0.

˛j



˛j



;

˛j

;(38)

˛j



˛j

j;

˛j

˛j

j



;(39)

Tensor product categoriﬁcations and the blob 2-category 767

˛j˛k



˛k

˛j



;

˛j˛k

˛k

˛j

;(40)

˛j˛k



˛k

˛j

Cıj;k X

aCbDj1˛j

˛k



:(41)

Multiplication is given by concatenation of diagrams that are read from bottom

to top, and it is zero if the labels do not match. The algebra TN



b.g/is ﬁnite-

dimensional and can be endowed with a Z-grading (we refer to [49, Deﬁnition 4.4]

for the deﬁnition of the grading).

In the case of

Dthe algebra T

b.g/contains a single red strand labeled 

and is isomorphic to the cyclotomic KLR algebra R.b/ for gin bstrands.

Deﬁnition 7.4. Fix a g-weight D.1; : : : ; jIj/with each ibeing a formal pa-

rameter. The dg-enhanced KLRW algebra T;



b.g/is deﬁned as in Deﬁnition 3.2,

with a blue strand labeled by and with the rred strands labeled by 1; : : : ; r

and the black strands labeled by simple roots. The black strands can carry dots

and be nailed on the blue strand:

˛j



with everything in homological degree 0, except that a nail is in homological

degree 1. The diagrams are taken up to graded braid-like planar isotopy, and are

required to satisfy the same local relations as TN



b.g/, together with the following

extra local relations:

˛j



˛j



;

˛j˛k



D 

˛j˛k



;

˛j



D0;

for all ˛j,˛k2I.

Note that we have an inclusion TN



b.g/T;



b.g/by adding a vertical blue

strand at the left of a diagram. The algebra T;



b.g/can be endowed with the

768 A. Lacabanne, G. Naisse, and P. Vaz

q-grading inherited from TN



b.g/. It can be additionally endowed with a k-grading

for each ˛k2Isuch that TN



b.g/T;



b.g/sits in k-degree zero for all k, and

degq;k0

@˛j



WD .0; 2ık;j /:

We usually consider T;



b.g/as a Z1CjIj-graded dg-algebra .T ;



b.g/; 0/ with

trivial diﬀerential. In the case of

D¿the algebra T;¿

b.g/contains a blue strand

labeled and is isomorphic to the b-KLR algebra Rb.b/ introduced in [34, §3.1].

The results of §3can be generalized to T;



b.g/. In particular, one can prove

it is free over kand that it admits a basis similar to the one in Theorem 3.11.

Moreover, by using induction and restriction functors that add a black strand, we

obtain a categorical action of gon Ddg.T ;



b.g/; 0/ (in the sense of [34]), which

categoriﬁes the Uq.g/-action on the tensor product of a universal Verma module

and several integrable modules.

Fix an integrable dominant weight of gand deﬁne a diﬀerential don T ;



b.g/

(after specialization of the j-grading to qj) by setting

d ˛j

!D

j

˛j



and d.t/ D0for all t2TN



b.g/T;



b.g/, and extending by the graded Leibniz

rule with respect to the homological grading. A straightforward computation

shows that dis well deﬁned.

Proposition 7.5. The dg-algebra .T ;



b.g/; d/is formal with

H.T ;



b.g/; d/ŠT.;

/

b.g/:

Proof. The proof follows by similar arguments as in [34, Theorem 4.4]. 

7.2.2. Dg-enhanced KLRW algebras for parabolic subalgebras. Let pg

be a parabolic subalgebra with partition IDI

ftIrof the set of simple roots,

and .; n/ D.i/i2I, with ia formal parameter if i2Ir, and iDqniwith

ni2Nif i2I

Tensor product categoriﬁcations and the blob 2-category 769

Introduce a diﬀerential d;n on T;



b.g/(after specialization of the j-grading

to qnjfor each ˛j2Ir) by setting

d;n ˛j

!D8

0if ˛j2Ir;

˛j



if ˛j2I

and d;n .t / D0for all t2TN



b.g/T;



b.g/, and extending by the graded

Leibniz rule with respect to the homological grading. As before, a straightforward

computation shows that it is well deﬁned.

Proposition 7.6. The dg-algebra .T ;



b.g/; d;n/is formal.

Proof. The proof follows by similar arguments as in [34, Theorem 4.4]. 

Deﬁnition 7.7. We deﬁne the dg-enhanced p-KLRW algebra as

T;



b.g;p/WD H.T ;



b.g/; d;n/:

Note that by Proposition 7.6 we have a quasi-isomorphism

.T ;



b.g;p/; 0/ Š.T  ;



b.g/; d;n/:

Similarly as above, Ddg.T ;



b.g/; d;n/categoriﬁes the tensor product of a par-

abolic Verma module and several integrable modules, and comes with a categor-

ical action of g.

7.3. Dg-enhanced quiver Schur algebras. In order to deﬁne a quiver Schur

algebra of type A1, we follow the approach of [23], which best suits our goals. We

actually use a slightly diﬀerent deﬁnition because theirs corresponds to a thick

version of KLRW algebra (see [23, §9.2]), and we want to relate it to the version

we use.

7.3.1. Cyclic modules and quiver Schur algebras. Recall that NHN

bŠT.N /

is the N-cyclotomic nilHecke algebra on bstrands. Fix r0and x

ND.N0;

N1; : : : ; Nr/2Nrsuch that PiNiDN. For D.b0; b1; : : : ; br/such that

PibiDb, we deﬁne the element

WD

iD1

.xNiCCN1

brCCbiC1C1xNiCCN1

brCCbiC1Cbi/2NHN

770 A. Lacabanne, G. Naisse, and P. Vaz

where we recall that xbis a dot on the bth black strand. Then, we consider the

cyclic right NHN

b-module deﬁned as

WD xx

NHN

The quiver Schur algebra (of type A1) is deﬁned as the Z-graded algebra:

bWD ENDNHN

bM

2Pr

qdegq.x x

/=2Yx

;

where END means the (Z-)graded endomorphism ring. The Z-graded algebra Qx

is isomorphic to Tx

b[49, Proposition 5.33]. The reduced quiver Schur algebra (of

type A1) is deﬁned as

redQWD ENDNHN

bM

2Pr; x

qdegq.x x

/=2Yx

;

where Pr; x

bWD ¹.b0; b1; : : : ; br/WbiNifor 0irº  Pr

b. It is Morita

equivalent to Qx

b(this can be shown by observing that if bi> Nifor some i, then

is isomorphic to a direct sum of elements in ¹qdegq.x x

0/=2Yx

0j02Pr; x

bº),

and thus to Tx

7.3.2. Dg-enhanced cyclic modules. Our goal is to construct a dg-enhancement

of Yx

over .T ;;

b; dN/, the dg-enhanced KLRW algebra without red strands. We

will simply write T

bfor T;;

b. Recall from Theorem 3.13 that .T 

b; dN/is quasi-

isomorphic to T.N /

bŠNHN

Let Tq`

bfor `2Zbe the algebra deﬁned similarly as T

b(see Deﬁnition 3.2)

except that the blue strand is labeled by q`, and the nail is in Z2-degree:

degq; 0

@q`

D.2`; 2/:

Whenever ``0and bb0, there is an inclusion of algebras

Tq`

b,! Tq`0

b0;(42)

given by ﬁrst turning any q`-nail into a q`0-nail by adding dots:

q`

7! ``0

q`0

Tensor product categoriﬁcations and the blob 2-category 771

so that the blue strand labeled q`becomes labeled q`0, and then adding b0b

vertical black strands at the right:

q`0

: : :

D7!

q`0

: : :

b0b

A straightforward computation shows that the map in (42) is well deﬁned, and

Theorem 3.11 shows that the map is injective.

By restriction, the inclusion Tq`

b,!Tq`0

b0deﬁnes a left action of Tq`

bon

any Tq`0

b0-module.

Deﬁnition 7.8. We deﬁne the right T

b-modules

WD TqNrN1

br˝TqNr1N1

   ˝TqN1

brCCb2

TqN1

brCCb1˝T

brCCb1

T

and

WD xx

z

:

Note that we can endow Gx

with either a diﬀerential of the form dN(as in §3.4)

or a trivial one, making it a right dg-module over .T 

b; dN/or .T 

b; 0/ respectively.

Example 7.9. Take for example rD2. Then, we picture Gx

in terms of diagrams

qN2N1

: : :

TqN2N1

TqN1

b1Cb2

T

b2b1

Note that whenever NC`0we can equip Tq`

bwith a diﬀerential dNgiven

dN0

@q`

WD NC`

q`

and it is compatible with the inclusion in (42).

772 A. Lacabanne, G. Naisse, and P. Vaz

We conjecture the following:

Conjecture 7.10. There is a quasi-isomorphism

.G x

; dN/'

! .Y x

; 0/:

Lemma 7.11. There is a decomposition as graded vector spaces

qn

: : :

n n

Tqn

kC1

qn

:::

n n

Tqn

NHkC1

: : :

n n

qn

Tqn

NHkC1

Proof. The claim follows from Theorem 3.11.

Proposition 7.12. Suppose and 0are such that biDb0

ifor all 0im

except iDjand iDjC1where they respect bjDb0

j1and bjC1Db0

jC1C1.

Then there is an inclusion of right dg-modules

,! Gx

0:

Proof. We can work locally, and thus we want to prove that

G2WD

qn

:::

n n :::

:::

Tqn

kC1

T



qn

:::

n:::

:::

Tqn

T

DW G1:

We apply Lemma 7.11 on Tqn

kC1inside G2. The left summand is clearly in G1.

For the right summand, it is less clear since the nails in T

ball acts by adding a

nail and ndots on the blue strand labeled qn. Thus, we want to show that



qn

2G1:

Tensor product categoriﬁcations and the blob 2-category 773

By (19), we have



qn



qn



`D0X

rCsDn1

  

qn

The term of the left is clearly in G1since there are ndots next to the nail, so that

it can be obtained from a nail in T

b. The terms on the right are also in G1since

we can slide the nail and crossings on the left to the top, into Tqn

k.

Consider .xn

1xn

k/T qn

k˝T

kT

b. We obtain an inclusion

.xn

1xn

k/T qn

k,! .xn

1xn

kxn

kC1/T qn

kC1;

of q-degree 2n by adding a vertical strand on the right on which we put ndots

(again, the fact it is an inclusion follows immediately from Theorem 3.11). In

turns, it gives rise to a map of right (dg-)modules

.xn

1xn

k/T qn

k˝T

kT

b! .xn

1xn

kxn

kC1/T qn

kC1˝T

kC1T

In terms of diagrams, we can picture the inclusion above as

qn

:::

n:::

:::

Tqn

T

7!

qn

:::

n n :::

:::

Tqn

T



qn

: : :

n n : : :

: : :

Tqn

kC1

T

This generalizes into the following proposition:

Proposition 7.13. Under the same hypothesis as in Proposition 7.12, we obtain

a map of right dg-modules

0! Gx

;

of q-degree 2NjC1. Diagrammatically, this map is given by gluing on top the

dots xNjC1

rCCb0

jC1C1.

774 A. Lacabanne, G. Naisse, and P. Vaz

7.3.3. Dg-quiver Schur algebra

Deﬁnition 7.14. We deﬁne the dg-quiver Schur algebras as

.dgQx

b; dN/WD ENDdg

.T 

b;dN/M

2Pr

qdegq.x x

/=2.G x

; dN/;

and

.dgQx

b; 0/ WD ENDdg

.T 

b;0/ M

2Pr

qdegq.x x

/=2.G x

; 0/;

where ENDdg is the Z2-graded (Z-graded in the ﬁrst case) dg-endomorphism ring

(see §3.1.2). We also deﬁne a reduced version as

.red

dg Qx

b; 0/ WD ENDdg

.T 

b;0/ M

2Pr; x

qdegq.x x

/=2.G x

; 0/:

Conjecture 7.15. There is a quasi-isomorphism

.dgQx

b; dN/'

! .Q x

b; 0/:

Our goal is to construct a graded map of algebras

T; x

b! dgQx

For D.b0; b1; : : : ; br/2Pr

b, we send

17! Id 2ENDT

b.G x

/dgQx

Dots on the ith black strand (resp. black/black crossings on the ith and .i C1/th

black strands) on 1is sent to multiplication on the left (i.e. gluing on top) by a

dot on the ith black strand (resp. crossing) on Gx

. These are indeed maps of right

T

b-modules since the dots and crossing commutes with xn

ixn

iC1for all n0.

Similarly, a nail on the blue strand labeled in T ; x

bis sent to multiplication on

the left by a nail on the blue strand labeled qNrN1in Gx

.

For black/red crossing i, if the red strand goes from bottom left to top right,

then we have 10i1where and 0are as in Proposition 7.12. Then, we associate

to it the map Gx

!Gx

0of Proposition 7.12. If the red strand goes from bottom

right to top left, then we have 1i10, and we associate to it the map Gx

0!Gx



of Proposition 7.13.

Tensor product categoriﬁcations and the blob 2-category 775

Proposition 7.16. The map deﬁned above gives rise to maps of Z-graded dg-

algebras

.T ; x

b; dN0/! .dgQx

b; dN/;

and of Z2-graded dg-algebras

.T ; x

b; 0/ ! .dg Qx

b; 0/:

Proof. We show the assignment given above is a map of algebras, the commuta-

tion with the diﬀerentials being obvious since the image by dN0of a nail on a blue

strand labeled consists of N0dots on the ﬁrst black strand; and the image by dN

of a nail on a blue strand labeled qNrN1consists of NNr    N1DN0

dots.

Thus, we need to prove the map respects all the deﬁning relations in Deﬁni-

tion 3.2. Relations in (18) and (19) are immediate by construction. The relations

in (20) follow from commutations of dots. Since the map in Proposition 7.13 is

multiplication by njC1dots and the map in Proposition 7.12 is an inclusion, we

have the relations in (21). For the left side of (22) both black/red crossings are

given by an inclusion, and thus commutes with the multiplication on the left by

the black/black crossing. For the right side, the black/red crossings give a multipli-

cation by xNjC1

ixNjC1

iC1, which commutes with the black/black crossing. For (23),

one the black/red crossing is an inclusion and the other one is multiplication by

xNjC1

ion both side of the equality, so that the relation follows from (19). Finally,

the relation in (26) is immediate by construction. 

Conjecture 7.17. The maps in Proposition 7.16 are isomorphisms.

We also conjecture that the reduced dg-quiver Schur algebra .red

dg Qx

b; 0/ is

dg-Morita equivalent to the non-reduced one .dgQx

b; 0/.

776 A. Lacabanne, G. Naisse, and P. Vaz

Appendices

A. Detailed proofs and computations

We give the detailed computations used to prove various results of the paper.

A.1. Proofs of §2

Lemma 2.4.The action of Brtranslates in terms of v-vectors of M˝Vras

: : : : : :

v.:::;bi1;bi;biC1;biC2;::: / 7! q1Œbiqv.:::;bi1CbiCbiC11;biC2;::: / ;(10)

: : : : : :

v7! qŒ2qv.:::;bi1;1;0;bi;::: / qv.:::;bi1C1;0;0;bi;::: / q v.:::;bi1;0;1;bi;::: / ;(11)

: : : W

v.b0;b1;::: / 7! .1qb0qŒb0q/v.0;b0Cb1;::: / Cq2Œb0qv.1;b0Cb11;::: / :(12)

Proof. We start with the cap. We have

v.:::;bi1;bi;biC1;biC2;:::/

DŒbiqv.:::;bi1Cbi1;1;biC1;biC2;:::/ Œbi1qv.:::;bi1Cbi;0;biC1;biC2;:::/;

and we easily check that

v.:::;bi1Cbi1;1;biC1;biC2;:::/ 7! q1v.:::;bi1CbiCbiC11;biC2;:::/;

v.:::;bi1Cbi;0;biC1;biC2;:::/ 7! 0:

We now turn to the cup. It suﬃces to do the computation for iDrC1because

of the recursive deﬁnition of v. By deﬁnition,

v.b0;:::;bn/7! qv.b0;:::;bn/˝v1;0 ˝F v1;0 Cv.b0;:::;bn/˝F v1;0 ˝v1;0 :

Since

v.b0;:::;bn/˝v1;0 ˝F v1;0

Dv.b0;:::;bn;0;1/ q2v.b0;:::;bnC1;0;0/ qv.b0;:::;bn/˝F v1;0 ˝v1;0 ;

Tensor product categoriﬁcations and the blob 2-category 777

and

v.b0;:::;bn/˝F v1;0 ˝v1;0 Dv.b0;:::;bn;1;0/ qv.b0;:::;bnC1;0;0/ ;

one ﬁnds the expected formula.

Finally, we ﬁnish with . Using the fact that is a morphism of Uq.sl2/-mod-

ules, it suﬃces to consider the case of the vector v.b0;b1/. One may check that

v.b0;b1/DŒb0qv.1;b0Cb11/ Œb01qv.0;b0Cb1/

and therefore

.v.b0;b1//DŒb0qFb0Cb11.v.1;0//Œb01qFb0Cb1.v.0;0//:

Using the deﬁnition of we have

.v.0;0//D1v.0;0/ ;

and

.v.1;0//Dq2v.1;0/ q. 1/v.0;1/:

Hence we deduce that

.v.b0;b1//

Dq2Œb0qv.1;b0Cb11/ .q. 1/Œb0qC1Œb01q/v.0;b0Cb1/:

We conclude by checking that 1qŒb0q1Œb01qD1qb0.

A.2. Proofs of §5

Lemma 5.7.The map kWpXk!Xkis surjective.

Proof. First, we recall the following well-known relation

D  (43)

which follows easily from (18) and (19). We also observe



(21)



(33)



:(44)

778 A. Lacabanne, G. Naisse, and P. Vaz

Then, we compute



(43)







and



(22)



(26)

D 



Thus, using (44) we obtain



Consequently, using Theorem 5.5, we deduce that Xkis generated as left .T ;r

b; 0/-

module by the elements

:::



˝N

1`; ;

: : : : : :



˝N

1`; ;

for all 0tk1. In particular, kis surjective. 

Lemma A.1. Suppose rD1and `D0. As a ZZ2-graded k-module, X1k;0

admits a decomposition

X1k;0 Š1q2k .T ;0

˚M

0t<k

p0

.q2pC12.kt / .X1k1;0 /˚2q2pC12.kCt/.X1k1;0/Œ1/:

Tensor product categoriﬁcations and the blob 2-category 779

Proof. It follows from Theorem 5.5 that we have a decomposition



: : :

Š: : :



˚M

0t<k

p0

@

:::

Xk1

˚: : :

: : :



Xk11

concluding the proof. 

Lemma 5.8.The sequence

0! Y1

! Y0

k

! Xk! 0;

is a short exact sequence of left .T ;r ; 0/-modules.

Proof. Since we already have a complex with an injection and a surjection, it is

enough to show that

gdim XkDgdim Y0

kgdim Y1

where gdim is the graded dimension in the form of a Laurent series in NJh˙1; ˙1;

q˙1K. We will show this by induction on k. When kD0, this is immediate.

Suppose it is true for k, and we will show it for kC1.

Let

Œˇ Cth

qWD 1qtChqt

q1qDq1qtChqt

1q2:

Note that

Œk C1qŒˇ CtCkh

rD0

Œˇ Ct2r h

q;(45)

Œk C1qDqŒkqCqk;(46)

Œˇ kC1h

qDq1Œˇ kh

qhqk;(47)

780 A. Lacabanne, G. Naisse, and P. Vaz

and

Œk C1qŒˇ kC1h

qDŒkqŒˇ kh

qCq1kŒˇ kh

qhqkŒk C1q:(48)

We ﬁrst restrict to the case `D0and rD1. By Lemma A.1 using (45),

followed by the induction hypothesis, we have

gdim X1kC1;0

D1q2.kC1/ gdim T;0

kC1Cq2kŒk C1qŒˇ kh

qgdim X1k;0

D1q2.kC1/ gdim T;0

kC1Cq2kŒk C1qŒˇ kh

..1qkChqŒkq/gdim T;1

k10;k hq2Œkqgdim T;1

k11;k1/:

By deﬁnition, we have

gdim Y0

kC1;0 D.1qkC1ChqŒk C1q/gdim T;1

kC110;kC1;

gdim Y1

kC1;0 Dhq2Œk C1qgdim T;1

kC111;k :

By Corollary 3.12,

gdim T;1

kC110;kC1DqkC1gdim T;0

kC1Cq2kŒk C1qŒˇ C1kh

qgdim T;1

k10;k ;

gdim T;1

kC111;k Dqkgdim T;0

kC1Cq2kŒˇh

qgdim T;1

k10;k

Cq2kŒkqŒˇ kh

qgdim T;1

k11;k1:

We now gather by gdim T;0

kC1, gdim T;1

k10;k and gdim T;1

k11;k1. For gdim T;0

kC1,

we verify that

1q2.kC1/ D.1qkC1CqŒk C1q/qkC1q2Œk C1qqk:

Gathering by gdim T ;1

k10;k , we obtain on one hand

q2kŒk C1qŒˇ kh

q.1qkChqŒkq/

DqkŒk C1qŒˇ kh

qCh2q12kŒkqŒk C1qŒˇ kh

q;(49)

and on the other hand

.1qkC1ChqŒk C1q/q2kŒk C1qŒˇ C1kh

qhq2Œk C1qq2kŒˇh

Dq1kŒk C1qŒˇ kC1h

qCh2q12k Œk C1qŒk C1qŒˇ kC1h

h2q22k Œk C1qŒˇh

Tensor product categoriﬁcations and the blob 2-category 781

DqkŒˇ kh

qŒk C1qhq12kŒk C1q

Ch2q12k Œk C1q.ŒkqŒˇ kh

qCq1kŒˇ kh

qhqkŒk C1q/

h2q22k Œk C1qŒˇh

using (47) and (48). We remark that the ﬁrst and third terms coincide with (49).

We gather the remaining terms, putting hq2k Œk C1qin evidence, so that we

obtain

qCqkŒˇ kh

qh2q1kŒk C1qq2Œˇh

q1q.q.q1q/ Cqk.1qkChqk/

h2q1k.qk1qkC1/q2.1Ch// D0:

Finally, for gdim T;1

k11;k1, we verify that

q2kŒk C1qŒˇ kh

q.hq2Œkq/D hq2Œk C1qq2k ŒkqŒˇ kh

concluding the proof in the case `D0and rD1.

The case ` > 0 comes from an induction on `and using the case `D0. Using

a similar decomposition as in Lemma A.1, we obtain

gdim Xk;` D1q2kC`gdim T;0

kC`Cq2`2k2ŒkqŒˇ kC1h

qgdim Xk1;`

Cq2k2` Œ`qŒˇ 2k `h

qgdim Xk;`1;

where Xk;` WD Xk1k;`. Similarly, one can compute

gdim T;1

kC`10;kC`DqkC`gdim T;0

kC`

Cq2k2` Œk C`qŒˇ k`h

qgdim T;1

kC`110;kC`1;

gdim T;1

kC`11;kC`1

DqkC`1gdim T;0

kC`Cq22.kC`/ Œˇh

qgdim T;1

kC`110;kC`1

Cq2k2` Œk C`1qŒˇ k`1h

qgdim T;1

kC`111;kC`2:

By the same reasons as above, the part in gdim T ;0

kC`annihilates each others. By

induction hypothesis we know that

gdim Xk;`1D.1qkChqŒkq/gdim T;1

kC`110;kC`1

hq2Œkqgdim T;1

kC`111;kC`2:

782 A. Lacabanne, G. Naisse, and P. Vaz

Using the fact that

Œk C`qŒˇ k`h

qDŒkqŒˇ kh

qCŒ`qŒˇ 2k `h

Œk C`1qŒˇ k`1h

qDŒk 1qŒˇ kC1h

qCŒ`qŒˇ 2k `h

together with the induction hypothesis, we cancel the part given by gdim.Xk;`1/

in Xk;` with the part given by

q2k2` Œ`qŒˇ 2k `h

qgdim T;1

kC`110;kC`1

in Y0

k;` minus the part given by

q2k2` Œ`qŒˇ 2k `h

qgdim T;1

kC`111;kC`2:

in Y1

k;`.

The remaining terms yields the same computations as for the case `D0

(replacing kC1by k), but shifting everything by q2`. Thus, it concludes the

case ` > 0.

The general case follows from a similar argument, using the fact that Xde-

composes similarly to T;r

bwhenever r > 1, that is as in Corollary 3.12, replacing

all Tby X. We leave the details to the reader. 

A.3. Proofs of §6

Lemma 6.12.As a right .T ;r ; 0/-module, 11;kC`1;Xis generated by the ele-

ments



:::

k1

˝N

1`; ;and



˝N

1`Ck1;:(37)

Proof. We prove this claim using an induction on k. The case kD1is obvious.

We suppose it is true for k1, and thus it is enough to show that we can generate

the element:



:::

k2

˝N

1`; :

Using (19), we have



: : :

k2



:::

k2





: : :

k1

Tensor product categoriﬁcations and the blob 2-category 783

The second term on the right-hand side is generated by the second element in (37).

For the ﬁrst term of the right-hand side, we slide the dot to the left using repeat-

edly (19):



: : :

k2



: : :

k1



k2

jD1



: : : : : :

Because of the symmetric of (44), the ﬁrst term on the right-hand side is generated

by the second element in (37). We now prove that every element of the sum on

the right-hand side is generated by elements in (37).

By applying the induction hypothesis, it suﬃces to show that for every 1

jk2, the elements



: : :

and



::: :::

are in the right module generated be the elements in (37), which is clear for the

ﬁrst diagram. Concerning the second one, we have by (19)



: : : : : :



: : : : : :





: : : : : :

For the ﬁrst term of the right-hand side, we again slide the dot to the left using (19)

and obtain



::: :::



: : : : : :



kj3

lD0



: : : : : : : : :

(18)



: : : : : :



kj3

lD0



: : : : : : : : :

Another application of the symmetry of (44) deals with the ﬁrst term, and every

term of the sum is handled trough a descending induction on j, noting that the

sum is zero if jDk2.

784 A. Lacabanne, G. Naisse, and P. Vaz

For the second term, we apply once again (18) and obtain



::: :::



: : : : : :

which has the desired form. 

A.3.1. Acyclicity of Cone.'/

Theorem 6.7.The map

'WD

kD0

.1/k'kWCone.q2XŒ1 u

! q2T;r

bŒ1/Œ1

! Cone.X ˝L

TX1˝u

! 1X/;

is a quasi-isomorphism.

The goal of this section is to prove Theorem 6.7, which we will achieve by

showing that Cone.'k/is acyclic. We have that Cone.'k/is given by the complex

X˝TY1

q2.Xk/Œ1 X ˝TY0

k1Xk

L`; q2.T ;r

b1k;`; /Œ1

1˝{k

u



u˝k

The map '1

kuis injective since uis injective by Corollary 5.6, and the map

u˝kis surjective. We want to ﬁrst show that '0

kC1˝{kis surjective on the

kernel of u˝k. This requires some preparation.

Lemma A.2. For k2, the local relation

: : :

1k2



k2

sD0

.1/s: : :

: : :

D.1/k1

: : :

k2

(50)

holds in T;r .

Tensor product categoriﬁcations and the blob 2-category 785

Proof. We prove the statement by induction on k2. If k2D0, then the claim

follows from (23). Suppose by induction that (50) holds for k3. We compute

:::

1k2

(23)

D: : :

1k2

: : :

1k2

(51)

and

: : :

1k2

(19), (18)

D:::

1k3

;(52)

:::

1k2

(21)

D:::

1k2

(23)

D: : :

1k2

(21)

D: : :

1k2

:(53)

Applying the induction hypothesis on (52), and inserting the result together

with (53) in (51) gives (50). 

Lemma A.3. We have

: : :

ztC2D::: (54)

for all t0.

Proof. We prove the statement by induction on t. The claim is clearly true for

tD0. Suppose it is true for t. We compute

:::

ztC3(34)

: : :

ztC2C

: : :

ztC2

786 A. Lacabanne, G. Naisse, and P. Vaz

(54)

D::: C: : :

(18), (19)

D: : :

concluding the proof. 

Lemma A.4. We have

: : :

ztC2D0;

for all t0.

Proof. It is a direct consequence of Lemma A.3 together with (18). 

Lemma A.5. We have

@

:::

k1

˝N

1`;



k2

sD0

.1/s.1 ˝{k1

:::



:::

˝N

1`;

D.1/k10





: : :

k1

˝N

1`; ;



˝N

1`Ck1;1

2.X ˝TY00

k/˚.X ˝TY0;k1

k/:

(55)

Proof. The case kD1is clear, thus we assume k > 1. First, let us write ?and ?s

for the inputs of '0

kand of 1˝{k1

kin (55), respectively. Then, on one hand, we

note that '0;t0

k.?/ D0by Lemma A.4 whenever t0¤k1, because of (18). For

Tensor product categoriﬁcations and the blob 2-category 787

t0Dk1, we obtain

'0;k1

k.?/ D:::

1k1



˝N

1`; (56)

using (18) and (19). Similarly, using Lemma A.4, we get

0.?/ D:::

1k1



˝N

1`; :(57)

On the other hand, we compute

.1 ˝{k1

k/.?s/

D.1/s

:::



:::

˝N

1`; ;: : :



: : :

˝N

1`;

D.1/s

:::



:::

˝N

1`; ;: : :



: : :

˝N

1`;

(58)

using (21) and (33). Then, we conclude by observing that (55) follows by applying

Lemma A.2 on (56), (57), and (58) together. 

788 A. Lacabanne, G. Naisse, and P. Vaz

Lemma A.6. As a left .T ;r ; 0/-module, ker.u ˝k/is generated by the elements





: : :

˝N

1`Ck1t; ;



˝N

1`Ck1;1

2.X ˝TY00

k/˚.X ˝TY0;t

(59)

for all 0tk1.

Proof. Let KX˝TY0

kbe the submodule generated by the elements in (59).

A straightforward computation shows that Kker.u ˝k/. Thus, we have a

complex

0! K ,! X˝TY0

u˝k

!! 1Xk! 0; (60)

where the left arrow is an injection and the right arrow is a surjection. Further-

more, by Theorem 5.5, we have that

KŠ1Y1

k; X ˝TY0

kŠ1Y0

Therefore, by Lemma 5.8 we obtain that the sequence in (60) is exact. In particular,

we have KDker.u ˝k/.

Proposition A.7. We have ker.u ˝k/Dim.'0

kC1˝{k/.

Proof. We will show by backward induction on tthat the elements (59) are all

in im.'0

kC1˝{k/. The case tDk1is Lemma A.5. The induction step is

essentially similar to the proof of Lemma A.5. In particular, we want to show

that (59) is in St0tim.'0

kC1˝{t0

k/. For this, we write

?WD



:::

˝N

1`; :

Tensor product categoriﬁcations and the blob 2-category 789

Then, we compute using Lemma A.4 and Lemma A.3

'0;t0

k.?/ D

0if t0< t,

: : :

1k1



˝N

1`; if t0Dt,

: : :

zkt0



˝N

1`; if t0> t ,

for 0t0k1, and

'00

k.?/ D  : : : : : :



˝N

1`; 

k1

t0DtC1

:::

zkt0



˝N

1`;:

Thus, since t0> t , by induction hypothesis we know that

k.?/ 

: : : : : :



˝N

1`; ;: : :

1k1



˝N

1`;

Cim.'0

kC1˝{k/2.X ˝TY0

0/˚.X ˝TY0;t

k/:

Then, by the same arguments as in Lemma A.5, that is using (50), we obtain (59).



790 A. Lacabanne, G. Naisse, and P. Vaz

Lemma A.8. The map '0

kis injective.

Proof. Since adding black/red crossings is injective, it is enough because of

Lemma A.3 to show that the left T;0

k-module map

T;0

k!

k1

tD0

q2.k1t / T;0

::: 7! 0

@

: : :

: : : 1

A0t<k

;

(61)

is injective. Since T;0

kis isomorphic to the dg-enhanced nilHecke algebra of [37],

we know by the results in [37, Proposition 2.5] that there is a decomposition

T;0

kŠM

t00

Pk;t 0; Pk;t 0WD M

p0



:::

NHk1k

;(62)

where the box labeled NHk1is the nilHecke algebra, and the circle labeled kis the

algebra generated by labeled ﬂoating dots in the rightmost region (see [37, §2.4]).

These ﬂoating dots correspond to combinations of nails, dots and crossings, giving

elements that are in the (graded with respect to the homological degree) center

of T;0

Furthermore, the map

NHk1! q2k2NHk;

: : :

NHk17!

: : :

NHk1

;

is injective (this can be deduced by sliding all dots to the bottom using (19), and

then using a basis theorem as for example in [21, Theorem 2.5] to see the map

takes the form of a column echelon matrix with 1as pivots).

Tensor product categoriﬁcations and the blob 2-category 791

Then, applying (61) on Pk;t 0yields



:::

NHk1k

7!

0if t < t0;

: : :

NHk1kif tDt0;

: : :

: : : : : :

NHk1

if t > t0:

Therefore, after decomposing T ;0

k, (61) yields a column echelon form matrix

with injective maps as pivots, and thus is injective. 

Proposition A.9. We have ker..1 ˝{k/C'0

k/Dim.'1

ku/.

Proof. First, recall that {kis injective (as explained in §5.4). Thus, both .1 ˝{k/

and '0

kare injective, and we get

ker..1 ˝{k/C'0

k/Šim.1 ˝{k/\im.'0

k/:

We observe that im.1 ˝{k/\im.'0

k/\.X ˝TY0;t

k/is generated by

'0;t

@

:::

˝N

1`; 1

D.1 ˝{t

k/0

@: : :

: : :

zkt



˝N

1`; 1

: : :

zkt



˝N

1`; :

and by

'0;t

@:::



˝N

1`; 1

D.1 ˝{t

k/0

@: : :

: : :

zkt



˝N

1`; 1

792 A. Lacabanne, G. Naisse, and P. Vaz

: : :

zkt



˝N

1`; :

Moreover, we have

'1;t

: : :



˝N

1`; 1

:::

zkt



˝N

1`; ;

: : :



˝N

1`; 1



:::

˝N

1`; :

The case with a nail is similar, concluding the proof. 

Proof of Theorem 6.7.Since '1

kuis injective, and u˝kis surjective, and by

Proposition A.7 and Proposition A.9, we conclude that Cone.'k/is acyclic for

all k. Consequently, 'is a quasi-isomorphism. 

A.3.2. The bimodule map Q'

Theorem 6.9.The map 'is a map of Z2-graded .T ;r ; 0/-.T ;r ; 0/-A1-bimod-

ules.

The goal of this section is to prove Theorem 6.9. To this end, we ﬁrst prove

that the map Q'0Wq2.T ;r

b/Œ1 !X˝TXis a map of bimodules.

Proposition A.10. We have

Q'00

@

:::

˝N

1`; 1

AD.1/k



: : :

Q'.k/

: : :

˝N

1`; ;

where

Q'.0/ WD

1

Q'.0/ WD 0;

Tensor product categoriﬁcations and the blob 2-category 793

Q'.1/ WD

1

Q'.1/ WD







;

Q'.t C2/ WD : : :

1

: : :

Q'.t C2/

: : :

WD : : :

1

: : :

Q'.t C1/

: : :

C: : :

1

: : :

Q'.t C1/

: : :

C: : :

1

: : :

Q'.t C1/

: : :

for all t0.

Proof. Recall that

Q'0WD .1 ˝/ ı'0:

Then, we obtain

.1 ˝/ ı'00

@

: : :

˝N

1`; 1

D.1/k

k1

tD0

::: :::

zkt



˝N

1`; ::: :::

zkt



˝N

1`; :

(63)

We prove the statement by induction on k. The claim is clearly true for kD0and

kD1. Suppose it is true for kC1, and we will show it is true for kC2.

794 A. Lacabanne, G. Naisse, and P. Vaz

By deﬁnition of Q'.k C2/ and using (19), we have

: : :

1

: : :

Q'.k C2/

: : :

D: : :

1

: : :

Q'.k C1/

: : :

C: : :

: : :

1

: : :

Q'.k C1/ :(64)

Applying the induction hypothesis on (64), we get

: : :

1

: : :

Q'.k C2/

: : :

k1

tD0

::: :::

zkC1t



C: : : : : :

zkC1t



C: : :



C: : :



(similar terms with the nail above).

Applying (34) on each pair of terms in the sum (including non-displayed terms)

gives the part for 0t < k 2in (63) for kC2. The last two terms (including

non-displayed terms) give tDkand tDkC1, since z2is a single crossing,

concluding the proof. 

Tensor product categoriﬁcations and the blob 2-category 795

Having Proposition A.10, proving Theorem 6.9 boils down to proving that the

left and right action by the same element of T ;r

bon

kC`CjjDb

.1/kQ'.k/ ˝N

1`;

coincide.

Lemma A.11. We have

: : :

1

: : :

Q'.t C1/

: : :

D  : : :

1

: : :

Q'.t /

: : :

D: : :

1

: : :

Q'.t C1/

: : :

for all t0.

Proof. We show the ﬁrst equality, and the second one follows by symmetry along

the horizontal axis of the deﬁnition of Q'.t C1/.

We prove the statement by induction on t. The case tD0follows from (44).

Suppose the claim is true for t0. We compute using (64)

: : :

1

: : :

Q'.t C2/

: : :

D: : :

1

: : :

Q'.t C1/

: : :

C: : :

1

: : :

Q'.t C1/

: : :

D  :::

1

:::

Q'.t C1/

:::

: : :

1

: : :

Q'.t /

: : :

:::

1

:::

Q'.t /

:::

where the last two terms annihilate each other, concluding the proof. 

Lemma A.12. We have

:::

1

:::

Q'.t C2/

:::

D: : :

1

: : :

Q'.t C1/

: : :

:::

1

:::

Q'.t /

:::

D:::

1

:::

Q'.t C2/

:::

for all t0.

796 A. Lacabanne, G. Naisse, and P. Vaz

Proof. By (64), we have

: : :

1

: : :

Q'.t C2/

: : :

D: : :

1

: : :

Q'.k C1/

: : :

C: : :

1

: : :

Q'.k C1/

: : :

We conclude by applying Lemma A.11.

Lemma A.13. We have

: : :

1

: : :

Q'.t C2/

: : :

D: : :

1

: : :

Q'.t C1/

: : :

D: : :

1

: : :

Q'.t C2/

: : :

for all t0.

Proof. This is immediate by applying (18) on the deﬁnition of Q'.t C2/.

Lemma A.14. We have

:::

1

:::

Q'.t C3/

:::

D: : :

1

: : :

Q'.t C3/

: : :

for all t0.

Proof. By (64) we have

: : :

1

: : :

Q'.t C3/

: : :

D: : :

1

: : :

Q'.t C2/

: : :

C:::

1

:::

Q'.t C2/

:::

Tensor product categoriﬁcations and the blob 2-category 797

Then, we compute

: : :

1

: : :

Q'.t C2/

: : :

D: : :

1

: : :

Q'.t C1/

: : :

C: : :

1

: : :

Q'.t C1/

: : :

C:::

1

:::

Q'.t C1/

:::

and

: : :

1

: : :

Q'.t C2/

: : :

D: : :

1

: : :

Q'.t C1/

: : : C: : :

1

: : :

Q'.t C1/

: : :

C: : :

1

: : :

Q'.t C1/

: : : :

Furthermore, we compute mainly using (18) and (19),

:::

1

:::

Q'.t C1/

::: D  : : :

1

: : :

Q'.t C1/

: : :

and

:::

1

:::

Q'.t C1/

::: D: : :

1

: : :

Q'.t C1/

: : :

798 A. Lacabanne, G. Naisse, and P. Vaz

In conclusion, we get

: : :

1

: : :

Q'.t C3/

: : :

D:::

1

:::

Q'.t C1/

:::

C:::

1

:::

Q'.t C1/

:::

C: : :

1

: : :

Q'.t C1/

: : :

C: : :

1

: : :

Q'.t C1/

: : :

which is symmetric with respect to taking the mirror image along the horizontal

axis. Therefore, we get the same a crossing at the bottom of Q'.t C3/, ﬁnishing

the proof. 

Lemma A.15. We have

: : :

1

: : :

Q'.t C1/

: : :

D  : : :

1

: : :

Q'.t /

: : :

for all t0.

Proof. We prove the statement by induction on t. The case tD0follows

from (33). We suppose the claim is true for t0. We compute using the mirror

of (64),

: : :

1

: : :

Q'.t C2/

: : :

D:::

1

:::

Q'.t C1/

::: C: : :

1

: : :

Q'.t C1/

: : :

Tensor product categoriﬁcations and the blob 2-category 799

D:::

1

:::

Q'.t C1/

::: C: : :

1

: : :

Q'.t C1/

: : :

: : :

1

: : :

Q'.t C1/

: : : :

Then, we have

:::

1

:::

Q'.t C1/

::: D  :::

1

:::

Q'.t /

:::

D0;

by induction hypothesis. Finally, we obtain

: : :

1

: : :

Q'.t C1/

: : : :::

1

:::

Q'.t C1/

::: D  : : :

1

: : :

Q'.t C1/

: : :

ﬁnishing the proof. 

Proposition A.16. The map Q'0is a map of dg-bimodules.

Proof. As already mentioned above, it is enough to show that the left and right

action by the same element of T;r

bon

kC`CjjDb

.1/kQ'.k/ ˝N

1`;

coincide. We obtain commutation with dots and crossings by induction on k,

using Lemmas A.11–A.15. The commutation with a nail also comes from a

straightforward induction on k, where the base case is immediate by (26). 

800 A. Lacabanne, G. Naisse, and P. Vaz

B. Homological toolbox

The goal of this section is to recall and brieﬂy explain the tools from homological

algebra which we use in this paper. The main references for this section are [15],

[46], and [33] (see also [47], [16] and [34, Appendix A]).

B.1. Derived category. Let .A; dA/be a Zn-graded dg-algebra (with the same

conventions as in §3.1).

The derived category D.A; dA/of .A; dA/is the localization of the category

.A; dA/-mod of Zn-graded (left) .A; dA/-dg-modules along quasi-isomorphisms.

It is a triangulated category with translation functor induced by the homological

shift functor Œ1, and distinguished triangles are equivalent to

.M; dN/f

! .N; dN/{N

! Cone.f / M Œ1

! .M; dN/Œ1;

for every maps of dg-modules fW.M; dM/!.N; dN/.

B.1.1. (Co)ﬁbrant replacements. Acoﬁbrant dg-module .P; dP/is a dg-mod-

ule such that Pis projective as graded A-module. Equivalently, it is a dg-module

.P; dP/such that for every surjective quasi-isomorphism .L; dL/'

!! .M; dM/,

every morphism .P; dP/!.M; dM/factors through .L; dL/. For any dg-module

.N; dN/, we have

HomD.A;dA/..P; dP/; .N; dN// ŠH0

0.HOM.A;dA/..P; dP/; .N; dN///:

Moreover, tensoring with a coﬁbrant dg-module preserves quasi-isomorphisms.

Given a left (resp. right) dg-module .M; dM/, there exists a coﬁbrant replace-

ment .pM; dpM/(resp. .M q; dMq/) together with a surjective quasi-isomorphism

MWpM'

M(resp. 0

MWMq'

M).

Moreover, the assignment M7! pM(resp. M7! Mq) is natural. Thus, we can

compute HomD.A;dA/..M; dM/; .N; dN// by taking

0.HOM.A;dA/..pM; dpM/; .N; dN/// ŠHomD.A;dA/..M; dM/; .N; dN//:

A dg-module .I; dI/is ﬁbrant if for every injective quasi-isomorphism

.L; dL/'

,! .M; dM/;

every morphism .L; dL/!.M; dM/extends to .M; dM/. Then,

HomD.A;dA/..M; dM/; .I; dI// ŠH0

0.HOM.A;dA/..M; dM/; .I; dI///:

Tensor product categoriﬁcations and the blob 2-category 801

Again, for every dg-module .M; dM/there exists a ﬁbrant replacement .iM; diM/

with an injective quasi-isomorphism {MW.M; dM/'

,! .iM; diM/.

B.1.2. Strongly projective modules. Let Rbe a unital commutative ring. The

following was introduced in [32], but we use the deﬁnition given in [5].

Deﬁnition B.1 ([5, Deﬁnition 8.17]). A dg-module .P; dP/over a dg-R-algebra

.A; dA/is strongly projective if it is a direct summand of some dg-module

.A; dA/˝R.Q; dQ/where .Q; dQ/is a .R; 0/-dg-module such that both H .Q; dQ/

and im.dQ/are projective R-modules.

Proposition B.2 ([5, Lemma 8.23]). Let .P; dP/be a strongly projective left

dg-module. For any right dg-module .M; dM/, we have an isomorphism

H..M; dM/˝.A;dA/.P; dP// ŠH.M; dM/˝H.A;dA/H.P; dP/:

B.1.3. A1-action. Let .B; dB/be a dg-bimodule over a pair of dg-algebras

.S; dS/-.R; dR/. As explained in [29, §2.3], there is (in general) no right

.R; dR/-action on pBicompatible with the left .S; dS/-action. However, there

is an induced A1-action (deﬁned uniquely up to homotopy), so that the quasi-

isomorphism BWpB'

!! Bcan be upgraded to a map of A1-bimodules.

Lemma B.3. Let .A; dA/be a dg-algebra, and let Uand Vbe dg-(bi)modules

over .A; dA/, with a ﬁxed coﬁbrant replacement VWpV!V. Suppose

.1 ˝pV/WU˝.A;dA/pV'

! U˝.A;dA/V

is a quasi-isomorphism. If fWZ!U˝.A;dA/pVis a map of complexes of graded

k-spaces, and fı.1 ˝pV/is a map of dg-(bi)modules, then there is an induced

map N

Z!U˝LVof A1-(bi)modules whose degree zero part is f.

Proof. We take N

fWD .1 ˝pV/1ı.f ı.1 ˝pV//, as a composition of maps

of A1-(bi)modules, since any map of (bi)module can be considered as a map of

A1-(bi)modules with no higher composition. 

Note that the equivalent statement also holds for a coﬁbrant replacement

Uq! U

such that

.U˝1/WUq˝.A;dA/V'

! U˝.A;dA/V

is a quasi-isomorphism.

802 A. Lacabanne, G. Naisse, and P. Vaz

B.2. Dg-derived categories. One of the issues with triangulated categories is

that the category of functors between triangulated categories is in general not

triangulated. To ﬁx this, we work with a dg-enhancement of the derived category.

In particular, this allows us to talk about distinguished triangles of dg-functors.

Recall that a dg-category is a category where the hom-spaces are dg-modules

over .k; 0/, and compositions are compatible with this structure (see [15, §1.2] for

a precise deﬁnition). Given such a dg-category Cwith hom-spaces HomC.X; Y / D

Lh2ZHomh.X; Y /; dX;Y , we can consider its underlying category Z0.C/,

which is given by the same objects as Cand hom-spaces

HomZ0.C/.X; Y / WD ker.dX;Y WHom0.X; Y / ! Hom1.X; Y //:

Similarly, the homotopy category H0.C/is given by

HomH0.C/.X; Y / WD H0.HomC.X; Y //:

Adg-enhancement of a category C0is a dg-category Csuch that H0.C/ŠC0.

The dg-derived category Ddg.A; dA/of a Zn-graded dg-algebra .A; dA/is the

Zn-graded dg-category with objects being coﬁbrant dg-modules over .A; dA/, and

hom-spaces being subspaces of the graded dg-spaces HOM.A;dA/from (16), given

by maps that preserve the Zn-grading:

HomDdg.A;dA/.M; N / WD HOM.A;dA/.M; N /0;

for .M; dM/and .N; dN/coﬁbrant dg-modules. By construction,

H0.Ddg.A; dA// ŠD.A; dA/:

Moreover, Ddg.A; dA/is a dg-triangulated category, meaning its homotopy cat-

egory is canonically triangulated (see [46] for a precise deﬁnition, or [34, Ap-

pendix A] for a summary oriented toward categoriﬁcation), and this triangulated

structure matches with the usual one on D.A; dA/.

B.2.1. Dg-functors. Adg-functor between dg-categories is a functor commut-

ing with the diﬀerentials. Given a dg-functor FWC!C0, it induces a functor on

the homotopy categories ŒF WH0.C/!H0.C0/. We say that a dg-functor is a

quasi-equivalence if it gives quasi-isomorphisms on the hom-spaces, and induces

an equivalence on the homotopy categories. We want to consider dg-category up

to quasi-equivalence. Let Hqe be the homotopy category of dg-categories up to

quasi-equivalence , and we write RHomHqe for the dg-space of quasi-functors be-

tween dg-categories (see [46], [47], or [34, Appendix A]). These quasi-functors

induce honest functors on the homotopy categories. Whenever C0is dg-triangu-

lated, then RHomHqe.C;C0/is dg-triangulated.

Tensor product categoriﬁcations and the blob 2-category 803

Remark B.4. The space of quasi-functors is equivalent to the space of strictly

unital A1-functors.

It is in general hard to understand the space of quasi-functors. However, by

the results of Toen [46], if kis a ﬁeld and .A; dA/and .A0; dA0/are dg-algebras,

then it is possible to compute the space of ‘coproduct preserving’ quasi-functors

RHomcop

Hqe.Ddg.A; dA/; Ddg.A0; dA0//, in the same way as the category of coprod-

uct preserving functors between categories of modules is equivalent to the cate-

gory of bimodules. Indeed, we have a quasi-equivalence

RHomcop

Hqe.Ddg.A; dA/; Ddg.A0; dA0// ŠDdg ..A0; dA0/; .A; dA//; (65)

where Ddg..A0; dA0/; .A; dA// is the dg-derived category of dg-bimodules. Com-

position of functors is equivalent to derived tensor product, and understanding the

triangulated structure of RHomcop

Hqe.Ddg.A; dA/; Ddg.A0; dA0// becomes as easy as

to understand D..A; dA/; .A0; dA0//. In particular, a short exact sequence of dg-bi-

modules gives a distinguished triangle of dg-functors.

B.3. Derived hom and tensor dg-functors. Let .R; dR/and .S ; dS/be dg-

algebras. Let Mand Nbe .R; dR/-module and .S ; dS/-module respectively. Let

Bbe a dg-bimodule over .S; dS/-.R; dR/. Then, the derived tensor product is

B˝L

.R;dR/MWD B˝pM;

and the derived hom space is

RHOM.S;dS/.B ; N / WD HOM.S ;dS/.B; iN /:

Note that we have quasi-isomorphisms as dg-spaces

B˝L

.R;dR/MŠBq˝.R;dR/pMŠBq˝.R ;dR/M;

RHOM.S;dS/.B ; N / ŠHOM.S ;dS/.pB; iN / ŠHOM.S ;dS/.pB; N /:

This deﬁnes in turns triangulated dg-functors

B˝L

.R;dR/./WDdg.R; dR/! Ddg.S; dS/;

and

RHOM.S;dS/.B; /WDdg.S ; dS/! Ddg.R; dR/:

They induce a pair of adjoint functors B˝L

.R;dR/./`RHOM.S;dS/.B; /

between the derived categories Ddg.R ; dR/and Ddg.S ; dS/.

804 A. Lacabanne, G. Naisse, and P. Vaz

B.3.1. Computing units and counits. The natural bijection

ˆB

M;N WHomD.S;dS/.B ˝L

.R;dR/M; N / '

! HomD.R;dR/.M; RHOM.S ;dS/.B ; N //;

is obtained by making the following diagram commutative:

HomD.S;dS/.B ˝L

.R;dR/M; N / HomD.R;dR/.M; RHOM.S;dS/.B ; N //

Hom.S;dS/.B ˝.R;dR/pM; iN / Hom.R;dR/.pM; HOM.S ;dS/.B; iN //

ˆB

M;N

ˆB

pM;iN

where ˆis deﬁned in (17).

For the sake of keeping notations short, we will write HOM instead of

HOM.S;dS/, and ˝instead of ˝.R;dR/, and similarly for the derived versions.

We are interested in computing the unit

MWM! RHOM.B ; B ˝LM /;

which is given by MDN

ˆB

M;B˝LM.IdB˝LM/. Composing with the isomorphisms

RHOM.B; B ˝LM / ŠHOM.B; i.B ˝pM // and pMŠM, we can compute

Mas

0

MDˆB

pM;i.B˝LM /.{B˝LM/WpM! HOM.B; i.B ˝pM //;

which gives

0

M.m/ D.b 7! {B˝pM.b ˝m//:

Using the quasi-isomorphisms

HOM.pB; i.B ˝LM //

HOM.pB; B ˝LM / HOM.B ; i.B ˝LM/ /

{B˝LMı

ıB

we can compute 0

Mthrough

00

MWpM! HOM.pB; B ˝pM /; 00

M.m/ WD .b 7! B.b/ ˝m/:

This is particularly useful, since it means we do not have to compute any ﬁbrant

replacement to understand M.

Tensor product categoriﬁcations and the blob 2-category 805

Similarly, for the counit

"MWB˝LRHOM.B; M / ! M;

we have "MD.N

ˆB

RHOM.B;M /;M /1.IdRHOM.B ;M //. We rewrite it as

MDˆ1

pHOM.B;iM /;iM.HOM.B;iM / /WB˝pHOM.B; iM / ! iM;

with "0

M.b ˝f / D.HOM.B;iM / .f //.b/. We consider the quasi-isomorphisms

Bq˝pHOM.B; iM /

B˝pHOM.B; iM / B q˝HOM.B; iM /

B˝1

1˝HOM.B;iM /

Therefore, we can compute "Mas

"00

MWBq˝HOM.B; iM / ! iM; "00

M.b ˝f / WD f . 0

B.b//;

where 0

BWBq'

!B.

If in addition Bis already coﬁbrant as left dg-module, then we can suppose

pBDBand BDIdB, and we obtain a commutative diagram

Bq˝HOM.B; iM / iM

Bq˝HOM.pB; iM / iM

Bq˝HOM.pB; M / M

"00

1˝.ıB/

(

"000

1˝.{Mı/

where

"000

MWBq˝HOM.B; M / ! M; "000

M.b ˝f / WD f . 0.b//:

This is useful, since it means we can compute "Musing "000

M, which does not require

any ﬁbrant replacement.

B.4. Asymptotic Grothendieck group. The usual deﬁnition of the Grothendieck

group of a triangulated category does not take into consideration relations com-

ing from inﬁnite iterated extensions. When Cis a triangulated subcategory

of a triangulated category Tadmitting countable products and coproducts, and

these preserves distinguished triangles, then there exists a notion of asymptotic

Grothendieck group K

0.C/of C, given by modding out relations obtained from

Milnor (co)limits (see §B.4.3 below) in the usual Grothendieck group K0.C/

(see [33, §8] for a precise deﬁnition).

806 A. Lacabanne, G. Naisse, and P. Vaz

B.4.1. Ring of Laurent series. We follow the construction of the ring of formal

Laurent series given in [2] (see also [33, §5]). The ring of formal Laurent series

k..x1; : : : ; xn// is given by ﬁrst choosing a total additive order on Zn. One says

that a cone CWD ¹˛1v1C    C ˛nvnW˛i2R0º  Rnis compatible with 

whenever 0vifor all i2 ¹1; : : : ; nº. Then, we set

k..x1; : : : ; xn// WD [

e2Zn

xekJx1; : : : ; xnK;

where kJx1; : : : ; xnKconsists of formal Laurent series in kJx1; : : : ; xnKsuch that

the terms are contained in a cone compatible with . It forms a ring when we

equip k..x1; : : : ; xn// with the usual addition and multiplication of series.

B.4.2. C.b.l.f. structures. We ﬁx an arbitrary additive total order on Zn. We

say that a Zn-graded k-vector space MDLg2ZnMgis c.b.l.f. (cone bounded,

locally ﬁnite) dimensional if

dim Mg<1for all g2Zn;

there exists a cone CMRncompatible with and e2Znsuch that

MgD0whenever ge…CM.

Let .A; dA/be a Zn-graded dg-algebra. Suppose that .A; d / is concentrated in

non-negative homological degrees, that is Ah

gD0whenever h < 0. The c.b.l.f.

derived category Dcblf .A; dA/of .A; dA/is the triangulated full subcategory of

D.A; dA/given by dg-modules having homology being c.b.l.f. dimensional for

the Zn-grading. There exists also a dg-enhanced version Dcblf

dg .A; dA/. We write

K

0.A; d / WD K

0.Dcblf.A; dA//:

Deﬁnition B.5. We say that .A; d / is a positive c.b.l.f. dg-algebra if

(1) Ais c.b.l.f. dimensional for the Zn-grading;

(2) Ais non-negative for the homological grading;

(3) A0

0is semi-simple;

(4) Ah

0D0for h > 0;

(5) .A; dA/decomposes a direct sum of shifted copies of modules PiWD Aeifor

some idempotent ei2A, such that Piis non-negative for the Zn-grading.

Tensor product categoriﬁcations and the blob 2-category 807

In a Zn-graded triangulated category C, we deﬁne the notion of c.b.l.f. direct

sum as follows:

take a ﬁnite collection of objects ¹K1; : : : ; Kmºin C;

consider a direct sum of the form

g2Zn

xg.K1;g˚    ˚ Km;g/; with Ki;gD

ki;g

jD1

KiŒhi;j;g;

where ki;g2Nand hi;j;g2Zsuch that

there exists a cone Ccompatible with , and e2Znsuch that for all jwe

have kj;gD0whenever ge…C;

there exists h2Zsuch that hi;j;ghfor all i; j; g.

If Cadmits arbitrary c.b.l.f. direct sums, then K

0.C/has a natural structure of

Z..x1; : : : ; xn//-module with

g2C

agxeCgŒX WD hM

g2C

xgCeX˚agi;

where X˚agDLjagj

`D1XŒ˛gand ˛gD0if ag0and ˛gD1if ag< 0.

Theorem B.6 ([33, Theorem 9.15]). Let .A; d / be a positive c.b.l.f. dg-algebra,

and let ¹Pjºj2Jbe a complete set of indecomposable coﬁbrant .A; d /-modules

that are pairwise non-isomorphic (even up to degree shift). Let ¹Sjºj2Jbe the set

of corresponding simple modules. There is an isomorphism

K

0.A; d / ŠM

j2J

Z..x1; : : : ; x`//ŒPj;

and K

0.A; d / is also freely generated by the classes of ¹ŒSjºj2J.

Proposition B.7 ([33, Proposition 9.18]). Let .A; d / and .A0; d 0/be two c.b.l.f.

positive dg-algebras. Let Bbe a c.b.l.f. dimensional .A0; d 0/-.A; d /-bimodule.

The derived tensor product functor

FWDcblf.A; d / ! Dcblf.A0; d 0/; F .X / WD B˝L

.A;d/ X;

induces a map

ŒF WK

0.A; d / ! K

0.A0; d 0/;

sending ŒX to ŒF .X /.

808 A. Lacabanne, G. Naisse, and P. Vaz

B.4.3. C.b.l.f. iterated extensions. Using the terminology of [17], recall that

the Milnor colimit MColimr0.fr/of a collection of arrows ¹Xr

! XrC1ºr2N

in a triangulated category Tis the mapping cone ﬁtting inside the following

distinguished triangle

r2N

1f

! a

r2N

Xr! MColimr0.fr/!

where the left arrow is given by the inﬁnite matrix

1fWD 0

1 0 0 0 

f01 0 0 

0f11 0 

:::::::::::::

Deﬁnition B.8. Let ¹K1; : : : ; Kmºbe a ﬁnite collection of objects in C, and let

¹Erºr2Nbe a family of direct sums of ¹K1; : : : ; Kmºsuch that Lr2NEris a c.b.l.f.

direct sum of ¹K1; : : : ; Kmº. Let ¹Mrºr2Nbe a collection of objects in Cwith

M0D0, such that they ﬁt in distinguished triangles

! MrC1! Er!

Then, we say that an object M2Csuch that MŠTMColimr0.fr/in Tis a

c.b.l.f. iterated extension of ¹K1; : : : ; Kmº.

Note that under the conditions above, we have

ŒM  DX

r0

ŒEr;

in the asymptotic Grothendieck group K0.C/.

Deﬁnition B.9. Let Tbe a Zn-graded (dg-)triangulated (dg-)category, and let

¹Xjºj2Jbe a collection of objects in T. The subcategory of Tc.b.l.f. generated

by ¹Xjºj2Jis the triangulated full subcategory CTgiven by all objects Y2T

such that there exists a ﬁnite subset ¹Xkºk2Ksuch that Yis isomorphic to a c.b.l.f.

iterated extension of ¹Xkºk2Kin T.

Thus, under the conditions above, K0.C/is generated as a Z..x1; : : : ; xn//-mod-

ule by the classes of ¹ŒXjºj2J.

Tensor product categoriﬁcations and the blob 2-category 809

B.4.4. Dg-functors. Let .R; dR/and .S; dS/be (Zn-graded) dg-algebras. The

situation of (65) in §B.2.1 restricts to the c.b.l.f. version Dcblf

dg of §B.4.2, so that

RHomcop

Hqe.Dcblf

dg .R; dR/; Dcblf

dg .S; dS// ŠDcblf

dg ..S; dS/; .R ; dR//:

Then, we obtain an induced map

K

0.RHomcop

Hqe.Dcblf

dg .R; dR/; Dcblf

dg .S; dS///r

! HomZ..x1;:::;xn// .K

0.R; dR/; K

0.S; dS//; (66)

by using Proposition B.7.

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2011, 243–302. MR 2762557 Zbl 1216.18013

[48] B. Webster, Tensor product algebras, grassmannians and khovanov homology. In

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mology. Papers from the summer school held as part of the Séminaire de Mathéma-

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treal, Montreal, QC, June 24–July 5, 2013. Contemporary Mathematics, 680. Centre

de Recherches Mathématiques Proceedings. American Mathematical Society, Provi-

dence, R.I., 2016, 23–58. MR 3591642 Zbl 1408.57017

[49] B. Webster, Knot invariants and higher representation theory. Mem. Amer. Math.

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Received June 9, 2020

Abel Lacabanne, Université Clermont Auvergne, CNRS, LMBP,

63000 Clermont-Ferrand, France

e-mail: abel.lacabanne@uca.fr

Grégoire Naisse, Max-Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn,

Germany

e-mail: gregoire.naisse@gmail.com

Pedro Vaz, Institut de Recherche en Mathématique et Physique,

Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve,

Belgium

e-mail: pedro.vaz@uclouvain.be

Categorification of infinite-dimensional

\mathfrak{sl}_2

-modules and braid group 2-actions I: tensor products

Preprint

Full-text available

Mar 2021

This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. \\ In this part, we categorify all tensor products of Verma modules and integrable modules for quantum

\mathfrak{sl_2}

. The categorification is given by derived categories of dg versions of KLRW algebras which generalize both the tensor product algebras of Webster, and the dg-algebras used by Lacabanne, the second author and Vaz. We compute a basis for these dgKLRW algebras by using rewriting methods modulo braid-like isotopy, which we develop in an Appendix.

Enhanced nilHecke algebras and baby Verma modules

Preprint

Full-text available

Aug 2023

We define a derivation on the enhanced nilHecke algebra yielding a p-dg algebra when working over a field of characteristic p. We define functors on the category of p-dg modules resulting in an action of small quantum sl2 on the Grothendieck group, which is isomorphic to a baby Verma module. We upgrade the derivation into an action of the Lie algebra sl(2).

U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

Preprint

Full-text available

Jul 2022

We introduce new

U_q\mathfrak{sl}_2

-invariant boundary conditions for the open XXZ spin chain. For generic values of q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for

q=e^{\frac{i\pi}{p}}

a 2p-th root of unity

-

to its p-dimensional analogue. Both cases are parametrised by a continuous "spin"

\alpha\in\mathbb{C}

. To motivate our construction, we first specialise to q=i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the

(\eta,\xi)

ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if

q^\alpha\notin\pm q^{\mathbb{Z}}

, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to

U_q\mathfrak{sl}_2

. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of q and

\alpha

. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

Schur-Weyl duality, Verma modules, and row quotients of Ariki-Koike algebras

Preprint

Full-text available

Apr 2020

We prove a Schur-Weyl duality between the quantum enveloping algebra of

\mathfrak{gl}_m

and certain quotient algebras of Ariki-Koike algebras, which we give explicitly. The duality involves several algebraically independent parameters and is realized through the tensor product of a parabolic universal Verma module and a tensor power of the natural representation of

\mathfrak{gl}_m

. We also give a new presentation by generators and relations of the generalized blob algebras of Martin and Woodcock as well as an interpretation in terms of Schur-Weyl duality by showing they occur as a particular case of our algebras.

Two-row Delta Springer varieties

Preprint

Jul 2024

We study the geometry and topology of

\Delta

-Springer varieties associated with two-row partitions. These varieties were introduced in recent work by Griffin-Levinson-Woo to give a geometric realization of a symmetric function appearing in the Delta conjecture by Haglund-Remmel-Wilson. We provide an explicit and combinatorial description of the irreducible components of the two-row

\Delta

-Springer variety and compare it to the ordinary two-row Springer fiber as well as Kato's exotic Springer fiber corresponding to a one-row bipartition. In addition to that, we extend the action of the symmetric group on the homology of the two-row

\Delta

-Springer variety to an action of a degenerate affine Hecke algebra and relate this action to a

\mathfrak{gl}_{2}

-tensor space.

Handlebody diagram algebras

Article

Full-text available

Jul 2022

In this paper, we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley–Lieb, blob, Brauer, BMW, Hecke and Ariki–Koike algebras. Moreover, motivated by Green–Kazhdan–Lusztig’s theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.

{U}_{\mathfrak{q}}{\mathfrak{sl}}_2 -invariant non-compact boundary conditions for the XXZ spin chain

Article

Full-text available

Nov 2022
J HIGH ENERGY PHYS

A bstract We introduce new

{U}_{\mathfrak{q}}{\mathfrak{sl}}_2

U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of

\mathfrak{q}

q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for

\mathfrak{q}={e}^{\frac{i\pi}{p}}

q = e iπ p a 2 p -th root of unity — to its p -dimensional analogue. Both cases are parametrised by a continuous “spin” α ∈ ℂ. To motivate our construction, we first specialise to

\mathfrak{q}

q = i , where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the ( η, ξ ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if

\mathfrak{q}

q α ∉ ±

\mathfrak{q}

q ℤ , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to

{U}_{\mathfrak{q}}{\mathfrak{sl}}_2

U q sl 2 . Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of

\mathfrak{q}

q and α . Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

Verma Howe duality and LKB representations

Preprint

Full-text available

Jul 2022

We establish a version of quantum Howe duality with two general linear quantum enveloping algebras that involves a tensor product of Verma modules. We prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for various subgroups of the braid group, including the pure braid group.

Handlebody diagram algebras

Preprint

Full-text available

May 2021

In this paper we study handlebody versions of classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer/BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.

Enhanced nilHecke algebras and baby Verma modules

Article

Mar 2025

HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

Article

Full-text available

Mar 2021

We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-sphere. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.

Braid group actions from categorical symmetric Howe duality on deformed Webster algebras

Article

Full-text available

Aug 2021
TRANSFORM GROUPS

We construct a 2-representation categorifying the symmetric Howe representation of

\mathfrak{gl}_m

using a deformation of an algebra introduced by Webster. As a consequence, we obtain a categorical braid group action taking values in a homotopy category.

Relative cellular algebras

Article

Full-text available

Mar 2021
TRANSFORM GROUPS

In this paper we generalize cellular algebras by allowing different partial orderings relative to fixed idempotents. For these relative cellular algebras we classify and construct simple modules, and we obtain other characterizations in analogy to cellular algebras. We also give several examples of algebras that are relative cellular, but not cellular: most prominently, the restricted enveloping algebra and the small quantum group for

\mathfrak{s}{\mathfrak{l}}_2

, and an annular version of arc algebras.

On 2-Verma modules for quantum $\mathfrak{sl}_2

Article

Full-text available

Apr 2017

We study the superalgebra

A_n

introduced by the authors in a previous paper and use it to construct a 2-Verma module for quantum

\mathfrak{sl}_2

. We prove a uniqueness result about 2-Verma modules on

\Bbbk

-linear 2-categories.

An approach to categorification of Verma modules

Article

Full-text available

Mar 2016
P LOND MATH SOC

We give a geometric categorification of the Verma modules

M(\lambda)

for quantum

\mathfrak{sl}_2

Annular Khovanov homology and knotted Schur-Weyl representations

Article

Full-text available

May 2015

Let

\mathbb{L}\subset A\times I

be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of

\mathfrak{sl}_{2}(\wedge )

, the exterior current algebra of

\mathfrak{sl}_{2}

. When

\mathbb{L}

is an m -framed n -cable of a knot

K\subset S^{3}

, its sutured annular Khovanov homology carries a commuting action of the symmetric group

\mathfrak{S}_{n}

. One therefore obtains a ‘knotted’ Schur–Weyl representation that agrees with classical

\mathfrak{sl}_{2}

Schur–Weyl duality when K is the Seifert-framed unknot.

Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras

Article

Mar 2021
PAC J MATH

On the Hochschild cohomology of differential graded categories

Conference Paper

Dec 2021

Bernhard Keller

The Soergel category and the redotted Webster algebra

Article

May 2016
J PURE APPL ALGEBRA

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.

Sutured annular Khovanov-Rozansky homology

Article

Jun 2015

We introduce an sl(n) homology theory for knots and links in the thickened annulus. To do so, we first give a fresh perspective on sutured annular Khovanov homology, showing that its definition follows naturally from trace decategorifications of enhanced sl(2) foams and categorified quantum gl(m), via classical skew Howe duality. This framework then extends to give our annular sl(n) link homology theory, which we call sutured annular Khovanov-Rozansky homology. We show that the sl(n) sutured annular Khovanov-Rozansky homology of an annular link carries an action of the Lie algebra sl(n), which in the n=2 case recovers a result of Grigsby-Licata-Wehrli.

Tensor product categorifications, Verma modules and the blob 2-category

Abstract

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