2-Verma modules

Abstract

We construct a categorification of (parabolic) Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras.

Full text

arXiv:1710.06293v2 [math.RT] 1 Jul 2019
2-VERMA MODULES
GR´
EGOIRE NAISSE AND PEDRO VAZ
Abstract. We construct a categorification of parabolic Verma modules for symmetriz-
able Kac–Moody algebras using KLR-like diagrammatic algebras. We show that our
construction arises naturally from a dg-enhancement of the cyclotomic quotients of the
KLR-algebras. As a consequence, we are able to recover the usual categorification of
integrable modules. We also introduce a notion of dg-2-representation for quantum Kac–
Moody algebras, and in particular of parabolic 2-Verma module.
Contents
1. Introduction 1
2. Quantum groups and Verma modules 5
3. The b-KLR algebras 9
4. Dg-enhancement 21
5. Categorical action 29
6. The categorification theorems 44
7. 2-Verma modules 51
Appendix A. Summary on the homotopy category of dg-categories and
pretriangulated dg-categories 54
References 63
1. Introduction
The study of categorical actions of (quantum enveloping algebras of) Kac–Moody al-
gebras leads to many interesting results. An impressive example is due to Chuang and
Rouquier [11], who introduced categorical actions of sl2to prove the Brou´e abelian defect
group conjecture for symmetric groups. Another interesting result is Webster’s construc-
tion of homological versions of quantum invariants of knots and links obtained by the
Reshetikhin–Turaev machinery [44].
Until recently, only categorifications of integrable representations of quantum Kac–
Moody algebras were known. These are given by additive (or abelian) categories, on
which the quantum group acts by (exact) endofunctors respecting certain direct sum
decompositions, corresponding to the defining relations of the algebra (see for exam-
ple [15,19,25,26,38]). In [34], the authors followed a slightly different approach to
construct a categorification of the universal Verma module Mpλqfor quantum sl2. The
1

2 GR ´
EGOIRE NAISSE AND PEDRO VAZ
construction of [34] is given in the form of an abelian, bigraded (super)category, where the
commutator relation takes the form of a (non-split) natural short exact sequence
0ÑFE ÑEF ÑQK ‘ΠQK´1Ñ0,
where Π is the parity shift functor, and Qa categorification of 1
q´q´1in the form of an infinite
direct sum. This category is obtained as a certain category of modules over cohomology
rings of infinite Grassmannianns and their Koszul duals. Categorification of Verma modules
appeared independently in the litterature with a strongly different flavor in [12] and in [5].
Studying the endomorphism ring of Fk:“F˝ ¨ ¨ ¨ ˝ Fyields a (super)algebra Akthat
extends the ubiquitous nilHecke algebra NHk. This superalgebra was studied by the authors
in the follow up [35], where it was used to construct an equivalent categorification of Verma
modules for quantum sl2. The supercenter of Akwas also studied in [4]. The definition of
the superalgebra Akand is supercenter were extended in [37] to the case of a Weyl group
of type B.
The superalgebra Akcomes equipped with a family of differentials dnfor ně0. The
corresponding dg-algebras are formal, with homology being isomorphic to the n-cyclotomic
quotients of the nilHecke algebra. These quotients are known to categorify the irreducible
integrable Uqpsl2q-representations Vpnqof highest weight n. We interpret this as a categori-
fication of the universal property of the Verma module Mpλq, that is there is a surjection
Mpλq։Vpnqfor all n. This also means the dg-algebra pAk, dnqcan be seen as a dg-
enhancement of the cyclotomic nilHecke algebra NHn
k, and in particular, of categorified
Vpnq.
In [22,24] and [38], Khovanov–Lauda and Rouquier introduced generalizations of the
nilHecke algebra for any Cartan datum. These algebras are presented in the form of braid-
like diagrams in [22,24], with strands labeled by simple roots and decorated with dots. It
is proved in [22,24,38] that KLR algebras categorify the half quantum group associated
with the input Cartan datum. Khovanov and Lauda conjectured that certain quotients
of these algebras categorify irreducible, integrable representations of the quantum group.
Due to the isomorphism between these quotient algebras and cyclotomic Hecke algebras
in type A(see [7,38]), these quotients have become known as cyclotomic KLR algebras.
The corresponding cyclotomic conjecture was first proved in [8,9,27] for some special
cases, and then for all symmetrizable Kac–Moody algebras by Kang–Kashiwara in [19],
and independently by Webster in [44].
In this paper, we introduce a version of the KLR algebra associated to a pair pp,gq,
where pis a (standard) parabolic subalgebra of a quantum Kac-Moody algebra g. This
construction generalizes the algebra Akfrom [34], which we view as associated to the
(standard) Borel subalgebra of sl2. The usual KLR algebra is recovered by taking p“g. .
We prove that certain ‘cyclotomic quotients’ of these p-KLR algebras categorify parabolic
Verma modules induced over the parabolic subalgebra p, with the cyclotomic quotient
depending on the highest weight. The proof goes by showing first that if p“bis the
(standard) Borel subalgebra of g, then the b-KLR algebra is equipped with a categorical
g-action similar to the one constructed in [35]. In particular, it categorifies the universal

2-VERMA MODULES 3
Verma module of g. Next, we show that the b-KLR algebra can be equipped with a family
of differentials, turning it into a dg-enhancement of the cyclotomic p-KLR algebras. This
induces a categorical g-action on the cyclotomic p-KLR algebra. In particular, we recover
the usual categorical action on cyclotomic KLR algebras, and we can reinterpret Kang–
Kashiwara’s proof of Khovanov–Lauda’s cyclotomic conjecture in terms of dg-enhanced
KLR algebras. The world of dg-categories also allows to reinterpret the usual categorical
sl2-commutator relation in terms of mapping cones. More precisely, the derived category
of dg-modules over the dg-enhanced KLR algebra comes equipped with functors Ei,Fiand
an autoequivalence Kifor all simple root αi, that categorifies the action of the Chevalley
generators Ei, Fiand of the Cartan element Ki“qHi
i. Then, the sl2-commutator relation
of the categorical action takes the form of a quasi-isomorphism of mapping cones
ConepFiEiÑEiFiq»
ÝÑ ConepQiKiÑQiK´1
iq,
where Qiis a direct sum of grading shift copies of the identity functor that categories
1
q´1
i´qi. Whenever Fiis locally nilpotent, ConepQiKiÑQiK´1
iqis quasi-isomorphic to a
finite direct sum of shifted copies of the identity functor, corresponding to the usual notion
of an integrable categorical g-action (as in [19] for example).
Categorification of parabolic Verma modules have found connections with topology in
the work of the authors in [33]. In particular, they have constructed Khovanov–Rozansky’s
triply graded link homology using parabolic 2-Verma modules of gl2k. On the decategorified
level, the connection between the HOMFPY-PT link polynomial and Verma modules was
not known before.
Outline of the paper. In Section 2, we recall the basics about quantum groups and their
parabolic Verma modules.
In Section 3, we introduce the b-KLR algebra Rb(Definition 3.3) as a diagrammatic
algebra over a unital commutative ring k, in the same spirit as Khovanov–Lauda’s [22].
We construct a faithful action on a polynomial ring and exhibit a basis, proving Rbis a
free k-module.
In Section 4, we introduce the p-KLR algebra Rpfor any (standard) parabolic subalgebra
pof g. We also introduce the corresponding N-cyclotomic quotient RN
p. We introduce a
differential dNon Rb, turning it into a dg-enhancement of RN
p. In particular, we prove the
following theorem:
Theorem 4.4.The dg-algebra pRbpmq, dNqis formal with homology
HpRbpmq, dNq – RN
ppmq.
In Section 5, we construct a categorical action of Uqpgqon Rb, where the action of
the Chevalley generators Fiand Eiis given by functors Fiand Eiwhich are defined in
terms of induction and restriction functors for the map that adds a strand labeled i. The
sl2-commutator relation takes the form of a non-split natural short exact sequence. Let

4 GR ´
EGOIRE NAISSE AND PEDRO VAZ
‘rβi´α_
ipνqsqiIdνbe an infinite direct sum of degree shifts of the identity functor that cat-
egorifies the power series pλiq´α_
ipνq
i´λ´1
iqα_
ipνq
iq{pqi´q´1
iq(see Eq. (25) in the beginning
of Section 5).
Corollary 5.2.There is a natural short exact sequence
0ÑFiEiIdνÑEiFiIdνÑ ‘rβi´α_
ipνqsqiIdνÑ0,
for all iPIand there is a natural isomorphism
FiEj–EjFi,
for all i‰jPI.
Fix pĂg, and let Ifbe the set of simple roots for which FiPp. Let ‘rnsqiIdνbe a finite
direct sum of degree shifts of the identity functor that categorifies the quantum integer
rnsqi. The categorical g-action on Rblifts to the dg-algebra pRb, dNq, and thus to RN
pby
Theorem 4.4. The short exact sequence of Corollary 5.2 lifts to a short of exact sequence
of complexes, inducing a long exact sequence in homology. This allows us to compute the
action of the functors of induction FN
iand restriction EN
ion RN
p:
Theorem 5.17.For iRIfthere is a natural short exact sequence
0ÑFN
iEN
iIdνÑEN
iFN
iIdνÑ ‘rβi´α_
ipνqsqiIdνÑ0,
and for iPIfthere are natural isomorphisms
EN
iFN
iIdν–FN
iEN
iIdν‘rni´α_
ipνqsqiIdν,if ni´α_
ipνq ě 0,
FN
iEN
iIdν–EN
iFN
iIdν‘rα_
ipνq´nisqiIdν,if ni´α_
ipνq ď 0.
Moreover, there is a natural isomorphism
FN
iEN
j–EN
jFN
i,
for i‰jPI.
In Section 6, we compute the asymptotic Grothendieck group of pRb, dNq. The asymp-
totic Grothendieck group is a refined version of Grothendieck group, that was introduced
by the first author in [32]. It allows taking in consideration infinite iterated extensions
of objects, such as infinite projective resolutions and infinite composition series (see Defi-
nition 6.2). Let MppΛ, Nqbe the parabolic Verma module of highest weight pΛ, Nq, and
M
ppΛ, Nqbe the c.b.l.f. derived category of pRb, dNq(see Section 6.1).
Theorem 6.12.The asymptotic Grothendieck group K∆
0p
M
ppΛ, Nqq is a Uqpgq-weight
module, with action of Ei, Figiven by rEis,rFis. Moreover, there is an isomorphism of
Uqpgq-modules
K∆
0p
M
ppΛ, Nqq bZQ–MppΛ, N q.
In Section 7, we introduce a notion of categorical dg-action of gon a pretriangulated dg-
category (Definition 7.2), and of (parabolic) 2-Verma module (Definition 7.6). In particular,
we show that
M
ppΛ, Nqadmits a dg-enhancement
M
p
dgpΛ, N qin the form of a dg-category.

2-VERMA MODULES 5
It yields an example of parabolic 2-Verma module, for which Theorem 6.12 takes the
following form:
Corollary 7.8.For all iPIthere is a quasi-isomorphism of cones
Cone`FN
iEN
iIdνÑEN
iFN
iIdν˘»
ÝÑ Cone`Qiλiq´α_
ipνq
iIdνÑQiλ´1
iqα_
ipνqIdν˘,
in
E
ndHqep
D
dgpRb, dNqq.
Finally, in Appendix Awe recall the construction of the homotopy category of dg-
categories up to quasi-equivalence, based on Toen [41]. We also recall how to compute the
(derived) dg-hom-spaces between pretriangulated dg-categories.
Acknowledgments. G.N. is a Research Fellow of the Fonds de la Recherche Scientifique
- FNRS, under Grant no. 1.A310.16. P.V. was supported by the Fonds de la Recherche
Scientifique - FNRS under Grant no. J.0135.16.
2. Quantum groups and Verma modules
We recall the basics about quantum groups and their (parabolic) Verma modules. Our
presentation is close to [18] and [29], where the proofs can be found. References for classical
results about Verma modules are [30] and [17] (and [2] for the quantum case).
2.1. Quantum groups. Ageneralized Cartan matrix is a finite dimensional square matrix
A“ taij ui,jPIPZ|I|ˆ|I|such that
‚aii “2 and aij ď0 for all i‰jPI;
‚aij “0ôaji “0.
One says that Ais symmetrizable if there exists a diagonal matrix Dwith positive entries
diPZą0for all iPI, such that DA is symmetric. A Cartan datum consists of
‚a symmetrizable generalized Cartan matrix A;
‚a free abelian group Ycalled the weight lattice;
‚a set of linearly independent elements Π “ tαiuiPIĂYcalled simple roots;
‚adual weight lattice Y_:“HompY, Zq;
‚a set of simple coroots Π_“ tα_
iuiPIĂY_;
such that
‚α_
ipαjq “ aij ;
‚for each iPIthere is a fundamental weight ΛiPYsuch that α_
jpΛiq “ δij for all
jPI.
The abelian subgroup X:“ÀiZαiĂYis called the root lattice. We also write X`:“
ÀiNαiĂXfor the positive roots. Given a Cartan datum, since Ais symmetrizable with
diaij “djaji , one can construct a symmetric bilinear form
p´|´q :YˆYÑZ,
respecting
‚ pαi|αiq “ 2diP t2,4,...u;
‚ pαi|αjq “ diaij P t0,´1,´2,...ufor all i‰j;

6 GR ´
EGOIRE NAISSE AND PEDRO VAZ
‚α_
ipyq “ 2pαi|yq
pαi|αiqfor all yPY.
In the end, a Cartan datum is completely determined by pI, X, Y, p´|´qq.
Definition 2.1. The quantum Kac–Moody algebra Uqpgqassociated to a Cartan datum
pI, X, Y, p´|´qq is the associative, unital Qpqq-algebra generated by the set of elements
Ei, Fiand Kγfor all iPIand γPY_, with relations for all iPIand γ, γ1PY_:
K0“1, KγKγ1“Kγ`γ1,
KγEi“qγpαiqEiKγ, KγFi“q´γpαiqFiKγ,
One also imposes the sl2-commutator relation for all i, j PI:
(1) EiFj´FjEi“δij
Ki´K´1
i
qi´q´1
i
,
where qi:“qdiand Ki:“Kα_
i.
Finally, there are the Serre relations for i‰jPI:
ÿ
r`s“1´aij
p´1qr„1´aij
rqi
Er
iEjEs
i“0,(2)
ÿ
r`s“1´aij
p´1qr„1´aij
rqi
Fr
iFjFs
i“0.(3)
This ends the definition of Uqpgq.
Given a sequence i“i1¨¨¨imof elements in I, we write Fi:“Fi1¨¨¨Fimand Ei:“
Ei1¨¨¨Eim. We write SeqpIqfor the set of such sequences. Any element of Uqpgqdecomposes
as a sum of elements FiKγEjwith i,jPSeqpIq.
The half quantum group U´
qpgqof Uqpgqis the subalgebra generated by the elements
tFiuiPI. As a Qpqq-vector space, it admits a basis given by a subset of tFiuiPSeqpIq.
2.2. Weight modules. Let Mbe an Uqpgq-module with ground ring RĄQpqq. Consider
aZ-linear functional
λ:Y_ÑRˆ,
where the group structure on Rˆis the product. For each such λand yPY, we call
pλ, yq-weight space the set
Mλ,y :“ tvPM|Kγv“λpγqqγpyqvfor all γPY_u.
Note that EiMλ,y ĂMλ,y`αiand FiMλ,y ĂMλ,y´αi. A weight module is a module that
decomposes as a direct sum of weight spaces. A highest weight module is a module M
such that M“Uqpgqvλfor some vλPMλ,0with Eivλ“0 for all iPI. In that case, we
call λthe highest weight and we have
M–à
yPX`
Mλ,´y.
as R-module.

2-VERMA MODULES 7
One says that a Uqpgq-module Mis integrable if for each vPMthere exists k"0 such
that Ek
iv“0 and Fk
iv“0 for all iPI. Any finite dimensional module is integrable, and
any integrable module is a weight module with λpΠ_q Ă Zrqs. We consider only type 1
modules, that is λpΠ_q Ă Nrqs.
Let Mbe a highest weight module with highest weight vector vλPMλ,0. Then we set
λi:“λpα_
iqfor each iPI. We are interested in λsuch that each λiis either λi“qnifor
some niPZor λiis formal. In that case, we write it λi“qβiwhere we interpret βias a
formal parameter.
2.2.1. Parabolic Verma modules. The (standard) Borel subalgebra Uqpbqof Uqpgqis gener-
ated by Kγand Eifor all γPY_and iPI. A (standard) parabolic subalgebra of Uqpgqis
a subalgebra containing Uqpbq. It is generated by Kγ, Eiand Fjfor all γPY_, i PIand
jPIffor some fixed subset IfĂI. The part given by Kγ, Ejand Fjfor jPIfis called
the Levi factor and written Uqplq. The nilpotent radical Uqpnqis generated by Eifor all
iPIr:“IzIf. Note that parabolic subalgebras are in bijection with partitions I“If\Ir.
Let Uqppqbe a parabolic subalgebra determined by I“If\Ir. For each iPIf, we choose
a weight niPN. For each jPIrwe choose a weight λjP tqβj, qnju. We write N“ tniuiPIf
and Λ “ tλjujPIr. Let VpΛ, N qbe the unique (type 1) integrable, irreducible representation
of Uqplqon the ground ring R“Qpq, Λq, and with highest weight λdetermined by
λpα_
kq “ #qni,if k“iPIf,
λj,if k“jPIr.
We extend it to a representation of Uqppqby setting UqpnqVpΛ, Nq “ 0.
Definition 2.2. The parabolic Verma module of highest weight pΛ, Nqassociated to
Uqppq Ă Uqpgqis the induced module
MppΛ, Nq:“Uqpgq bUqppqVpΛ, N q.
Whenever Uqppq Ĺ Uqpgq, we have that MppΛ, Nqis an infinite dimensional module.
Moreover, for all parabolic Verma modules, there is a Qpqq-linear surjection
U´
qpgq bQpqqR։MppΛ, Nq.
Example 2.3. If Uqppq “ Uqpbq, then N“ H, and VpΛ, N q – Qpq, ΛqvΛis 1-dimensional,
and such that
EivΛ“0, KγvΛ“ź
jPI
λγpΛjq
jvΛ.
In this case, we simply call it Verma module, and denote it MbpΛq. If λj“qβis formal
for all jPIr, then we call it the universal Verma module.
Example 2.4. If Uqppq “ Uqpgq, then Λ “ H and MppΛ, N q – VpNqis an integrable,
irreducible Uqpgqrepresentation.

8 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Since qis a generic parameter we can apply Jantzen’s criterion [17, Theorem 9.12], thanks
to the results in [2]. We obtain that MppΛ, N qis irreducible whenever λjR tqn|nPNufor
all jPIr. If λj“qnjfor njPN, then MppΛ, N qcontains a non-trivial, proper submodule,
which is isomorphic to MppΛnj
´nj´2, Nqfor Λnj
´nj´2given by exchanging qnjwith q´nj´2in
Λ. Moreover, the quotient
MppΛ, Nq
MppΛnj
´nj´2, Nq–Mp`jpΛztqnju, N \ tnjuq,
is isomorphic to the parabolic Verma module associated to the parabolic subalgebra p`j
given by adding jto If, that is generated by pand Fj.
Furthermore, whenever λj“qβjis formal, there is a surjective map
evnj:MppΛ, Nq։MppΛnj
βj, Nq,
for all njPZ, given by evaluating βj“nj.
These two facts together allow us to define a partial order on parabolic Verma mod-
ules. For this, we say that there is an arrow from MppΛ, N qto Mp1pΛ1, N1qif we have an
evaluation map evnjsuch that
evnjpMppΛ, Nqq – Mp1pΛ1, N 1q,
or if there is a short exact sequence
0ÑMppΛnj
´nj´2, Nq Ñ MppΛ, N q Ñ Mp1pΛ1, N1q Ñ 0.
For parabolic Verma modules Mand M1we say that Mis bigger than M1if there is a
chain of arrows from Mto M1. In that case, there is an M2, which is either trivial or a
parabolic Verma module, and a short exact sequence
0ÑM2ÑevpMq Ñ M1Ñ0,
where ev is a composition of evaluation maps evnj. With this partial order, the universal
Verma module is a maximal element and each integrable, irreducible module is a minimum.
This also means that we can recover any parabolic Verma module from the universal one.
2.2.2. The Shapovalov form. Let ρ:Uqpgq Ñ Uqpgqop be the Qpqq-linear algebra anti-
involution given by
ρpEiq:“q´1
iK´1
iFi, ρpFiq:“q´1
iKiEi, ρpKγq:“Kγ,(4)
for all iPIand γPY_.
Definition 2.5. The Shapovalov form
p´,´q :MppΛ, Nq ˆ MppΛ, N q Ñ Qpq, Λq,
is the unique bilinear form respecting
‚ pvΛ,N , vΛ,N q “ 1, for vΛ,N the highest weight vector;
‚ puv, v1q “ pv, ρpuqv1qwhere ρis defined in (4);
‚fpv, v1q “ pf v, v1q “ pv, f v1q,
for all v, v1PMppΛ, N q, u PUqpgqand fPQpq, Λq.

2-VERMA MODULES 9
2.2.3. Basis. Since parabolic Verma modules are highest weight modules, they admit at
least one basis given in terms of elements of the form FivΛ,N for iPSeqpIq, where vΛ,N is a
highest weight vector. In particular, as R-modules they are all submodules of U´
qpgqbQpqqR,
meaning that these basis lives in a subset of tFivΛ,N |iPSeqpIqu modded out by the
Serre relations. We call such a basis an induced basis and write it tvΛ,N “m0, m1,...u.
Any element in such basis takes the form Fi“Fbr
ir¨¨¨Fb1
i1for some i1,...,irPIand
b1,...,brPN, with iℓ‰iℓ`1. Replacing each Fb
iby the divided power Fpbq
i:“Fb
i{prbsqi!q
yields another basis tvΛ,N “m1
0, m1
1,...uthat we refer as a canonical basis. Whenever
MppΛ, Nqis irreducible, the Shapovalov form is non-degenerate. Therefore, in this case,
for each canonical basis there is a dual canonical basis uniquely determined by
pm1
i, mjq “ δij .
3. The b-KLR algebras
Fix once and for all a Cartan datum pI, X, Y, p´|´qq, and let
dij :“ ´α_
ipαjq P N.
For νPX`we write
ν“ÿ
iPI
νi¨αi, νiPN,
and we set |ν|:“řiνi, and Supppνq:“ ti|νi‰0u.
We also fix a choice of scalars in a commutative, unital ring kas introduced in [39].
Following the conventions in [10], it consists of:
‚tij Pkˆfor all i, j PI;
‚stv
ij Pkfor i‰j, 0 ďtădij and 0 ďvădj i;
‚riPkˆfor all iPI,
respecting
‚tii “1;
‚tij “tji whenever dij “0;
‚stv
ij “svt
ji ;
‚stv
ij “0 whenever tpαi|αiq ` vpαj|αjq ‰ ´2pαi|αjq.
In addition, whenever tă0 or vă0, we put stv
ij :“0. Thus we have spq
ij “0 for pądij
or qądji . We will also write sdij 0
ij :“tij and s0dji
ij :“tji. Hence if pαi|αjq “ 0 we get
s00
ij “s00
ji “tij “tj i.
Definition 3.1 ([22,38]).The Khovanov–Lauda–Rouquier (KLR) algebra Rpmqis the
k-algebra generated by braid-like diagrams on mstrands, read from bottom to top, such
that
‚two strands can intersect transversally;
‚strands can be decorated by dots (we use a dot with a label kto denote kconsecutive
dots on a strand);

10 GR ´
EGOIRE NAISSE AND PEDRO VAZ
‚each strand is labeled by a simple root, written iPI, that we (usually) write at
the bottom;
‚multiplication is given by concatenation of diagrams, which preserves the labeling
(i.e. connecting two strands with different labels gives zero);
‚diagrams are taken modulo isotopies and the following local relations:
ij
“$
’
’
’
’
’
&
’
’
’
’
’
%
0 if i“j,
ř
t,v
stv
ij
i
t
j
vif i‰j,
(5)
for all i, j PI,
ij
“
ijij
“
ij
(6)
i i
“
i i
`ri
i i
,
i i
“
i i
`ri
i i
(7)
for all i‰jPI,
ik
j
´
ik
j
“
$
’
’
’
’
’
’
’
&
’
’
’
’
’
’
’
%
0 if i‰k,
riř
t,v
stv
ij ř
u`ℓ“
t´1
j
v
i
u
i
ℓotherwise,
(8)
for all i, j, k PI. In addition, Rpmqis Z-graded by setting
degq¨
˝ij˛
‚:“ ´pαi|αjq,degq¨
˝i˛
‚:“ pαi|αiq.
Remark 3.2. Note that in Eq. (5) and Eq. (8), the sum ř
t,v
stv
ij can be restricted to the
finite number of pairs t, v PNsuch that tpαi|αiq ` vpαj|αjq “ ´2pαi|αjq. Moreover, it
contains at least two non-zero elements with invertible coefficients, given by t“dij , v “0
and t“0, v “dji.
As proven in [22,24] (see also [38]), these algebras categorify the half quantum group
U´
qpgqassociated to pI , X, Y, p´|´qq, as a (twisted) bialgebra. The multiplication and
comultiplication are categorified using respectively induction and restriction functors, ob-
tained by putting diagrams side by side.

2-VERMA MODULES 11
For each non-negative integral highest weight N:“ tniPN|iPIu, there is a N-
cyclotomic quotient RNpmqof Rpmqgiven by modding out the two-sided ideal generated
by all diagrams of the form
i
ni
j
...
k
“0.
As first conjectured in [22] and proved in [19] and independently in [44], these cyclotomic
quotients categorify the irreducible integrable Uqpgq-module of highest weight N, where
the action of Fi(resp. Ei) is given by induction (resp. restriction) along the map RpmqãÑ
Rpm`1qthat adds a vertical strand with label iat the right.
3.1. b-KLR algebra. Our first goal is to construct a dg-enhancement of the cyclotomic
KLR algebras RNpmq, in the same spirit as in [35]. We introduce the following algebra:
Definition 3.3. The b-KLR algebra Rbpmqis the k-algebra generated by braid-like dia-
grams on mstrands, read from bottom to top, such that
‚two strands can intersect transversally;
‚strands can be decorated by dots;
‚regions in-between strands can be decorated by floating dots, which are labeled by
a subscript in Iand a superscript in N;
‚each strand is labeled by a simple root, written iPI;
‚multiplication is given by concatenation of diagrams, which preserves the labeling;
‚diagrams are taken modulo isotopies that preserves the relative height of the floating
dots, and modulo the KLR relations Eq. (5–8) and the following local relations:
¨¨¨
a
ib
j
“ ´ ¨¨¨
a
i
b
j
a
i
a
i
“0,(9)
meaning floating dots anti-commute with each other for all i, j PIand a, b PN,
i
a
j“
$
’
’
’
’
’
’
&
’
’
’
’
’
’
%
i
a´1
i´
i
a´1
iif i“jand aą0,
ř
t,v
p´1qvstv
ij
i
t
a`v
jif i‰j,
(10)
ij
a
j“
ij
a
j`ÿ
t,v
stv
ij ÿ
u`ℓ“
v´1
p´1qu
i
t
j
ℓ
a`u
jif i‰j,(11)

12 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Moreover, a floating dot in the left-most region is zero
a
i
jk
...
ℓ
“0.
Given a diagram, it is sometimes useful to decorate some of its regions with an element
K:“řiPIki¨αiPX`, where kidenotes the number of strands with label iat the left of
the region. The algebra Rbis Z1`|I|-graded (a q-grading and a λk-grading for each kPI)
with
degq¨
˝ij˛
‚:“ ´pαi|αjq,degq¨
˝i˛
‚:“ pαi|αiq,
degλk¨
˝ij˛
‚:“0,degλk¨
˝i˛
‚:“0,
and
degq˜a
i
K¸:“ p1`a´α_
ipKq ` kiqpαi|αiq,
degλk˜a
i
K¸:“2δik.
This ends the definition of Rb.
3.2. Tightened basis. Before going any further, let us introduce some useful notations
borrowed from [22]. First, let Rbpνqbe the subalgebra of Rbpmqgiven by diagrams where
there are exactly νistrands labeled i, for each iPI. We also denote Seqpνqthe set of
all ordered sequences i“i1i2¨¨¨imwith ikPIand iappearing νitimes in the sequence.
The symmetric group Smacts on Seqpνqwith the simple transposition σkPSmacting on
i“i1i2¨¨¨imPSeqpνqby permuting ikand ik`1. Sometimes, for K“řiPIki¨αiPX`,
we abuse notation by writing σKinstead of σ|K|.
For i“i1i2¨¨¨imPSeqpνq, let 1iPRbpνqbe the idempotent given by mvertical strands
with labels i1, i2,...,im, that is
1i:“
i1i2im
¨¨¨
We have 1i1j“δij for all i,jPSeqpνq, and so there is a decomposition of k-modules
Rbpνq – à
i,jPSeqpνq
1jRbpνq1i.
Our goal is to construct a basis of 1jRbpνq1ias k-module.

2-VERMA MODULES 13
3.2.1. An action of Rbon a polynomial space. We construct a polynomial representation
of Rbwith a similar flavor as in [22,§2.3]. We fix νPX`with |ν| “ m. For each iPIwe
define
Qi:“krx1,i,...,xνi,is b Ź‚xω1,i ,...,ωνi,iy.
We write QI:“ÂiPIQi, where bmeans the supertensor product in the sense that
ωℓ,iωℓ1,j “ ´ωℓ1,j ωℓ,i for all i, j PIand xi,ℓ commutes with everything. Thus, QIis a
supercommutative superring. Then we construct the ring
Qν:“à
iPSeqpνq
QI1i,
where the elements 1iare central idempotents. It is Z1`|I|-graded by setting
degqpxℓ,iq “ pαi|αiq,degqpωℓ,i q “ p1´ℓqpαi|αiq,
degλjpxℓ,iq “ 0,degλjpωℓ,iq “ 2δij .
We first construct an action of the symmetric group Smon Qνby letting the simple
transposition
σk:QI1iÑQI1ski,
to act by sending
xp,i1iÞÑ $
’
’
&
’
’
%
xp`1,i1ski,if ik“ik`1“iand p“#tsďk|is“iu,
xp´1,i1ski,if ik“ik`1“iand p“1`#tsďk|is“iu,
xp,i1ski,otherwise,
for iPI, p P t1,...,νiuand i“i1. . . im, and by sending
ωp,i1iÞÑ #pωp,i ` pxp,i ´xp`1,i qωp`1,iq1ski,if ik“ik`1“iand p“#tsďk|is“iu,
ωp,i1ski,otherwise,
which we extend to Qνby setting σkpfgq:“σkpfqσkpgqfor all f, g PQν.
Proposition 3.4. The procedure described above yields a well-defined action of Smon Qν.
Proof. The proof is a straightfoward computation. We leave the details to the reader. 
Then, we define inductively the element ωa
p,j PQIfor aPNas
ω0
p,j :“ωp,j , ωa`1
p,j :“ωa
p´1,j ´xp,j ωa
p,j.
For K“řiPIki¨iPX`such that kiďνi, we define ωa
jpKq P QIinductively as
ωa
jpKq:“$
’
’
&
’
’
%
0,if kj= 0,
ωa
kj,j ,if ki“0 for all i‰j,
ř
t,v
p´1qtstv
ij xt
ki,iωa`v
jpK´iq,otherwise,
where K´iis a shorthand for K´1¨αi.

14 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Lemma 3.5. The element ωa
jpKqis well-defined.
Proof. Take i‰i1‰jPIsuch that kią0 and ki1ą0. We can suppose by induction that
ωb
jpK´i´i1qis well-defined for all bě0. Then we have
ÿ
t,v
p´1qtstv
ij xt
ki,i ÿ
t1,v1
p´1qt1st1v1
i1jxt1
ki1,i1ωa`v
jpK´i´i1q
“ÿ
t1,v1
p´1qt1st1v1
i1jxt1
ki1,i1ÿ
t,v
p´1qtstv
ij xki,iωa`v
jpK´i1´iq,
for all i‰i1‰jPI.
It will be useful to give ωa
jpKqa non-inductive expression. We write Kzj:“ři‰jki¨αii.
For a given non-negative integer niPNwe define
(12) εj
ni,ipxki,i q:“ÿ
|Vi|“ni˜ki
ź
ℓ“1
svℓtℓ
ji xtℓ
ℓ,i¸PPi,
with the sum being over all partitions Vi:v1` ¨ ¨ ¨ ` vki“visuch that pαi|αiq|vℓpαj|αjq
for each ℓP t1,...,kiu, and with tℓ:“´2pαi|αjq´vℓpαj|αjq
pαi|αiq. This is a symmetric polynomial of
q-degree ´2kipαi|αjq ´ vipαj|αjqwhenever it is non zero. Clearly, we can suppose vℓďdj i,
and therefore we can also suppose that niďdj iki. For nPNwe define
(13) εj
vpxKq:“ÿ
|V|“n˜ź
i‰j
εj
ni,ipxki,i q¸PPI,
with the sum being over all partitions V:ři‰jni“n. Notice that εj
vpxKqis a polynomial
of q-degree p´α_
jpKzjq ´ nqpαj|αjq.
Lemma 3.6. We have
(14) ωa
jpKq “
´α_
jpKzjq
ÿ
n“0
p´1qnωa`n
kj,j εj
npxKq P PI.
Proof. A straightforward computation shows that the RHS of Eq. (14) respects the recur-
sive definition of ωa
jpKq, which proves the equality. 
We now have all the tools we need to define an action of Rppνqon Pν. First, we choose
an arbitrary orientation iÐjor iÑjfor each pair of distinct i, j PI. Then, we let
aPRppνq1jact as zero on PI1iwhenever j‰i. Otherwise, we declare that
‚the dot
i
K
acts as multiplication by xki`1,i1i;

2-VERMA MODULES 15
‚the floating dot
a
j
K
acts as multiplication by ωa
jpKq1i;
‚the crossing
i j
K
acts as
f1iÞÑ ri
f1i´sKpf1iq
xki,i ´xki`1,i
,if i“j,
f1iÞÑ ˜ÿ
t,v
stv
ij xt
ki,ixv
kj`1,j¸sKpf1iq,if iÑj,
f1iÞÑ sKpf1iq,if iÐj.
Proposition 3.7. The rules above define an action of Rbpνqon Qν.
Proof. We have to check the validity of the KLR relations Eq. (5–8) and of the relations
involving floating dots Eq. (9–11), as well as the relations coming from regular isotopies.
We start by proving the KLR relations. Clearly Eq. (5), Eq. (6) and Eq. (7) are satisfied.
The case i‰kof Eq. (8) is also straightforward. For iÐjand k“iwe compute the
action of the LHS of Eq. (8) on fPQνas
fÞÑ `ÿ
t,v
stv
ji ytxv
1˘σ1B2σ1pfq ´ σ2B1`ÿ
t,v
stv
ji ytxv
2σ2pfq˘
“`ÿ
t,v
stv
ji ytxv
1˘f´σ1σ2σ1pfq
x1´x2
´`řt,v stv
ji ytxv
2˘f´`řt,v stv
ji ytxv
1˘σ2σ1σ2pfq
x1´x2
“ÿ
t,v
stv
ji ytxv
1f´xv
2
x1´x2
“ÿ
t,v
stv
ji ytÿ
r`s“
v´1
xr
1xs
2,
where x1, x2correspond with the xki,i, xki`1,i and ywith xkj,j. What remains coincides
with the RHS of Eq. (8). A similar computation applies for the case iÑj.
For the relations involving floating dots, we remark that Eq. (9) follows from the super-
commutativity of Qν, and ωa
jpKqrespects Eq. (10) by construction. For relation Eq. (11),
we apply the action of the LHS on some fPQνand we obtain
fÞÑ `ÿ
t,v
stv
ji ytxv˘ωa
jpK`jqf,

16 GR ´
EGOIRE NAISSE AND PEDRO VAZ
and for the RHS we obtain
fÞÑ `ωa
jpK`i`jq ` ÿ
t,v
stv
ij ÿ
u`ℓ“
v´1
p´1quωa`u
jpKqxtyℓ˘f
“`ÿ
t,v
p´1qvstv
ij xtωa`v
jpK`jq ` ÿ
t,v
stv
ij ÿ
u`ℓ“
v´1
p´1quωa`u
jpKqxtyℓ˘f.
Then we compute
ωa`v
jpK`jq “ `ÿ
u`ℓ“
v´1
p´1qv´1´uyℓωa`u
jpKq˘` p´1qvyvωa
jpK`jq,
which implies that the action of the RHS of Eq. (11) coincides with the one of the LHS.
The only non trivial relation coming from regular isotopies we need to verify is given
by the commutation of a floating dot and a crossing at its left. This is a consequence of
the fact that Eq. (12) is a symmetric polynomial, which commutes with divided difference
operators. 
3.2.2. Left-adjusted expressions. Recall from [35,§2.2.1] that a reduced expression σir¨¨¨σi1
of wPSnis left-adjusted if ir` ¨ ¨ ¨ ` i1is minimal. Equivalently, it is left-adjusted if and
only if
min
tPt0,...,ruσit¨¨¨σi1pkq ď min
tPt0,...,ruσjt¨¨¨σj1pkq,
for all kP t0,...,nuand all other reduced expression σjr¨¨¨σj1“w. In this condition, we
write
minwpkq:“min
tPt0,...,ruσit¨¨¨σi1pkq.
Note that a left adjusted expression always exists and is unique up to distant permutation
(σiσjØσjσifor |i´j| ą 1), so that minwpkqis well-defined. In particular, one can obtain
a left-adjusted reduced expression for any permutation by taking its representative in the
coset decomposition
(15) Sn“
n
ğ
a“1
Sn´1σn´1¨¨¨σa,
applied recursively. If we think of a reduced expression in terms of string diagrams, then
it is left-adjusted if all strings are pulled as far as possible to the left.
Example 3.8. The permutation p1 3 2 4q P S4admits as left-adjusted reduced expres-
sion the word σ1σ2σ1σ3σ2which comes from the summand S2σ3σ2in the first step of the
recursive decomposition (15). Note that σ1σ2σ3σ1σ2is also left-adjusted while σ2σ1σ2σ3σ2
and σ2σ1σ3σ2σ3are not. In terms of string diagrams, we consider as example the following
reduced expression of the permutation w“ p14352q P S5:

2-VERMA MODULES 17
It is not left-adjusted since the 4th strand (read at the bottom) can be pulled to the left.
Hence we obtain the following left-adjusted minimal presentation:
Suppose σir¨¨¨σi1is a left-adjusted reduced expression of w. Then we can choose for
each kP t1,...,nuan index tkP t1,...,rusuch that
σitk¨¨¨σi1pkq “ minwpkq.
Clearly this choice is not necessarily unique and we can have tk“tk1for k‰k1. However, it
defines a partial order ăon the set t1,...,nuwhere kăk1whenever tkďtk1. We extend
this order arbitrarily and we write ătfor it. There is a bijective map s:t1,...,nu Ñ
t1,...,nuwhich sends kăk1to spkq ătspk1q, so that tspkqďtspk1q. In terms of string
diagrams, the map stells us in which order the strands attain their (chosen) leftmost
position while reading from bottom to top. In particular, spkqgives the starting point of
the strand that attains its leftmost position in kth position.
Example 3.9. Consider again the following left-adjusted string diagram:
Both the 1st and 3th strand attain their leftmost position at the bottom of the diagram,
thus we can choose sp1q “ 1 and sp2q “ 3. Then both the 2nd and 4th strand attain their
leftmost position, thus we can take sp3q “ 4 and sp4q “ 2. Finally, the 5th strand attains
its leftmost position and we put sp5q “ 5.
For kP t1,...,n`1u, we put
wk:“σitspkq¨¨¨σitspk´1q,
where it is understood that tsp0q:“0 and tspn`1q:“r. It defines a partition of the reduced
expression of σir¨¨¨σi1“w. Moreover, it is constructed so that
wk¨¨¨w1pspkqq “ minwpspkqq,
for all 1 ďkďn.
Example 3.10. Consider again w“σ1σ2σ1σ3σ2with i1“2, i2“3, i3“1, i4“2, i5“1.
We can choose for example t1“0, t2“0, t3“3 and t4“5. Then we can put sp1q “ 1
(or 2), sp2q “ 2 (or 1), sp3q “ 3 and sp4q “ 4, with w1“1, w2“1, w3“σ1σ3σ2and
w4“σ1σ2.

18 GR ´
EGOIRE NAISSE AND PEDRO VAZ
3.2.3. A generating set. We say that a floating dot is tight if it is placed directly at the
right of the left-most strand, and has superscript 0. We can also suppose it has the same
subscript as the label of the strand at its left (otherwise it would slide to the left and be
zero).
Lemma 3.11. The algebra Rbpνqis generated by KLR elements (i.e. dots and crossings)
and tight floating dots.
Proof. We first compute
i i
a
i“
i i
a
i´
i i
a
i
(16)
for all aě0, i PI. Eq. (16) , together with Eq. (11) and Eq. (10) allows to bring all
floating dots to the left. Then applying Eq. (10) recursively allows to transform all floating
dots with superscript bigger than zero into dots and tight floating dots. 
We write ωfor a tight floating dot, τafor a crossing between the ath and pa`1qth
strands (counting from left), and xafor a dot on the ath strand, where we suppose the
label of the strands given by the context, in the form of an idempotent 1i. We also define
the tightened floating dot in Rbpmqas θa:“τa´1¨¨¨τ1ωτ1¨¨¨τa´1, or diagrammatically
θa:“...
...
...
...
a
We also write θ0
a:“θaand θ´1
a:“1.
Lemma 3.12. Tigthened floating dots anticommute with each others, up to adding terms
with a smaller number of crossings, that is
θaθb“ ´θbθa`R, pθaq2“0`R1,
where R(resp. R1) possesses strictly less crossings than θaθb(resp. pθaq2), for all 1ď
a, b ďm.
Proof. We first compute that
k ℓ
a
i
b
j
`
k ℓ
a
i
b
j
“0,(17)

2-VERMA MODULES 19
for all i, j, k, ℓ PIand a, b PN. Then we obtain
(8)
“`R0
(17)
“ ´ `R0`R1,
where both R0and R1have less crossings. 
Fix i,jPSeqpνq. Since they are both sequences of the same elements, there is a subset
jSiĂSmof permutations wPSmsuch that ik“jwpkqfor all kP t1,...,mu. Given such
a permutation wPjSi, we can choose a left-adjusted reduced expression. It comes with a
partition wm`1¨¨¨w2w1“wand a bijection s:t1,...,mu Ñ t1,...,mu, such that
wk¨¨¨w1pspkqq “ minwpspkqq,
for all 1 ďkďm. Then, consider the collection of elements
jBi:“ xam
m¨¨¨xa1
1τwm`1θℓm
minwpspmqqτwm¨¨¨θℓ2
minwpsp2qqτw2θℓ1
minwpsp1qqτw1|
aiPN, ℓiP t0,´1u, w PjSi(
(18)
in 1jRbpmq1i. Diagrammatically, elements in jBican be constructed using the following
algorithm:
(1) choose a permutation wPjSi, consider its corresponding string diagram and make
it left-adjusted by bringing all strands to the left;
(2) for each strand, choose whether we want to add a floating dot. If so, add a tightened
floating dot where the strand attains its left-most position by pulling the strand to
the far left and adding the floating dot immediately at its right;
(3) for each strand, choose a number of dots to add at the top of the diagram.
Proposition 3.13. Elements in jBigenerate 1jRbpmq1ias a k-vector space.
Proof. The proof is an induction on the number of crossings. By Lemma 3.11, we can
assume that all floating dots are tight. By Eq. (6) and Eq. (7) we can bring all the dots to
the top of any diagram, at the cost of adding diagrams with fewer crossings. Moreover, all
braid isotopies hold up to adding terms with a lower amount of crossings thanks to Eq. (5)
and Eq. (8).
We claim that we can also assume that there is at most one floating dot at the immediate
right of each strand. Indeed, suppose there are two tight floating dots at the right of the
same strand. Then we can apply a braid-isotopy to bring it as most as possible to the
left, until it is possibly blocked by other tight floating dots. We depict it by the following

20 GR ´
EGOIRE NAISSE AND PEDRO VAZ
picture:
...
...
“
...
...
`terms with
fewer crossings,
where the dashed strand in red represents the one we want to pull, and the boxes represent
other elements in Rbpmq. If there is no floating dot in-between, then it is zero by Eq. (9).
Otherwise, we apply Eq. (17) to jump the bottom floating dot over all the floating dots
in-between, until we have two floating dots in the same region at the top, which is zero
by Eq. (9). This proves the claim.
Finally, we observe that given a strand with a single tight floating dot, we can tighten
it by braid isotopy, until we end up with a tightened floating dot. Since by Lemma 3.12
tightened floating dots anticommute, this concludes the proof. 
3.2.4. The basis theorem.
Proposition 3.14. The action in Proposition 3.7 is faithful.
Proof. The proof is inspired by [39, Proposition 3.8] (see also [22, Theorem 2.5] for a differ-
ent approach). We claim that elements of jBiact as linearly independent endomorphisms
on Pν. The action yields morphisms
PI1iÑPI1j,
that we will consider as endomorphisms of PI.
First we extend the scalars to kpx1,i,...,xνi,iqin Pifor all iPI. We claim that different
choices of wPjSiand ℓiP t´1,0ugive linearly independent operators. Notice that since
i,jis fixed, wis given by choices of permutations between strands of the same label.
Since crossings between strands with different labels act as multiplication by a polynomial,
we can ignore them as being multiplication by a scalar. By [35, Corollary 3.9], we know
that different choices of permutations and tightened floating dots for strands with label
iact as linearly independent operators on Pi, hence proving our claim. Finally, taking
into account the multiplication by the polynomial given by the choice of the aiPNas in
Eq. (18) concludes the proof. 
Theorem 3.15. The k-module 1jRbpmq1iis free with basis jBi.
Proof. It follows from Proposition 3.13 and Proposition 3.14.
From this, we also deduce the following:

2-VERMA MODULES 21
Corollary 3.16. The b-KLR algebra admits a presentation given by the KLR-generators
and tight floating dots, subjected to the KLR-relations Eq. (5–8) together with
j i
i
j
`
j i
i
j
“0,
i
i
i“0,
for all i, j PIr.
4. Dg-enhancement
We fix a subset IfĂIand consider the associated parabolic subalgebra Uqppq Ă Uqpgq
as defined in Section 2.2. For each jPIf, we also choose a weight njPN, and write
N:“ tnjujPIf.
Definition 4.1. The p-KLR algebra Rppmqis given by forgetting the λj-degree for each
jPIfin Rbpmqand modding out by
j
j i1
...
im´1
“0,
for all jPIf. The N-cyclotomic quotient RN
ppmqof Rppmqis given by modding out by
j
nj
i1
...
im´1
“0,
for all jPIf.
In particular, Rgpmqis the usual KLR algebra Rpmq(see Definition 3.1). Its N-
cyclotomic quotient RN
gpmqis also the usual cyclotomic quotient of the KLR algebra.
Taking If“ H gives p“band we recover Definition 3.3.
We equip Rbpmqwith a homological Z-grading, denoted h, by setting
degh¨
˝ij˛
‚:“0,degh¨
˝i˛
‚:“0,degh˜a
i
K¸“1,
for all i, j PI. Then, we equip Rbpmqwith a differential dNby setting
dN¨
˝ij˛
‚:“dN¨
˝i˛
‚:“0,

22 GR ´
EGOIRE NAISSE AND PEDRO VAZ
and
dN¨
˚
˚
˚
˝j
j i1
...
im´1
˛
‹
‹
‹
‚:“$
’
’
’
&
’
’
’
%
0,if jRIf,
p´1qnj
j
nj
i1
...
im´1
,if jPIf.
Thanks to Lemma 3.11 and extending by the Leibniz rule, this is enough to turn Rbpmq
into a dg-algebra. From this, we derive that for jPIrwe have
dN˜a
j
K¸“ p´1qnj´kj`1`a
´α_
jpKzjq
ÿ
r“0
h
nj`a´kj`1`rpxkj,jqεj
rpxKq,
where xℓ,i is a dot on the ℓth strand with label i,
h
nis the nth complete homogeneous
polynomial, and εj
rpxKqis defined in Eq. (13).
Definition 4.2. We refer to the dg-algebra pRbpmq, dNqas dg-enhanced KLR algebra.
Proposition 4.3. If nj´νj´α_
jpνzjq ă 0, then pRbpνq, dNqis acyclic.
Proof. Taking a:“ ´pnj´νj´α_
jpνzjq ` 1qand considering the floating dot placed on the
far right with subscript jand superscript ayields
dN˜a
j
ν¸“ p´1qα_
jpνzjq.
Thus, HpRbpνq, dNq – 0. 
Our goal for the rest of the section will be to prove the following:
Theorem 4.4. The dg-algebra pRbpmq, dNqis formal with homology
HpRbpmq, dNq – RN
ppmq.
4.1. Proof of Theorem 4.4.Denote 1pm,iq:“řjPIm1ji, or diagrammatically
1pm,iq:“ÿ
pj1,...,jmqPIm
j1jm
¨¨¨
i
.
It is an idempotent of Rbpm`1q. We also define 1pν,iq:“řjPSeqpνq1jifor νPX`. The
algebra Rbpmqacts on 1pm,iqRbpm`1qby first adding a vertical strand labeled iat the
right of DPRbpmqand then multiplying on the left in Rbpm`1q.
We now introduce some other diagrammatic notations as in [35,§3.1]. We draw Rbpmq
(viewed as Rbpmq-Rbpmq-bimodule) as a box labeled by m
Rbpmq “
...
...
m

2-VERMA MODULES 23
and bm:“ bRbpmqbecomes stacking boxes on top of each other. Moreover, when Rbpm`1q
is viewed as a left Rbpmq-module, as a right Rbpmq-module or as an Rbpmq-Rbpmq-bimodule,
we draw respectively
...
m`1...
m`1
...
...
m`1
Lemma 4.5. As a left Rbpmq-module, 1pm,iqRbpm`1qis free with decomposition
m`1
à
a“1à
ℓě0`Rbpmq1pm,iqτm¨¨¨τaxℓ
a‘Rbpmq1pm,iqτm¨¨¨τaθℓ
a˘,
where θℓ
a:“τa´1¨¨¨τ1ωxℓ
1τ1¨¨¨τa´1.
We draw this as
i
...
m`1–
m`1
à
a“1à
ℓě0
¨
˚
˚
˚
˚
˚
˝
...
...
m
ℓ
i
a
‘
...
...
m
ℓ
i
a
˛
‹
‹
‹
‹
‹
‚
Proof. By Theorem 3.15 we obtain
i
...
m`1–
m`1
à
a“1à
ℓě0
¨
˚
˚
˚
˚
˚
˝
...
...
mℓ
i
a
‘
...
...
mℓ
i
a
˛
‹
‹
‹
‹
‹
‚
We then apply Eq. (6) and Eq. (7) to bring all the dots to the desired position. It is a
triangular change of basis, concluding the proof. 
From now on, we will draw boxes with label ‘m, dN’ to denote the dg-algebra pRbpmq, dNq.
Similarly, a box with label Hpmqdenotes its homology HpRbpmq, dNq. Then, the decom-
position in Lemma 4.5 lifts directly to a direct sum decomposition of dg-modules whenever
iRIf. Otherwise, for iPIf, it lifts to the mapping cone
i
...
m`1, dN
–Cone ¨
˚
˚
˚
˚
˚
˝
m`1
à
a“1à
ℓě0
...
...
m, dN
ℓ
i
a
¯
dN
ÝÑ
m`1
à
a“1à
ℓě0
...
...
m, dN
ℓ
i
a
˛
‹
‹
‹
‹
‹
‚

24 GR ´
EGOIRE NAISSE AND PEDRO VAZ
where the map ¯
dNis induced by the differential of pRbpm`1q, dNq.
We will prove Theorem 4.4 using induction on the number of strands m. Therefore, we
can assume pRbpmq, dNqto be formal. Recall the following result of homological algebra:
Proposition 4.6. Let pA, dAqbe a dg-algebra, pM, dMqbe a right pA, dAq-module, and
pN, dNqa left one. If pM, dMqis formal and pN, dNqis cofibrant, then we have
H`pM, dMq bpA,dAqpN, dNq˘–H`HpM, dMq bpA,dAqpN, dNq˘.
Proof. Tensoring with a cofibrant dg-module preserves quasi-isomorphisms. 
We obtain an exact sequence
m`1
à
a“1à
ℓě0
...
...
Hpmq
ℓ
i
a
¯
dN
ÝÑ
m`1
à
a“1à
ℓě0
...
...
Hpmq
ℓ
i
a
ÝÑ i
...
Hpm`1qÑ0,(19)
thanks to Proposition 4.6.
Proposition 4.7. The exact sequence Eq. (19)is a short exact sequence, with ¯
dNbeing
injective.
Theorem 4.4 above is a direct consequence of Proposition 4.7. Therefore, we now focus
on proving this proposition. This is in fact similar to Kang–Kashiwara’s [19, Eq. (4.13)],
with basically only a change of basis, and thus we will follow the same ideas. We introduce
the equivalent of ‘ga’ from the reference and draw it as an undercrossing:
ij
:“
$
’
’
’
’
’
’
&
’
’
’
’
’
’
%
ij
if i‰j,
ri
i i
´ri
i i
´
i i
2´
i
2
i
`2
i i
if i“j.
In order to shorten out our diagrams, we introduce the convenient notation
i i
:“
i i
´
i i
It respects the relation
(20)
i i
“
i i

2-VERMA MODULES 25
We also have that
(21)
i i
“ri
i i
´
i i
Lemma 4.8. ([19, Lemma 4.12]) Undercrossings respect the following relations:
i j
“
i j i j
“
i j
i j k
“
i j k
for all i, j, k PI.
Still as in [19], in order to construct a nearly inverse for ¯
dN, we define the map
P:
m`1
à
a“1à
ℓě0
...
...
Hpmq
ℓ
i
a
ÝÑ
m`1
à
a“1à
ℓě0
...
...
Hpmq
ℓ
i
a
as multiplication on the left (or diagrammatically stacking above) with the element
r
θm`1:“
i
i...
...
...
Lemma 4.9. The map Pdefined above is a map of HpRbpmq, dNq-modules.
Proof. Crossings and dots slide over the upper part of the pm`1qth strand in r
θm`1at the
cost of adding diagrams with floating dots that are killed in HpRbpmq, dNq, and then slide
over the lower part thanks to Lemma 4.8. Tight floating dots with subscript jRIfalso
slide over r
θm`1thanks to Eq. (11). 
Lemma 4.10. The composition P˝¯
dNis given on HpRbpmq, dNq bm1pν,iqRbpm`1qby
multiplication by
(22) r2νi
i
2νi´α_
ipνq
ÿ
p“0
xni`p
m`1εi
ppxνq,
where εi
ppxνqis as in Eq. (13).

26 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Proof. The proof is similar to [19, Theorem 4.15]. We have
(23)
...
...
Hpmq
ℓ
i
P˝¯
dN
ÞÝÝÝÝÑ
...
...
Hpmq
ℓ`ni
i
We prove by induction on the number of strands mthat
. . .
ni
”r2νi
i
2νi´α_
ipνq
ÿ
p“0
xni`p
m`1εi
ppxνq,
where ”means equality up to adding elements killed in the quotient HpRbpmq, dNq –
RN
ppmq. If m“0, then it is trivial. Thus we suppose by induction that it holds for m´1.
We fix the label of the strands on the diagram above as ijwith j“j1¨¨¨jmPSeqpνq, and
we consider the different possible cases.
If jm‰i, then the result follows by applying Eq. (5) with Lemma 4.8, and using the
induction hypothesis.
If jm“i, we first observe that
(24)
ii
“ri
i i
Then we need to consider jm´1. If m“1, we have that
ii
ni
“ri
i
ni
i
´ri
i
ni
i
“ri
i
ni
i
´ri
i
ni`1
i
`r2
i
i
ni
i
´r2
i
i i
ni
Moreover, we observe that
i i
ni”0,

2-VERMA MODULES 27
which finishes the case m“1. For jm´1“i, we have
. . .
i ii
ni
“r2
i
i ii
ni
´ri
i ii
ni
using Eq. (24), Eq. (21) and Lemma 4.8. Using Eq. (20) followed by Lemma 4.8 and
Eq. (24) we obtain
r2
i
i ii
ni
“ri
i ii
ni
Keeping in mind Eq. (23), we have
i i i
”
i ii
P
i
Hpmq
by Eq. (8) and Eq. (7). This means we can apply the induction hypothesis to get
ri
i ii
ni
”r3
i
i ii
ni
Similarly, we have
ri
i ii
ni
“ri
i ii
ni
“
iii
ni
”r2
i
iii
ni
“r3
i
i ii
ni

28 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Putting these two results together and using Eq. (7), we obtain
i ii
ni
. . . ”r2
i
i ii
ni
. . .
which concludes this case.
For the final case jm´1“j‰i, we compute
jii
“ri
iji
“ri
iji
`r2
iÿ
t,v
stv
ij ÿ
u`ℓ“
t´1i
u
j
v
i
ℓ
using Eq. (8). Then we obtain for the first term on the RHS of the second equality, using
the induction hypothesis together with Eq. (5)
ri
iji
“
jii
”r2
i
iji
“r2
iÿ
t,v
stv
ij
ij
v
i
t
On the other hand, we have for all t, v that
ÿ
u`ℓ“
t´1i
u
j
v
i
ℓ“
i
t
j
v
i
´
ij
v
i
t
Putting these results together with the case jm‰iyields
jii
ni
”r2
i
jii
ni
which concludes the proof. 
Proof of Proposition 4.7.The polynomial Eq. (22) is monic (up to invertible scalar) with
leading terms xni`2νi´α_
ipνq
m`1. Therefore, multiplication by Eq. (22) yields an injective map.
Thus Lemma 4.10 tells us that P˝¯
dNis injective, and so is ¯
dN.
As a consequence, this also ends the proof of Theorem 4.4.

2-VERMA MODULES 29
5. Categorical action
For each iPIthere is a (non-unital) inclusion RbpmqãÑRbpm`1q1pm,iq, given by adding
a vertical strand with label ito the right of a diagram DPRbpmq:
j1j2... jm
DÞÑ
j1j2... jm
D
i
This gives rise to induction and restriction functors
Indm`i
m:Rbpmq-mod ÑRbpm`1q-mod,
Indm`i
mp´q – Rbpm`1q1pm,iqbm´,
Resm`i
m:Rbpm`1q-mod ÑRbpmq-mod,
Resm`i
mp´q – 1pm,iqRbpm`1q bm`1´.
which are adjoint.
We write
Rξi
bpνq:“Rbpνq b krξis – à
ℓě0
q2
iRbpνq,
with degqpξiq “ pαi|αiq. We will prove the following theorem in the next subsection:
Theorem 5.1. There is a short exact sequence
0Ñq´2
iRbpνq1pm´1,iqbm´11pm´1,iqRbpνq Ñ 1pν,iqRbpm`1q1pν,iq
ÑRξi
bpνq ‘ λ2
iq´2α_
ipνq
iRξi
bpνqr1s Ñ 0
of Rbpmq-Rbpmq-bimodules for all iPI. Moreover, there is an isomorphism
q´pαi|αjqRbpνq1pm´1,iqbm´11pm1,jqRbpνq – 1pν,jqRbpm`1q1pν,iq,
for all i‰jPI.
As we will see in the proof of Theorem 5.1, we can picture these facts as a short exact
sequence of diagrams
j
i
...
m
...
m
...
ãÑ
i
j
m`1
...
...
։à
ℓě0
i
ℓ
j
m
...
...
‘
i
ℓ
j
m
...
...
where the cokernel vanishes whenever i‰j. We write πfor the projection
π:
i
i
m`1
...
...
։à
ℓě0
i
ℓ
i
m
...
...

30 GR ´
EGOIRE NAISSE AND PEDRO VAZ
We write Idν:“Rbpνq bmp´q and we define
Fi:“à
mě0
Indm`i
m,Ei:“à
mě0à
|ν|“m
λ´1
iqα_
ipνq
iResm`i
mIdν`i.
These are exact functors thanks to Lemma 4.5. Define
(25) ‘rβi´α_
ipνqsqiIdν:“à
ℓě0
q1`2ℓ
i`λ´1
iqα_
ipνq
iIdν‘λiq´α_
ipνq
iIdνr1s˘.
It is a categorification of the fraction λiq´α_
ipνq
i´λ´1
iqα_
ipνq
i
qi´q´1
i
. We obtain:
Corollary 5.2. There is a natural short exact sequence
0ÑFiEiIdνÑEiFiIdνÑ ‘rβi´α_
ipνqsqiIdνÑ0,
for all iPIand there is a natural isomorphism
FiEj–EjFi,
for all i‰jPI.
Proposition 5.3. For each i, j PIthere is a natural isomorphism
tpdij `1q{2u
à
a“0„dij `1
2aqi
F2a
iFjFdij `1´2a
i–
tdij {2u
à
a“0„dij `1
2a`1qi
F2a`1
iFjFdij ´2a
i,
and in particular for pαi|αjq “ 0we have FiFj1ν–FjFi1ν.By adjunction, the same
isomorphism exists for the Ei,Ej.
Proof. Similarly as in the case of the usual KLR algebras, it follows from Eq. (7) and Eq. (8)
(the proof of [24, Proposition 6] we can be applied directly). 
5.1. Proof of Theorem 5.1.By symmetry along the horizontal axis, we obtain a decom-
position of Rbpm`1qas a right Rbpmq-module similar to the one of Lemma 4.5. Note that
the left and right decompositions are not compatible, and therefore we do not have a de-
composition as a Rbpmq-Rbpmq-bimodule. However, the surjection Rbpm`1q։q2ℓRbpmq
that projects on the summand Rbpmqxℓ
m`1, given by taking a“m`1 in Lemma 4.5, is a
(left-invertible) map of bimodules.
We define the map
πℓ
L: 1pν,iqRbpm`1q1pν,iq։λ2
iq2ℓ´2α_
ipνq
iRbpνqr1s,
as the projection map on the summand Rbpmqθℓ
n`1in the left decomposition of Rbpm`1q
as Rbpmq-module in Lemma 4.5. Similarly, let
πℓ
R: 1pν,iqRbpm`1q1pν,iq։λ2
iq2ℓ´2α_
ipνq
iRbpνqr1s,
be the projection map on θℓ
n`1Rbpmqin the right decomposition.
Lemma 5.4. We have
πℓ
Lpyq “ p´1qdeghpyqπℓ
Rpyq
for all yPRbpm`1q.

2-VERMA MODULES 31
Proof. We can suppose y“θℓ
m`1y1with y1PRbpmq. We want to prove that y“
p´1qdeghpyqy1θℓ
m`1`y0for some y0RRbpmqθℓ
m`1. For this, it is enough to show that
y1θm`1zy2“ p´1qdeghpzqy1zθn`1y2`z0where y1, y2PRbpmq,z0RRbpmqθℓ
m`1and zis any
generator of Rbpmq(i.e. crossing, dot or tight floating dot).
If z“xaand is on a strand labeled j‰i, then it slides freely over θm`1thanks to
Eq. (6). If the strand is labeled i, then we compute
i
ℓ
i
(7)
“
i
ℓ
i
`ri
i
ℓ
i
´r´1
iℓ
i
i
(8)
“
i
ℓ
i
`ri
i
ℓ
i
´r´1
i
ℓ
i
i
`R,
where the double strands represent multiple parallel strands (the number depending on m
and a), and Ris a sum of terms of the following form:
ℓ
and its mirror along the horizontal axis. Note that it is implicitly assumed that each of
these diagrams have the element y1at the top and y2at the bottom. Using Lemma 4.5,
we can rewrite the composition of the last three terms in the equation above with y2as
elements in ‘n
a“1‘pě0Rbpm´1qτmτm´1¨¨¨τaxp
aĆRbpmqθℓ
m`1. Hence they form the term
z0.
If z“τiis a crossing, then we obtain the desired property by Eq. (8), and applying a
similar reasoning as above.
Finally if z“ωand is at right of a strand labeled j‰i, it follows directly from Eq. (17).
Otherwise, if the strand is labeled i, we compute
i i
ℓp9,7q
“
i i
ℓ
(17)
“ ´
i i
ℓ
“ ´
i i
ℓ`r´1
iÿ
r`s“
ℓ´1i
r
i
s
Then for all r, s ě0 we compute using Eq. (7) again
i
r
i
s“
i
r
i
s

32 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Looking at these elements in the global picture yields
i
r
i
s
“
i
s
i
r
`R
which is an element not contained in Rbpmqθℓ
n`1for the same reasons as before. We see
that together they form the element z0, concluding the proof. 
We now have all the ingredients we need to prove Theorem 5.1.
Proof of Theorem 5.1.We first construct an injective map
(26) uij :q´pαi|αjqRbpνq1pm´1,iqbm´11pm1,jqRbpνqãÑ1pν,j qRbpm`1q1pν,iq
of Rbpmq-Rbpmq-bimodules, by setting (as in [19, Proposition 3.3])
uijpxbm´1yq:“xτmy.
In terms of diagrams, it consists of adding a crossing at the right
j
i
...
m
...
m
...
uij
ÞÝÝÝÑ
j
i
...
m
...
m
...
Then we construct a surjective map
1pν,iqRbpm`1q1pν,iq։Rξi
bpνq ‘ λ2
iq´2α_
ipνq
iRξi
bpνqr1s,
by projecting onto the direct summands Àℓě0xℓ
m`1Rbpmq ‘ θℓ
m`1Rbpmqof the decompo-
sition of Rbpm`1qas right Rbpmq-module. By Lemma 5.4 we know that this is a map of
Rbpmq-Rbpmq-bimodules. Finally, exactness follows directly from Lemma 4.5, since
Rbpνq1pm´1,iqbm´11pm1,jqRbpνq
–Rbpνq1pm´1,iqbm´1`m
à
a“1à
ℓě0
pRbpm´1q1pm,iqτm´1¨¨¨τaxℓ
a1j
‘Rbpm´1q1pm,iqτm´1¨¨¨τaθℓ
a1jq˘,
and so
uijpRbpνq1pm´1,iqbm´11pm1,jqRbpνqq
–
m
à
a“1à
ℓě0
pRbpm´1q1pm,iqτmτm´1¨¨¨τaxℓ
a1j
‘Rbpm´1q1pm,iqτmτm´1¨¨¨τaθℓ
a1jq.

2-VERMA MODULES 33
We remark that whenever i‰j, we have
à
ℓě0
1pν,jqxℓ
m`1Rbpmq1pν,iq‘1pν,jqRbpmqθℓ
m`11pν,iq“0,
and thus uij is an isomorphism, concluding the proof. 
5.2. Long exact sequence. We want to lift Theorem 5.1 to the dg-world of pRbpmq, dNq,
and study the long exact sequence that it induces. Therefore we define
yN:à
ℓě0
i
ℓm
...
...
Ñà
ℓě0
i
ℓ
m
...
...
as the Rbpmq-Rbpmq-bimodule map given by
yN
¨
˚
˚
˚
˚
˚
˝i
a...
...
. . .
˛
‹
‹
‹
‹
‹
‚
:“π¨
˚
˚
˚
˚
˚
˝i
ni`a...
...
...
˛
‹
‹
‹
‹
‹
‚
Pà
ℓě0
i
ℓ
m
...
...
whenever iPIf, and yN“0 for iRIf. Then we define
`Rξi
bpνq‘λ2
iq´2α_
ipνq
iRξi
bpνqr1s, dN˘
:“Cone ´pλ2
iq´2α_
ipνq
iRξi
bpνqr1s, dNqyn
ÝÑ pRξi
bpνq, dNq¯,
and
pRbpνq1pm´1,iqbm´11pm´1,iqRbpνq, dNq
:“ pRbpνq1pm´1,iq, dNq bpRbpm´1q,dNqp1pm´1,iqRbpνq, dNq.
Proposition 5.5. There is a short exact sequence of dg-bimodules
0Ñq´2
ipRbpνq1pm´1,iqbm´11pm´1,iqRbpνq, dNq Ñ p1pν,iqRbpm`1q1pν,iq, dNq
Ñ`Rξi
bpνq ‘ λ2
iq´2α_
ipνq
iRξi
bpνqr1s, dN˘Ñ0
for all iPI. Moreover, there is an isomorphism
q´pαi|αjqpRbpνq1pm´1,iqbm´11pm1,jqRbpνq, dNq – p1pν,j qRbpm`1q1pν,iq, dNq
for all i‰jPI.
Proof. It is a straightforward consequence of Theorem 5.1.
In order to understand the consequences of this short exact sequence in homology, we
need to compute the homology
H`Rξi
bpνq ‘ λ2
iq´2α_
ipνq
iRξi
bpνqr1s, dN˘,

34 GR ´
EGOIRE NAISSE AND PEDRO VAZ
for all iPIf.
Therefore, we want to compute the projection of the element
¯π¨
˚
˚
˚
˚
˚
˝i
p...
...
...
˛
‹
‹
‹
‹
‹
‚
Pà
ℓě0
i
ℓ
Hpmq
...
...
for all pěni. Note that we project on the homology of pRbpmq, dNq. This will ease some
of the computations we need to do. We write ¯πwhen we take the composite of πwith the
projection on the homology of pRbpmq, dNq. More precisely, ¯πis given by
¯π:“1bπ:
i
i
Hpmq
m`1
...
...
...
։à
ℓě0
i
ℓ
i
Hpmq
m
...
...
...
Similarly, we write ¯yN.
Lemma 5.6. If pě2νi, then
π¨
˚
˚
˚
˚
˝i
p. . .
. . .
. . .
ν
˛
‹
‹
‹
‹
‚
”ζ
i
p´α_
ipνq
. . . `
p´α_
ipνq´1
à
ℓ“0i
ℓ
m
. . .
. . .
for some invertible element ζPkˆ. If pă2νi, then
π¨
˚
˚
˚
˚
˝i
p...
...
...
ν
˛
‹
‹
‹
‹
‚
P
p´α_
ipνq
à
ℓ“0i
ℓ
m
. . .
. . .
Proof. The proof is an induction on m. If m“0, then it is trivial. Suppose the statement
holds for m´1. We fix the labels of the strands as the bottom as j“j1¨¨¨jmPSeqpνq.
If j1“i, then we compute
i i
p. . .
. . .
. . .
(7)
“riÿ
r`s
“p´1
i
r
i
s. . .
. . .
. . .
´r2
iÿ
r`s
“p´2
pr`1q
i
r
i
s...
...
...

2-VERMA MODULES 35
Then using Eq. (8) we have
ii
. . .
. . .
. . .
“
i i
. . .
. . .
. . .
`ÿ
jk‰i
stv
ijkÿ
t,v ÿ
r`s“
t´1
i
rv
jki
s
so that, since sădijk, we obtain by the induction hypothesis
π¨
˚
˚
˚
˚
˚
˝ii
. . .
. . .
. . .
˛
‹
‹
‹
‹
‹
‚
P
p´α_
ipνq´1
à
ℓ“0
ii
ℓ
m
. . .
. . .
Moreover, still by the induction hypothesis, we have
ÿ
r`s
“p´3
pr`2qπ¨
˚
˚
˚
˚
˝i
r`1
i
s. . .
. . .
. . .
˛
‹
‹
‹
‹
‚
P
p´α_
ipνq´1
à
ℓ“0
ii
ℓ
m
...
...
Finally, if pě2νi, by the induction hypothesis we get for s“p´2,
π¨
˚
˚
˚
˚
˝i i
s. . .
. . .
. . .
˛
‹
‹
‹
‹
‚
”ζ1
ii
s´α_
ipν´iq
. . . `
s´α_
ipν´iq´1
à
ℓ“0
ii
ℓ
m
. . .
. . .
which concludes the case by observing that s´α_
ipν´iq “ p´α_
ipνq, and taking ζ“r2
iζ1.
If pă2νithe claim is immediate by the induction hypothesis.
For the case j1“j‰i, we use Eq. (5) and then the induction hypothesis to get
π¨
˚
˚
˚
˚
˚
˝ji
p. . .
. . .
. . .
˛
‹
‹
‹
‹
‹
‚
“ÿ
t,v
stv
ij π¨
˚
˚
˚
˚
˚
˝j
v
i
t`p...
...
...
˛
‹
‹
‹
‹
‹
‚
”tij
ji
dij `p. . .
. . .
. . .
`
p´α_
ipνq´1
à
ℓ“0
ji
ℓ
m
...
...

36 GR ´
EGOIRE NAISSE AND PEDRO VAZ
where we recall that sdij 0
ij “tij . We conclude by applying the induction hypothesis, ob-
serving that dij `p´α_
ipν´jq “ p´α_
ipνq.
Consider also the following result, which is akin to [19, Lemma 5.4].
Lemma 5.7. We have for kăk1and t“k1´k,
¯yN
¨
˚
˚
˚
˚
˚
˝i
k1. . .
. . .
. . .
˛
‹
‹
‹
‹
‹
‚
”
ii
t
¯yNpξk
iq
. . .
. . .
`
t´1
ÿ
ℓ“0
ii
ℓ
Hpmq
. . .
. . .
where
ii
¯yNpξk
iq
. . .
. . .
“¯yN
¨
˚
˚
˚
˚
˚
˝i
k. . .
. . .
. . .
˛
‹
‹
‹
‹
‹
‚
Proof. First we observe that
i
ni`k1. . .
. . .
. . .
“
i
ni`k
t
. . .
. . .
. . .
P
i
i
Hpmq
m`1
. . .
. . .
. . .
using Eq. (7) and Eq. (6), and the fact that nidots on the left strand is annihilated in
HpRbpmq, dNq.
Then using Lemma 4.5 we obtain
(27)
i
ni`k...
...
...
“
ii
ψk
ϕk
...
...
...
`
ii
¯yNpξk
iq
. . .
. . .
for some ϕk, ψkPRbpmq. We conclude by observing that
(28)
i
t
i
ϕk
ψk
. . .
. . .
. . .
“
i
t
i
ϕk
ψk
. . .
. . .
. . .
´riÿ
r`s
“t´1
i
r
i
s
ϕk
ψk
...
...
...
thanks to Eq. (7). 

2-VERMA MODULES 37
Proposition 5.8. Putting ρi:“ni´α_
ipνq, we have
¯yN
¨
˚
˚
˚
˚
˚
˝i
k. . .
. . .
. . .
ν
˛
‹
‹
‹
‹
‹
‚
”ζ
ii
k`ρi
. . . `
k`ρi´1
à
ℓ“0
ii
ℓ
Hpmq
. . .
. . .
which is 0whenever k`ρiă0, and where ζPkˆ.
Proof. If niě2νi, then the result follows from Lemma 5.6. Otherwise, we take k1“2νi´ni
and the result follows from Lemma 5.6 for kěk1. Suppose kăk1and put t“k1´k.
Then by Lemma 5.7 we obtain
¯yN
¨
˚
˚
˚
˚
˚
˝i
k1. . .
. . .
. . .
˛
‹
‹
‹
‹
‹
‚
”
ii
t
¯yNpξk
iq
...
...
`
t´1
ÿ
ℓ“0
ii
ℓ
Hpmq
. . .
. . .
Therefore, we have
ii
t
¯yNpξk
iq
. . .
. . .
”ζ
ii
k1`ρi
. . . `
maxpk1`ρi,tq´1
à
ℓ“0
ii
ℓ
Hpmq
...
...
From this, we deduce
ii
¯yNpξk
iq
...
...
”ζ
ii
k`ρi
. . . `
k`ρi´1
à
ℓ“0
ii
ℓ
Hpmq
...
...
which concludes the proof. 
We now have all the tools we need to compute the homology of the cokernel of the short
exact sequence of Proposition 5.5.
Proposition 5.9. There is an isomorphism of RN
ppνq-RN
ppνq-bimodules
H`Rξi
bpνq ‘ λ2
iq´2α_
ipνq
iRξi
bpνqr1s, dN˘
–#Àρi´1
ℓ“0q2ℓ
iRN
ppνq,if ρiě0,
λ2
iq´2α_
ipνq
iÀ´ρi´1
ℓ“0q2ℓ
iRN
ppνqr1s,if ρiď0,
where ρi“ni´α_
ipνq.

38 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Proof. First suppose ρiě0. Then Proposition 5.8 tells us that ¯yNpξk
iqis a monic poly-
nomial (up to invertible scalar) with leading terms ξk`ρi
i. This gives us the first case. If
ρiď0, then we have ¯yNpξk
iq “ 0 for kă ´ρi. Moreover, ζ´1¯yNpξ´ρi
iq “ 1, and in general
¯yNpξk
iqis a monic polynomial with leading term ξk`ρi
ifor ką ´ρi. This concludes the
proof. 
5.3. Strongly projective dg-modules. The following notions were originally introduced
by Moore [31]. We use the presentation given in [40], which is best suited for our notations.
Definition 5.10 ([40, Definition 8.5]).Let pR, 0qbe a ring Rviewed as a dg-Z-algebra
concentrated in degree zero. An pR, 0q-module pQ, dQqis strongly projective if HpQ, dQq
and im dQare both projective R-modules.
Lemma 5.11 ([43, Theorem 9.3.2]).Let pP, dPqbe a strongly projective right pR, 0q-module
and pN, dNqany left pR, 0q-module, then
H`pP, dPq bpR,0qpN, dNq˘–HpP, dPq bRHpN, dNq.
Definition 5.12 ([40, Definition 8.17]).Let pA, dAqbe a dg-R-algebra. A left (resp. right)
pA, dAq-module pP, dPqis strongly projective if it is a dg-direct summand of pA, dAq bpR,0q
pQ, dQq(resp. pQ, dQq bpR,0qpA, dAq) for some strongly projective pR, 0q-module pQ, dQq.
Proposition 5.13 ([40, Lemma 8.23]).If pP, dPqis a strongly projective right pA, dAq-
module and pN, dNqis any left pA, dAq-module, then
H`pP, dPq bpA,dAqpN, dNq˘–HpP, dPq bHpA,dAqHpN, dNq.
Note that if pP, dPqis a strongly projective pA, dAq-module, then HpP, dPqis a projective
HpA, dAq-module. Indeed, we can assume pP, dPq “ pA, dAq bpR,0qpQ, dQq, and we have
HpP, dPq – HpA, dAq bRHpQ, dQq.
Since HpQ, dQqis a projective R-module, it is a direct summand of a free R-module F.
Therefore HpP, dPqis a direct summand of HpA, dAqbRF, which is a free HpA, dAq-module.
Remark 5.14. This result does not hold in general. As a counterexample we can take
pA, dq “ pQrx, ys,0qand consider the dg-module pX, dXqgiven by the mapping cone
ConepQrx, ysx´y
ÝÝÑ Qrx, ysq. In this case we have that HpX, dXq – Qrx, ys{px´yqbut
HppX, dXq bpA,dqpX, dXqq – Qrx, ys ‘ Qrx, ysr1s.
5.3.1. Strong projectivity of Rbpm`1q.Our next goal is to show the following:
Proposition 5.15. The pRbpmq, dNq-module p1pm,iqRbpm`1q, dNqis strongly projective.
It is obvious for iRIfby Lemma 4.5, and thus we can assume iPIf. We first construct
the mapping cone
pQ,dQq:“
Cone`m`1
à
a“1à
ℓě0
Rgpmq1pν,iqτm¨¨¨τaθℓ
a
dQ
ÝÑ
m`1
à
a“1à
ℓě0
Rgpmq1pν,iqτm¨¨¨τaxℓ
a˘,

2-VERMA MODULES 39
where we think of τm¨¨¨τaθℓ
aas a formal symbol that represents a degree shift corresponding
to the degree of the element 1pν,iqτm¨¨¨τaθℓ
ain Rbpm`1q. The map dQis given by first
embedding Rgpmqinto Rbpm`1qthrough the diagrams
Rgpmq1pν,iqτm¨¨¨τaθℓ
aãÑ
...
...
m
ℓ
i
a
then applying dNof pRbpm`1q, dNq, then decomposing the image in the left-decomposition
Àm`1
a“1Àℓě0Rbpmq1pm,iqτm¨¨¨τaxℓ
a, and finally projecting unto the part in homogical degree
zero of Rbpmq, which is trivially isomorphic to Rgpmq. Moreover, pRbpmq, dNqis a (right)
module over pRg,0qwhich acts by gluing KLR diagrams on the bottom. Then we have, as
pRbpmq, dNq-modules
pRbpm`1q, dNq – pRbpmq, dNq bpRgpmq,0qpQ, dQq.
Therefore, we want to show that pQ, dQqis strongly projective as pRgpmq,0q-module. We
write
Q1rξis:“
m`1
à
a“1à
ℓě0
Rgpmq1pν,iqτm¨¨¨τaθℓ
a,
Q0rξis:“
m`1
à
a“1à
ℓě0
Rgpmq1pν,iqτm¨¨¨τaxℓ
a,
where we identify ξiwith xain Q0, and ξℓ
iwith xℓ
1in θℓ
a. Note that dQis not krξis-linear.
Lemma 5.16. The map
dQ:Q1rξis Ñ Q0rξis
defined above is injective.
Proof. Recall the map Pof Lemma 4.9 given by multiplication by r
θm`1. Since floating
dots are also annihilated in Rgpmq, multiplication by r
θm`1also defines a map
P1:Q0rξis Ñ Q1rξis.(29)
We reconsider the proof of Lemma 4.10 to show that P1˝dQis injective. First, we introduce
an order on the summands of Q1rξis “ Àm`1
a“1Àℓě0Rgpmq1pν,iqτm¨¨¨τaθℓ
aby declaring that
Rgpmq1pν,iqτm¨¨¨τaθℓ
aăRgpmq1pν,iqτm¨¨¨τaθℓ1
a,
Rgpmq1pν,iqτm¨¨¨τaθℓ
aăRgpmq1pν,iqτm¨¨¨τa1θℓ2
a1,
for all aąa1,ℓăℓ1, and for all ℓ2. In other words, if there are more crossings under
the floating dot, then the term is smaller. If there is the same amount of crossings, then
we consider the amount of dots at the left of the floating dot, and lesser dots meaning a
smaller term.

40 GR ´
EGOIRE NAISSE AND PEDRO VAZ
We claim that if ZPRgpmq1pν,iqτm¨¨¨τaθℓ
athen
P1˝dQpZq “ r2νi
i
2νi´α_
ipνq
ÿ
p“0
Zxni`p
m`1εi
ppxνq ` H,
where HăZxℓ`ni`2νi´α_
ipνq
m`1. This implies that P1˝dQis in echelon form (with pivot being
invertible scalars), and thus is injective. By consequence, so is dQ.
In order to prove our claim, we need to tweak the proof of Lemma 4.10. We need to
keep track of the terms that are annihilated when working over the cyclotomic quotient,
and show these appear as lower terms in the order defined above. The case jm‰iremains
the same. The case jm“iand m“1 becomes
ii
p
“r2
i
i
p
i
´r2
i
i
p
i
`ri
i
p
i
´ri
i
p`1
i
where p“ni`ℓ. The first term is the leading term. The second term possesses less dots
on the left of the floating dot, and so it is smaller. If a“0, then the last two terms possess
one more crossing at the bottom of the floating dot, and therefore they are smaller. If
a“1, then they are annihilated by Eq. (5). Finally the two remaining cases jm´1‰iand
jm´1follow from the same arguments as in the proof of Lemma 4.10, with the lower terms
in the induction hypothesis only adding lower terms because:
ii
“
ii
by (8), and,
i i
“0,
by Eq. (8) and Eq. (5). This concludes the proof of the claim, and therefore of the
proposition. 
Proof of Proposition 5.15.The proof is a revisit of the proof of [19, Lemma 4.18] that
applies to our particular case.
Recall the map P1from Eq. (29). We know that P1˝dQis given by multiplying by a
monic polynomial with leading term xni`2νi´α_
ipνq
m`1plus some remaining map giving lower
terms. In particular, it is injective and we have a short exact sequence
0ÑQ1rξisP1˝dQ
ÝÝÝÑ Q1rξis Ñ cokpP1˝dQq Ñ 0.

2-VERMA MODULES 41
Moreover, since P1˝dQis in echelon form, it means that cokpP1˝dQqis a projective Rgpmq-
module. Thus the sequence splits as Rgpmq-modules with splitting map σ:Q1rξis Ñ Q1rξis,
and we get σ˝P1˝dQ“IdQ1rξis. Then, the short exact sequence
0Q1rξisQ0rξisHpQ, dQq,0,
dQ
σ˝P1
obtained thanks to Lemma 5.16 splits with splitting map given by σ˝P1. Since Q0rξisis
a projective Rgpmq-module, so is HpQ, dQq. Finally, dQpQ1rξisq is also projective since dQ
is injective and Q1rξisis projective. 
5.4. Functors. We define for all iPIthe functors
FN
ip´q :“à
mě0
RN
ppm`1q1pm,iqbRN
ppmqp´q,
EN
ip´q :“à
mě0à
|ν|“m
λ´1
iqα_
ipνq
i1pν,iqRN
ppm`1q bRN
ppm`1qp´q,
where we interpret λi“qniwhenever iPIf. Thanks to Proposition 5.15, these are exact.
For nPN, we write
‘rnsqiIdν:“
n´1
à
ℓ“0
q1´n`2ℓ
iIdν,
for the finite direct sum that categorifies rnsqi.
Theorem 5.17. For iRIfthere is a natural short exact sequence
(30) 0 ÑFN
iEN
iIdνÑEN
iFN
iIdνÑ ‘rβi´α_
ipνqsqiIdνÑ0,
and for iPIfthere are natural isomorphisms
(31) EN
iFN
iIdν–FN
iEN
iIdν‘rni´α_
ipνqsqiIdν,if ni´α_
ipνq ě 0,
FN
iEN
iIdν–EN
iFN
iIdν‘rα_
ipνq´nisqiIdν,if ni´α_
ipνq ď 0.
Moreover, there is a natural isomorphism
(32) FN
iEN
j–EN
jFN
i,
for i‰jPI.
Proof. The short exact sequence Eq. (30) and the isomorphism Eq. (32) are immediate
consequences of Proposition 5.5 and Proposition 5.15. For the isomorphisms Eq. (31),
Proposition 5.5 and Proposition 5.15 give a long exact sequence of RN
ppνq-RN
ppνq-bimodules.
By Proposition 5.9 it truncates to a short exact sequence
0ÑFN
iEN
iIdνÑEN
iFN
iIdνÑ ‘rρisIdνÑ0,
if ρi“ni´α_
ipνq ě 0, and a short exact sequence
0Ñ ‘r´ρisIdνÑFN
iEN
iIdνÑEN
iFN
iIdνÑ0,

42 GR ´
EGOIRE NAISSE AND PEDRO VAZ
if ρi“ni´α_
ipνq ď 0. In the first case, we can identify
‘rρisIdν–q1´ρi
i
ρi´1
à
ℓ“0
ii
ℓ
ν
...
...
and the map EN
iFN
iIdνÑ ‘rρisqiIdνis induced by the projection π. Thus the sequence
splits with the splitting map ‘rρisqiIdνÑEN
iFN
iIdν,given by the sum of maps RN
ppνqξℓÑ
RN
ppν`iqthat add a vertical strand labeled icarrying ℓdots at the right of a diagram in
RN
ppνq. In the second case, we also identify
‘r´ρisqiIdν–q1`ρi
i
´ρi´1
à
ℓ“0
ii
ℓ
ν
. . .
. . .
Moreover the map ‘r´ρisIdνÑFN
iEN
iIdνis induced by the connecting homorphism δ.
Using the notations of Eq. (27) it takes the form
δ¨
˚
˚
˚
˚
˚
˝ii
k
. . . ˛
‹
‹
‹
‹
‹
‚
“u´1
ij
¨
˚
˚
˚
˚
˚
˝i
k. . .
. . .
. . .
˛
‹
‹
‹
‹
‹
‚
“
i
ϕk
ψk
...
where uij is the monomorphism defined in Eq. (26), and 0 ďkă ´ρi. We also note
that Eq. (28) tells us that
(33)
i
ϕk`t
ψk`t
... “
i
t
ϕk
ψk
...
Moreover since ¯yNpξ´ρi
iq “ ζand ¯yNpξℓ
iq “ 0 for ℓă ´ρi, we obtain by Eq. (28) again that
(34)
i
ϕk
ψk
...
i
“#´r´1
iζ, if k“ ´ρi´1,
0,if kă ´ρi´1.

2-VERMA MODULES 43
As in [19, Proof of Theorem 5.2], we construct a map Φ: FN
iEN
iIdνÑ ‘r´ρisqiIdνinduced
by the morphism of bimodules
Φ :
i
x
y
...
...
... ÞÑ ÿ
r`s“
´ρi´1
i
r
i
s
x
y
...
...
...
for all x, y PRN
ppνq.
Then we compute
Φ˝δ¨
˚
˚
˚
˚
˚
˝ii
k
. . . ˛
‹
‹
‹
‹
‹
‚
(33)
“ÿ
r`s“
´ρi´1
ii
s
ϕk`r
ψk`r
. . .
. . .
. . .
(34)
“ ´r´1
iζ
ii
k
. . . `
´ρi´1
ÿ
r“´ρi´k
ii
´ρi´1´r
ϕk`r
ψk`r
. . .
. . .
. . .
Therefore, Φ ˝δis given by a triangular matrix with invertible elements on the diagonal,
and thus is an isomorphism. In particular, δis left invertible, concluding the proof. 
Corollary 5.18. For iPIf, then 1νEiand Fi1νare biadjoint (up to shift).
Proof. By the results in [6], we know the splitting map EN
iFN
iIdνÑFN
iEN
iIdνof Theo-
rem 5.17 together with the unit and counit of the adjunction Fi%Eiallow to construct a
unit and counit for the adjunction Ei%Fi.
Proposition 5.19. For each i, j PIthere is a natural isomorphism
tpdij `1q{2u
à
a“0„dij `1
2aqi
pFN
iq2aFN
jpFN
iqdij `1´2a
–
tdij {2u
à
a“0„dij `1
2a`1qi
pFN
iq2a`1FN
jpFN
iqdij ´2a.
By adjunction, the same isomorphism exists for the EN
i,EN
j.
Proof. This follows from Proposition 5.3.
In particular, there is a strong 2-action of the 2-Kac–Moody algebra of [23,38] associated
to xEi, Fi, KiyiPIfon ‘νPX`RN
ppνq-mod through FN
i,EN
i.

44 GR ´
EGOIRE NAISSE AND PEDRO VAZ
5.5. A differential on RN
p.We fix a subset IfĂI1
fĂIand consider the parabolic
subalgebras Uqppq Ă Uqpp1q Ă Uqpgq. For each jPI1
fzIfwe choose a weight n1
jPN. For
jPIfwe take n1
j:“njPN, and we write N1:“ tn1
jujPI1
f. Then we equip the cyclotomic
p-KLR algebra RN
ppmqwith a differential dN
N1which is zero on dots and crossings and
dN
N1¨
˚
˚
˚
˝j
j i1
...
im´1
˛
‹
‹
‹
‚:“$
’
’
’
&
’
’
’
%
0,if jRI1
f,
p´1qnj
j
nj
i1
...
im´1
,if jPI1
fzIf.
Theorem 5.20. The dg-algebra pRN
ppmq, dN
N1qis formal with homology
HpRN
ppmq, dN
N1q – RN1
p1pmq.
Proof. We have RN
ppmq – HpRbpmq, dNqand RN1
p1pmq – HpRbpmq, dN1qby Theorem 4.4.
Moreover, dN
N1can be lifted to Rbpmq. We split the homological grading of Rbpmqin three:
a first one that counts the amount of floating dots with subscript in If, a second one for the
floating dots with subscript in I1
fzIf, and a third one for IzI1
fthat we ignore for the moment.
Then, we have that dN
N1has degree p0,´1qand dNhas degree p´1,0q, and they commute
with each other. Thus we have a (bounded) double complex pRb, dN, dN
N1qwith total com-
plex being pRb, dN1q, since dN1“dN`dN
N1. In particular, there is a spectral sequence from
HpRN
ppmq, dN
N1qto HpRb, dN1q – RN1
p1pmq. Now, Theorem 4.4 tells us that HpRb, dNqis con-
centrated in homological degree zero (for the first homological grading). Thus, the spectral
sequence converges at the second page, and in particular HpRN
ppmq, dN
N1q – RN1
p1pmq.
We interpret this result as a categorical version of the fact that if there is an arrow from a
parabolic Verma module MppΛ, Nqto Mp1pΛ1, N 1q(see Section 2.2), then there is a surjec-
tion MppΛ, Nq։Mp1pΛ1, N 1q. Indeed, in that case there is a surjective quasi-isomorphism
pRN
p, dN1q»
ÝÑ pRN1
p1,0q, inducing equivalences of derived categories that commute up to
quasi-isomorphism with the categorical actions of Uqpgq.
6. The categorification theorems
Recall that the k-algebra of formal Laurent series kppx1,...,xnqq (as constructed in [3],
see also [32,§5]) is given by first choosing a total additive order ăon Zn. One says that
a cone C:“ tα1v1` ¨ ¨ ¨ ` αnvn|αiPRě0u Ă Rnis compatible with ăwhenever 0 ăvifor
all iP t1,...,nu. Then we set
kppx1,...,xnqq :“ď
ePZn
xekăJx1,...,xnK,
where kăJx1,...,xnKconsists of formal Laurent series in kJx1,...,xnKsuch that the terms
are contained in a cone compatible with ă. It forms a ring when we equip kppx1,...,xnqq
with the usual addition and multiplication of series. Requiring that all series are contained
in cones compatible with ăensures that the product of two elements in kppx1,...,xnqq

2-VERMA MODULES 45
is well-defined. Indeed, under these conditions, any coefficient in the product can be
determined by summing only a finite amount of terms.
6.1. C.b.l.f. derived category. We fix an arbitrary additive total order ăon Zn. We
say that a Zn-graded k-vector space M“ÀÀgPZnMgis c.b.l.f. (cone bounded, locally
finite) dimensional if
‚dim Mgă 8 for all gPZn;
‚there exists a cone CMĂRncompatible with ăand ePZnsuch that Mg“0
whenever g´eRCM.
In other words, Mis c.b.l.f. dimensional if and only if
gdimqM:“ÿ
gPZn
xgdimpMgq P xekăJx1,...,xnK.
Let pA, dqbe a Zn-graded dg-k-algebra, where A“Àph,gqPZˆZnAh
g, and dpAh
gq Ă Ah´1
g.
Suppose that pA, dqis concentrated in non-negative homological degrees, that is Ah
g“0
whenever hă0. Let
D
pA, dqbe the derived category of pA, dq. Let
D
lf pA, dqbe the full
triangulated subcategory of
D
pA, dqconsisting of pA, dq-modules having homology being
c.b.l.f. dimensional for the Zn-grading. We call
D
lf pA, dqthe c.b.l.f. derived category of
pA, dq.
As in [32], we say that pA, dqis a positive c.b.l.f. dimensional dg-algebra if
(1) Ac.b.l.f.dimensional for the Zn-grading;
(2) Ais non-negative for the homological grading;
(3) A0
0is semi-simple;
(4) Ah
0“0 for hą0;
(5) pA, dqdecomposes a direct sum of shifted copies of relatively projective modules
Pi:“Aeifor some idempotent eiPA, such that Piis non-negative for the Zn-
grading.
Remark 6.1. As explained in [32, Remark 9.5], condition (4.) cannot be respected when-
ever Pi:“Aeiis acyclic. However, in this case there is a quasi-isomorphism pA, dq»
ÝÑ
pA{AeiA, dqand we can weaken hypothesis (4.) so that it is respected only after removing
all acyclic Pi. This is the case of pRb, dNq.
6.1.1. Asymptotic Grothendieck group. As already observed in [1] (see also [34, Appendix]),
one caveat of the usual definition of the Grothendieck group is that it does not allow to
take into consideration infinite iterated extensions of objects. We need to introduce new
relations in the Grothendieck groups to handle such situations. One solution is to use
asymptotic Grothendieck groups, as introduced by the first author in [32].
Let
C
be a triangulated subcategory of some triangulated category
T
. Suppose
T
admits
countable products and coproducts, and these preserve distinguished triangles. Let K∆
0p
C
q
be the triangulated Grothendieck group of
C
.

46 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Recall the Milnor colimit MColimrě0pfrqof a collection of arrows tXr
fr
ÝÑ Xr`1urPNin
T
is the mapping cone fitting inside the following distinguished triangle
ž
rPN
Xr
1´f‚
ÝÝÝÑ ž
rPN
XrÑMColimrě0pfrq Ñ
where the left arrow is given by the infinite matrix
1´f‚:“¨
˚
˚
˝
1 0 0 0 ¨¨¨
´f01 0 0 ¨¨¨
0´f11 0 ¨¨¨
.
.
.............
˛
‹
‹
‚
There is a dual notion of Milnor limit. Consider a collection of arrows tXr`1
fr
ÝÑ Xrurě0
in
T
. The Milnor limit is the object fitting inside the distinguished triangle
MLimrě0pfrq Ñ ź
rě0
Xr
1´f‚
ÝÝÝÑ ź
rě0
XrÑ
Definition 6.2. The asymptotic triangulated Grothendieck group of
C
Ă
T
is given by
K∆
0p
C
q:“K∆
0p
C
q{Tp
C
q,
where Tp
C
qis generated by
rYs ´ rXs “ ÿ
rě0
rErs
whenever both Àrě0Conepfrq P
C
and Àrě0ErP
C
, and
Y–MColim`X“F0
f0
ÝÑ F1
f1
ÝÑ ¨ ¨ ¨ ˘,
is a Milnor colimit, or
X–MLim`¨¨¨ f1
ÝÑ F1
f0
ÝÑ F0“Y˘,
is a Milnor limit, and
rErs “ rConepfrqs P K∆
0p
C
q,
for all rě0.
In a Zn-graded triangulated category
T
, we define the notion of c.b.l.f. direct sum as
follows:
‚take a a finite collection of objects tK1,...,Kmuin
T
;
‚consider a direct sum of the form
à
gPZn
xgpK1,g‘ ¨ ¨ ¨ ‘ Km,gq,with Ki,g“
ki,g
à
j“1
Kirhi,j,gs,
where ki,gPNand hi,j,gPZsuch that:
‚there exists a cone Ccompatible with ă, and ePZnsuch that for all jwe have
kj,g“0 whenever g´eRC;
‚there exists hPZsuch that hi,j,gěhfor all i, j, g.

2-VERMA MODULES 47
If
T
admits arbitrary c.b.l.f. direct sums, then K0p
T
qhas a natural structure of
Zppx1,...,xnqq-module with ÿ
gPC
agxe`grXs:“ rà
gPC
xg`eX‘ags,
where X‘ag“À|ag|
ℓ“1Xrαgsand αg“0 if agě0 and αg“1 if agă0.
Theorem 6.3 ([32, Theorem 9.15]).Let pA, dqbe a positive c.b.l.f. dg-algebra, and let
tPjujPJbe a complete set of indecomposable relatively projective pA, dq-modules that are
pairwise non-isomorphic (even up to degree shift). Let tSjujPJbe the set of corresponding
simple modules. There is an isomorphism
K∆
0`
D
lf pA, dq˘–à
jPJ
Zppx1,...,xℓqqrPjs,
and K∆
0`
D
lf pA, dq˘is also freely generated by the classes of trSjsujPJ.
Proposition 6.4 ([32, Proposition 9.18]).Let pA, dqand pA1, d1qbe two c.b.l.f. positive
dg-algebras. Let Bbe a c.b.l.f. dimensional pA1, d1q-pA, dq-bimodule. The derived tensor
product functor
F:
D
lf pA, dq Ñ
D
lf pA1, d1q, F pXq:“BbL
pA,dqX,
induces a continuous map
rFs:K∆
0p
D
lf pA, dqq Ñ K∆
0p
D
lf pA1, d1qq.
We will need the following definitions in Section 7:
Definition 6.5. Let tK1,...,Kmube a finite collection of objects in
C
, and let tErurPN
be a family of direct sums of tK1,...,Kmusuch that ÀrPNEris a c.b.l.f. direct sum of
tK1,...,Kmu. Let tMrurPNbe a collection of objects in
C
with M0“0, such that they fit
in distinguished triangles
Mr
fr
ÝÑ Mr`1ÑErÑ
Then we say that an object MP
C
such that M–
T
MColimrě0pfrqin
T
is a c.b.l.f.
iterated extension of tK1, . . . , Kmu.
Definition 6.6. We say that
V
is c.b.l.f. generated by tXjujPJfor some collection of
elements XjP
V
if for any object Yin
V
we can take a finite set tYkukPKof retracts
YkĂXjksuch that Yis isomorphic to a c.b.l.f. iterated extension of tYkukPK.
6.2. Categorification. In this section we assume that Rbpνqis a k-algebra over a field k.
We also choose an abritrary order ăfor constructing Zppq, Λqq such that 0 ăqăλifor all
formal λi“qβiPΛ. We assume that the parabolic Verma module Mppλ, N qis constructed
over the ground ring R:“QppΛ, qqq (instead of QpΛ, qq).
Every idempotent of Rbpνqis the image of an idempotent of the classical KLR algebra
Rgpνqunder the obvious inclusion RgpνqãÑRbpνq. Thanks to [22, Section 2.5] we know
all the idempotents of Rgpνq. We define the element
ei,n :“τϑnxn´1
1xn´2
2¨¨¨xn´11ii¨¨¨iPRgpnq,

48 GR ´
EGOIRE NAISSE AND PEDRO VAZ
where ϑnis the longest element in Sn. Let Seqdpνqbe the set of expressions ipm1q
1ipm2q
2¨¨¨ipmrq
r
for different rPNand mℓPNsuch that řr
ℓ“1mℓ¨αiℓ“ν. For each iPSeqdpνqwe define
the idempotent
ei:“ei1,m1bei2,m2b ¨ ¨ ¨ b eir,mrPRgpνq,
where xbymeans we put the diagram of xat the left of the one of y. Identifying eiwith
its image in Rbpνq, as in [22], we define a projective left Rbpνq-module
Pi:“Rbpνqei,
Then we put
xiy:“ ´
r
ÿ
ℓ“1
mℓpmℓ´1q
2dia.
and we define ˜
Pi:“q´xiyPi.
When writing ...i... and ...j... we mean we take two sequences i1ii2and i1ji2
in Seqdpνqthat coincide everywhere except on iand j. From the decomposition of the
nilHecke algebra [22,§2.2] we get an isomorphism of RN
p-modules
˜
P...im... – ‘prmsqi!q˜
P...ipmq....
Mimicking the arguments in [22, Proposition 2.13] and [24, Proposition 6] we have the
following:
Proposition 6.7. There are isomorphisms
tpdij `1q{2u
à
a“0
˜
P...ip2aqjipdij `1´2aq... –
tdij {2u
à
a“0
˜
P...ip2a`1qjipdij ´2aq...
for all i‰jPI.
Equipping Rbpνqwith dNinduces a differential on ˜
Pi, and Proposition 6.7 holds for the
dg-version p˜
Pi, dNq. We put
M
ppΛ, Nq:“à
mě0
D
lf pRbpmq, dNq,
with the particular case
M
pΛqmeaning p“band N“ H, and therefore dN“0. Note
that
D
lf pRbpmq, dNq –
D
lf pRN
ppmq,0q.
Proposition 6.8. There is an isomorphism of Qppq, Λqq-modules
U´
qpgq bQpqqQppq, Λqq – K∆
0p
M
pΛqq bZQ,
and a Qppq, Λqq-linear surjection
U´
qpgq bQpqqQppq, Λqq ։K∆
0p
M
ppΛ, Nqq bZQ,
both sending Fpm1q
i1Fpm2q
i2¨¨¨Fpmrq
irto rp ˜
Pi, dNqs for i“ipm1q
1ipm2q
2¨¨¨ipmrq
r.
Proof. Since projective modules of Rbpνqare in bijection with the ones of the classical KLR
algebra Rgpνqand respect the categorified Serre relations (see Proposition 6.7), both claims
are a direct consequence of the main results in [22,24], together with Theorem 6.3.

2-VERMA MODULES 49
Consider the subring Ppνqof Rbpνqconsisting of dots on vertical strands (without float-
ing dots). It admits an action of the symmetric group permuting the strands (with labels)
and dots on them (not to be confused with the action of Smon Pνfrom Section 3.2). We
write Sympνq:“PpνqSmfor the subring of invariants under this action. Clearly it lies in
the center of Rbpνqbut this inclusion is strict (see [35] or [4] for a study of the center in
the case of sl2).
The supercenter of Rbpνqcontains SympνqbÂiPIŹ‚x˜ω0
i,...,˜ωνi´1
iywhere ˜ωa
iis a floating
dot with subscript i, superscript aand placed in the rightmost region:
˜ωa
i:“... a
i
We conjecture that the supercenter contains no other elements.
Conjecture 6.9. There is an isomorphism of rings
ZpRbpνqq – Sympνq b â
iPIŹ‚x˜ω0
i,...,˜ωνi´1
iy,
where ZpRbpνqq is the supercenter of Rbpνq.
In general Rp,µpνqis not a free module over Sympνq b ÂiPIŹ‚x˜ω0
i,...,˜ωνi´1
iy, but we
have the following.
Proposition 6.10. Rbpνqis a free module over Sympνqof rank 2mpm!q2.
Proof. It follows from Theorem 3.15 and the fact Ppνqis a free module of rank m! on
Sympνq.
Since Sympνqlies in the center of Rbpνq, any simple Rbpνq-module is annihilated by
Sym`pνq, where Sym`pνqconsists of the elements in Sympνqwith non-zero degree. In
particular, a simple Rbpνq-module must be a finite dimensional Rbpνq{ Sym`pνqRbpνq-
module. Since Rbpνq{ Sym`pνqRbpνqhas finite dimension over k, we only have finitely
many simple modules, up to shift and isomorphism. We define for each iPSeqdpνqthe
simple module
Si:“Pi{Sym`pνqPi,
which is the unique simple quotient of Pi. We put ˜
Si:“q´xiySi. It lifts to a dg-version
p˜
Si, dNq.
By Lemma 4.5 and Proposition 5.15 we know that EiIdνand FiIdνare exact. Moreover,
they respect the conditions of Proposition 6.4. Therefore, they induce maps
rEiIdνs:K0`
D
lf pRbpνq, dNq˘ÑK∆
0`
D
lf pRbpν´iq, dNq˘,
rFiIdνs:K0`
D
lf pRbpνq, dNq˘ÑK∆
0`
D
lf pRbpν`iq, dNq˘.
Then Theorem 4.4, Theorem 5.17 and Proposition 5.19 tell us that K∆
0p
M
ppΛ, Nqq is an
Uqpgq-weight module. By Proposition 6.8 we know that K∆
0p
M
ppΛ, Nqq is cyclic as Uqpgq-
module, with highest weight generator given by the class of pRbp0q, dNq – pk,0q. Thus
K∆
0p
M
ppΛ, Nqq is a highest weight module.

50 GR ´
EGOIRE NAISSE AND PEDRO VAZ
As in [22], let ψ:Rbpνq Ñ Rbpνqop be the map that takes the mirror image of diagrams
along the horizontal axis. Given a left pRbpνq, dNq-module M, we obtain a right pRbpνq, dNq-
module Mψwith action given by mψ¨r:“ p´1qdeghprqdeghpmqψprq ¨ mfor mPMand
rPRbpνq. Then, we define the bifunctor
p´,´q :
M
ppΛ, Nq ˆ
M
ppΛ, Nq Ñ
D
lf pk,0q,pM, M 1q:“MψbL
pRb,dNqM1,
where bLis the derived tensor product.
Proposition 6.11. The bifunctor defined above respects:
‚ ppRbp0q, dNq,pRbp0q, dNqq – pk,0q;
‚ pIndm`i
mM, M 1q – pM, Resm`i
mM1qfor all M, M 1P
M
ppΛ, Nq;
‚ p‘fM, M 1q – pM, ‘fM1q – ‘fpM, M 1qfor all fPZppq, Λqq.
Proof. Straightforward. 
Comparing Proposition 6.11 with Definition 2.5, we deduce that p´,´q is a categorifi-
cation of the Shapovalov form on K∆
0p
M
ppΛ, Nqq. Moreover, it turns ˜
Siinto the dual of
˜
Pifor each iPSeqdpνq. Recall MppΛ, N qis the parabolic Verma module, and we assume
Λ“ tqβi|iPIrucontains only formal weights.
Theorem 6.12. The asymptotic Grothendieck group K∆
0p
M
ppΛ, Nqq is a Uqpgq-weight
module, with action of Ei, Figiven by rEis,rFis. Moreover, there is an isomorphism of
Uqpgq-modules
K∆
0p
M
ppΛ, Nqq bZQ–MppΛ, N q.
Proof. We already proved the first claim above. Because of Proposition 4.3, for iPIf,
both rFisand rEisact as locally nilpotent operators. In particular, the Uqplq-submodule of
K∆
0p
M
ppΛ, Nqq given by
Uqplq bUqpgqrpRbp0q, dNqs,
is an integrable module for the Levi factor Uqplq. Since it is an integrable cyclic weight
module, it must be isomorphic to VpΛ, N q(see [29]). Therefore, there is a surjective
Uqpgq-module morphism
γ:MppΛ, Nq։K∆
0p
M
ppΛ, Nqq bZQ.
Since MppΛ, Nqis irreducible and γis non-zero, it must be an isomorphism. 
Let mr“FivΛ,N be an induced basis element of MppΛ, N qwith iPSeqpνq. Then the
isomorphism of Theorem 6.12 identifies mrwith the class rpRbpνq, dNq1is. Similarly, let
m1
s“FjvΛ,N for jPSeqdpνqbe a canonical basis element, and let msbe its dual in
the dual canonical basis. Then the isomorphism identifies m1
swith rp ˜
Pj, dNqs and mswith
rp ˜
Sj, dNqs. Moreover, computing the c.b.l.f. composition series of ˜
Pi(see [32,§7]) or taking
the c.b.l.f. cell module replacement of ˜
Si(see [32,§9]) gives a categorical version of the
change of basis between canonical and dual canonical basis elements.

2-VERMA MODULES 51
7. 2-Verma modules
Let kbe a field of characteristic 0. Let
V
Pdg-catkbe a Z-graded pretriangulated dg-
category (see Definition A.23). Let
E
ndHqep
V
q:“
RH
omHqep
V
,
V
qbe the dg-category
of quasi-endofunctors on
V
(see Appendix A.5.1).
Remark 7.1. For example,
V
could be the dg-category
D
dgpR, dqof cofibrant dg-modules
over a dg-algebra pR, dq(see Definition A.15). Then, the subcategory of
E
ndHqep
V
qcon-
sisting of coproduct preserving quasi-functors would be given by the dg-enhanced derived
category of dg-bimodules
D
dgppR, dqop b pR, dqq (see Theorem A.21).
Let Qi:“Àℓě0q1`2ℓ
iId. It is a categorification of qi
1´q2
i“1
q´1
i´qi. We start by introducing
a notion of dg-categorical action and dg-2-representation.
Definition 7.2. Aweak dg-categorical Uqpgq-action on
V
is a collection of quasi-endofunctors
Fi,Ei,KγPZ0p
E
ndHqep
V
qq for all iPIand γPY_such that
‚there are isomorphisms
K0–Id,KγKγ1–Kγ`γ1,
KγEi–qγpαiqEiKγ,KγFi–q´γpαiqFiKγ,
where qdenotes the shift in the q-grading;
‚there is a quasi-isomorphism
(35) Cone`FiEj
uij
ÝÝÑ EjFi˘»
ÝÑ δij Cone`QiKi
hi
ÝÑ QiK´1
i˘,
where Ki:“Kα_
i;
‚there are isomorphisms
tpdij `1q{2u
à
a“0„dij `1
2aqi
F2a
iFjFdij `1´2a
i–
tdij {2u
à
a“0„dij `1
2a`1qi
F2a`1
iFjFdij ´2a
i,
tpdij `1q{2u
à
a“0„dij `1
2aqi
E2a
iEjEdij `1´2a
i–
tdij {2u
à
a“0„dij `1
2a`1qi
E2a`1
iEjEdij ´2a
i,
for all i‰jPI.
We say a weak dg-categorical Uqpgq-action on
V
is a dg-categorical action if in addition
‚Fiis left adjoint to q´1
iKiEiin Z0p
E
ndHqep
V
qq;
‚there is a map of algebras
Rgpiq Ñ Z0pENDpFiqq :“à
zPZ
Z0pHompFi, qzFiqq,
with Fi:“Fi1¨¨¨Fim, for all iPSeqpmq, inducing a surjection
Rgpiq bkZ0pEND
V
pMqq ։Z0pEND
V
pFiMqq,
for all MP
V
;
‚
V
is dg-triangulated (i.e. H0p
V
qis idempotent complete).
Such a
V
carrying a dg-categorical action is called a dg-2-representation of Uqpgq.

52 GR ´
EGOIRE NAISSE AND PEDRO VAZ
The following notions are dg-2-categorical lifts of the notions of weight module and
integrable module.
Definition 7.3. We say that a dg-2-representation
V
is a weight dg-2-representation if
there is a map
λ:Y_Ñ
E
ndHqep
V
q,
where λpγqcommutes with the grading shift qfor all γPY_and λpγq ˝ λpγ1q – λpγ`γ1q,
such that
V
–à
yPY
V
λ,y,Kγ|
V
λ,y p´q “ λpγqqγpyqp´q.
Definition 7.4. We say that a weight dg-2-representation
V
is i-integrable if
‚λpα_
iq “ qnifor some niPN;
‚there is a quasi-isomorphism
(36) Cone`QiKi
V
λ,y
hi
ÝÑ QiK´1
i
V
λ,y˘»
ÝÑ ‘rni´α_
ipyqsqiId,
where ‘rmsqiId :“ ‘r´msqiIdr1swhenever mă0;
‚Fiand Eiare locally nilpotent.
Under some mild hypothesis, this definition recovers the notion of integrable 2-represen-
tation from [38] and [10].
Proposition 7.5. Suppose
V
is i-integrable for all iPI. Also suppose that there is some
MP
V
λ,0such that EiM“0for all iPIand End
V
pMq – pk,0q, and H0p
V
qis c.b.l.f.
generated by tFiMuiPSeqpIq. Then H0p
V
qcarries an integrable categorical Uqpgq-action in
the sense of [38].
Proof. First, by adjunction, Eq. (35) and Eq. (36), we have
gdimqH0pEND
V
pFiMqq – gdimqRN
gpiq,
for all iPSeqpIq. In particular, we have that xni
11iacts by 0 on H0pEnd
V
pFiMqq for all
iPI, and x11iacts non-trivially whenever nią1. Thus, there is a map
γ:RN
gpiq Ñ H0pEND
V
pFiMqq.
Since γis surjective, we obtain RN
gpiq – H0pEND
V
pFiMqq, and the result follows from
Theorem 5.17.
For a Zn-graded dg-algebra pA, dqwe put
D
lf
dgpA, dqfor the dg-category having as objects
the one in
D
lf pA, dq X
D
dgpA, dqand the hom-spaces inherited from
D
dgpA, dq. It is a dg-
enhancement of the c.b.l.f. derived category of pA, dq.
Definition 7.6. Aparabolic 2-Verma module for pis a ZˆZ|Ir|-graded weight dg-2-
representation
V
such that
‚the highest weight space
V
λ:“
V
λ,0–
D
lf
dgpk,0q;
‚there exists M– pk,0q P
V
λsuch that
V
λ,y is c.b.l.f. generated by tFiMuiPSeqpyq
for all ´yPX`, and
V
λ,y “0 otherwise ;

2-VERMA MODULES 53
‚
V
is i-integrable for all iPIf;
‚hj“0 and λα_
j“λj(the degree shift) for all jRIf;
‚for each jRIf,njPNand iPSeqpIq, after specializing λj“qnj, there exists
a differential dnjanticommuting with the differential dof `End
V
pFiMq, d˘such
that the triangulated dg-category generated by c.b.l.f. iterated extension of the
representable modules of
V
nj:“ÀiPSeqpIq`End
V
pFiMq, d `dnj˘is j-integrable
with λpα_
jq “ qnj.
Proposition 7.7. Let
V
be a parabolic 2-Verma module. There is an isomorphism
pRbpiq, dNq – END
V
pFiMq,
in
D
pk,0qfor M– pk,0q P
V
λ.
Proof. First, by adjunction together with Eq. (35) and Eq. (36) we have
(37) END
V
pFiMq – HOM
V
pM, q´1
iKiEiFiMq – RN
ppiq,
in
D
pk,0qfor all iPI. Also,
(38) gdimqH˚pEND
V
pFiMqq “ gdimqRN
ppiq,
for all iPSeqpIq. In particular, there is a relation up to homotopy
α
j i
i
j
`β
j i
i
j
“0,(39)
in END
V
pFiFjMqfor all i, j PIr, identifying the diagrams with the image of the KLR
elements under the surjection Rgpijq։Z0pEND
V
pFiFjMqq, and the floating dot coming
from the isomorphism Eq. (37). Then, the existence of dniand dnjforces to have α“β.
Thus, by Corollary 3.16, there is an A8-map
pRbpiq, dNq Ñ END
V
pFiMq.
By Eq. (38), we conclude it is a quasi-isomorphism. Thus, there exists an isomorphism
pRbpiq, dNq – END
V
pFiMqin
D
pk,0q.
Using Theorem A.21 we can think of FN
iand EN
ifrom Section 5.4 as quasi-endofunctors
of
D
dgpRb, dnq. By Proposition 5.5 we obtain immediately the following:
Corollary 7.8. For all iPIthere is a quasi-isomorphism of cones
Cone`FN
iEN
iIdνÑEN
iFN
iIdν˘»
ÝÑ Cone`Qiλiq´α_
ipνq
iIdνÑQiλ´1
iqα_
ipνqIdν˘,
in
E
ndHqep
D
dgpRb, dNqq.

54 GR ´
EGOIRE NAISSE AND PEDRO VAZ
Together with Proposition 5.19, it means that the dg-enhancement
M
p
dgpΛ, N qof
M
ppΛ, Nq
(obtained by replacing
D
lf pRbpmq, dNqwith
D
lf
dgpRbpmq, dNq) is a weight dg-2-representation
of Uqpgq, where
λpα_
iq:“#λi,whenever iPIr,
qni,whenever iPIf.
Then, by Theorem 5.17, we obtain that
M
p
dgpΛ, N qis a parabolic 2-Verma module.
Corollary 7.9. Let
V
be a parabolic 2-Verma module. There is a quasi-equivalence
M
p
dgpΛ, N q»
ÝÑ
V
.
Proof. Since
V
λ,y is c.b.l.f. generated by ÀiPSeqpyqFiM, we have that
V
λ,y is completely
determined as dg-category by END
V
pFiMq. Thus we conclude by using Proposition 7.7.

Remark 7.10. A parabolic 2-Verma module can also be given a ‘2-categorical’ interpreta-
tion as an p8,2q-category where the hom-spaces are stable p8,1q-categories. For this, it is
enough to see
D
dgpRbpνq, dNqas 0-cells in the p8,2q-category of A8-categories constructed
by Faonte [13], and replace
H
omHqe by the dg-nerve of Lurie [28]. Thanks to [14], we know
that this is a stable p8,1q-category.
Appendix A. Summary on the homotopy category of dg-categories and
pretriangulated dg-categories
We gather some useful results on the homotopy category of dg-categories. References for
this section are [20] and [41]. We also suggest [21] and [42] for nice surveys on the subject.
Our goal is to recall how to construct a category of dg-categories up to quasi-equivalence,
so that the space of functors between two ‘triangulated categories’ is ‘triangulated’.
A.1. Dg-categories. Recall the definition of a dg-category:
Definition A.1. Adg-category
A
is a k-linear category such that:
‚Hom
A
pX, Y qis a Z-graded k-vector space ;
‚the composition
Hom
A
pY, Z q bkHom
A
pX, Y q´˝´
ÝÝÝÝÑ Hom
A
pX, Zq,
preserves the Z-degree ;
‚there is a differential d: Hom
A
pX, Y qiÑHom
A
pX, Y qi´1such that
d2“0, dpf˝gq “ df ˝g` p´1q|f|f˝dg.
Remark A.2. We use a differential of degree ´1 to match the conventions used in the
rest of the paper.
Example A.3. Any dg-algebra pA, dqcan be seen as a dg-category BA with a single
abstract object ‹and HomB Ap‹,‹q :“ pA, dq.

2-VERMA MODULES 55
Example A.4. Let
C
be an abelian, Grothendieck, k-linear category. Consider the cate-
gory Cp
C
qof complexes in
C
, and define Cdgp
C
qas the category where
‚objects are complexes in
C
;
‚hom-spaces are homogeneous maps of Z-graded modules;
‚the differential d: HomCdgp
C
qpX‚, Y ‚qiÑHomCdg p
C
qpX‚, Y ‚qi´1is given by
df :“dY˝f´ p´1q|f|f˝dX.
This data forms a dg-category.
Given a dg-category
A
, one defines
(1) the underlying category Z0p
A
qas
‚having the same objects as
A
;
‚HomZ0p
A
qpX, Y q:“ker`Hom
A
pX, Y q0d
ÝÑ Hom
A
pX, Y q´1˘;
(2) the homotopy category H0p
A
q(or r
A
s) as
‚having the same objects as
A
;
‚HomH0p
A
qpX, Y q:“H0pHom
A
pX, Y q, dq.
Example A.5. For
C
as in Example A.4, we have Z0pCdgp
C
qq – Cp
C
qand H0pCdgp
C
qq –
Komp
C
qthe homotopy category of complexes in
C
.
A.2. Category of dg-categories.
Definition A.6. Adg-functor F:
A
Ñ
B
is a functor between two dg-categories such
that Fpd
A
fq “ d
B
pF f q. We write rFs:H0p
A
q Ñ H0p
B
qfor the induced functor.
We write dg-cat for the category of dg-categories, where objects are dg-categories and
hom-spaces are given by dg-functors.
Let F, G :
A
Ñ
B
be a pair of dg-functors between dg-categories. One defines
H
ompF, Gq
as the Z-graded k-module of homogeneous natural transformations equipped with the dif-
ferential induced by dPHom
B
pF X, GX qfor all XP
A
. Then, we put HompF, Gq:“
Z0p
H
ompF, Gqq.
Definition A.7. A dg-functor
A
Ñ
B
is a quasi-equivalence if
‚F: Hom
A
pX, Y q»
ÝÑ Hom
B
pF X, F Y qis a quasi-isomorphism for all X, Y P
A
;
‚ rFs:H0p
A
q Ñ H0p
B
qis essentially surjective (thus an equivalence).
One defines the dg-category
H
omp
A
,
B
qof dg-functors between
A
and
B
as
‚objects are dg-functors
A
Ñ
B
;
‚hom-spaces are Hom
H
om p
A
,
B
qpF, Gq:“
H
ompF, Gq.
There is also a notion of tensor product of dg-categories
A
b
B
defined as
‚objects are pairs XbYfor all XP
A
and YP
B
;
‚hom-spaces are Hom
A
b
B
pXbY, X 1bY1q:“Hom
A
pX, X1q bkHom
B
pY, Y 1qwith
composition
pf1bg1q ˝ pfbgq:“ p´1q|g1||f|pf1˝fq b pg1˝gq;

56 GR ´
EGOIRE NAISSE AND PEDRO VAZ
‚the differential is dpfbgq:“df bg` p´1q|f|fbdg.
Then, there is a bijection
Homdg-catp
A
b
B
,
C
q – Homdg-catp
A
,
H
omp
B
,
C
qq.
This defines a symmetric closed monoidal structure on dg-cat. However, the tensor product
of dg-categories does not preserve quasi-equivalences.
A.3. Dg-modules. Let
A
be a dg-category. The opposite dg-category
A
op is given by
‚same objects as in
A
;
‚Hom
A
op pX, Y q:“Hom
A
pY, X q;
‚composition g˝
A
op f:“ p´1q|f||g|f˝
A
g.
Aleft (resp. right) dg-module Mover
A
is a dg-functor
M:
A
ÑCdgpkq,(resp. N:
A
op ÑCdg),
where Cdgpkqis the dg-category of k-complexes. The dg-category of (right) dg-modules is
A
op -mod :“
H
omp
A
op, Cdg pkqq. The category of (right) dg-modules is Cp
A
q:“Z0p
A
-modq,
and it is an abelian category. The derived category
D
p
A
qis the localization of Z0p
A
op -modq
along quasi-isomorphisms.
Moreover, for any XP
A
there is a right dg-module
X^:“Hom
A
p´, Xq.
One calls such dg-module representable. Any dg-module quasi-isomorphic to a repre-
sentable dg-module is called quasi-representable. It yields a dg-enriched Yoneda embedding
A
Ñ
A
op -mod .
Example A.8. Let pA, dqbe a dg-algebra. Then Z0pBAq-mod – pA, dq-mod and
D
pBAq –
D
pA, dq. The unique representable dg-module HomB Ap´,‹q is equivalent to
the free module pA, dq.
A.4. Model categories. We recall the basics of model category theory from [16]. Model
category theory is a powerful tool to study localization of categories. For example, we can
use it to compute hom-spaces in a derived category. We will mainly use it to describe the
homotopy category of dg-categories up to quasi-equivalence.
Let Mbe a category with limits and colimits.
Definition A.9. Amodel category on Mis the data of three classes of morphisms
‚the weak equivalences W;
‚the fibrations F ib;
‚the cofibrations Cof ;
satisfying
‚for Xf
ÝÑ Yg
ÝÑ ZPM, if two out of three terms in tf, g, g ˝fuare in W, then so is
the third;

2-VERMA MODULES 57
‚stability along retracts:W, F ib and Cof are stable along retracts, that is if we have
a commutative diagram
X Y X
X1Y1X1
IdX
gfg
IdX1
and fPW, F ib or Cof then so is g.
‚factorization: any Xf
ÝÑ Yfactorizes as p˝iwhere pPF ib and iPC of XWor
pPF ib XWand iPC of , and the factorization is functorial in f;
‚lifting property: given a commutative square diagram
A X
B Y
Cof Qi pPF ib
Dh
with iPCof and pPF ib, if either iPWor pPW, then there exists h:BÑX
making the diagram commute.
We tend to think about fibrations as ‘nicely behaved surjections’, and cofibrations as
‘nicely behaved injections’.
The localization H opMq:“W´1Mof Malong weak equivalences is called the homotopy
category of M. It has a nice description in terms of homotopy classes of maps between
fibrant and cofibrant objects.
Definition A.10. If H Ñ XPCof , then we say Xis cofibrant. If YÑ ˚ P F ib, then Y
is fibrant.
One says that f„g, that is f:XÑYis homotopy equivalent to g:XÑY, if there
is a commutative diagram
X
CpXqY
X
IdX
if
h
F ibXWQp
p
IdX
jg

58 GR ´
EGOIRE NAISSE AND PEDRO VAZ
where i\j:X\XÑCpXq P Cof. One calls CpXqthe cylinder object of X. When Xis
cofibrant and Yfibrant, then „is an equivalence relation on HomMpX, Y q. Moreover, we
have
HomHopMqpX, Y q – HomMpX, Y q{ „
whenever Xis cofibrant and Yfibrant. Note that any XPMadmits a cofibrant replace-
ment QX since we have a commutative diagram
HX
QX
Cof QipPF ibXW
Similarly, any YPMadmits a fibrant replacement RY .
Let Mcf be the full subcategory of Mgiven by objects that are both fibrant and cofibrant.
Let Mcf { „ be the quotient of Mcf by identifying maps that are homotopy equivalent.
Then, the localization functor MÑH opMqrestricts to Mcf , inducing an equivalence of
categories
Mcf { „ »
ÝÝÑ HopMq.
Example A.11. Let Cpkqbe the category of complexes of k-modules. It comes with a
model category structure where Wis the quasi-isomorphisms, F ib is the surjective maps,
and Cof is given by the maps respecting the lifting property. All objects are fibrant
and the cofibrant objects are essentially the complexes of projective k-modules. Then
HopCpkqq –
D
pkq.
A model category on Mis a Cpkq-model category if it is (strongly) enriched over Cpkq,
and the models are compatible (see [42,§3.1] for a precise definition). This definition
means that we have:
‚a tensor product ´b´:Cpkq ˆ MÑM;
‚an enriched dg-hom-space
H
omMpX, Y q P Cpkqfor any X, Y PMcompatible with
the tensor product:
HomMpEbX, Y q – HomCpkqpE,
H
omMpX, Y qq;
‚HopMqis enriched over
D
pkq – HopCpkqq;
‚a derived hom-functor
RH
omMpX, Y q:“
H
omMpQX, RY q P
D
pkq,
where QX is a cofibrant replacement of X, and RY a fibrant replacement of Y;
‚HomHopMqpX, Y q – H0p
RH
omMpX, Y qq.
Note that in particular for X, Y PMcf we have HomHopMqpX, Y q – H0p
H
ompX, Y qq.
Example A.12. Let
A
be a dg-category. There is a Cpkq-model category on
A
-mod
where Wis given by the quasi-isomorphisms, F ib are the surjective morphisms, and C of
is given by the maps respecting the lifting property. Then Hop
A
-modq –
D
p
A
q.

2-VERMA MODULES 59
Remark A.13. In the Cpkq-model category
A
-mod, all objects are fibrant. Moreover,
Pis cofibrant if and only if for all surjective quasi-isomorphism f:L»
ÝÑ X(i.e. map in
WXF ib) then there exists h:PÑLsuch that the following diagram commutes:
HL
P X
»
Dh
In a practical way, cofibrant dg-modules are quasi-isomorphic to direct summand of dg-
modules admitting a (possibly infinite) exhaustive filtration where all the quotients are
free dg-modules.
Definition A.14. For MaCpkq-model category, let M(resp. IntpMq) be the dg-category
with
‚the same objects as M(resp. Mcf );
‚HomMpX, Y q:“
H
omMpX, Y q.
.
Then, we have H0pIntpMqq – HopMq, and we say that IntpMqis a dg-enhancement of
HopMq.
Definition A.15. We write
D
dgp
A
q:“Intp
A
-modq
for the dg-enhanced derived category of
A
.
Note that
D
dgp
A
qis a dg-enhancement of
D
p
A
qsince we have H0p
D
dgp
A
qq –
D
p
A
q.
Example A.16. Let Rbe a k-algebra viewed as a dg-category with trivial differential.
Then we have that
D
dgpRqis the dg-category of complexes of projective R-modules.
A.5. The model category of dg-categories. Let Wbe the collection of quasi-equivalences
in dg-cat. Let F ib be the collection of dg-functors F:
A
Ñ
B
in dg-cat such that
(1) FX,Y : Hom
A
pX, Y q։Hom
B
pF X, F Y qis surjective;
(2) for all isomorphism v:FpXq»
ÝÑ YPH0p
B
qthere exists an isomorphism u:X»
ÝÑ
Y0PH0p
A
qsuch that rFspuq “ v.
This defines a model structure on dg-cat where everything is fibrant. One calls
Hqe :“Hopdg-catq
the homotopy category of dg-categories (up to quasi-equivalence).
How can we compute HomHqep
A
,
B
q? It appears that constructing a cofibrant replace-
ment for
A
is in general a difficult problem. However, we can do the following:
(1) replace
A
by a k-flat quasi-equivalent dg-category
A
1: meaning it is such that
Hom
A
1pX, Y q bk´
preserves quasi-isomorphisms (e.g. when Hom
A
1pX, Y qis cofibrant in Cpkq, i.e. a
complex of projective k-modules);

60 GR ´
EGOIRE NAISSE AND PEDRO VAZ
(2) define Repp
A
,
B
qas the subcategory of
D
p
A
op b
B
qwith FPRepp
A
,
B
qif and
only if for all XP
A
there exists YP
B
such that
XbLF–
D
p
B
qY_,
(in other words, Fis a dg-bimodule sending representable
A
-modules to quasi-
representable
B
-modules);
(3) then
HomHqep
A
,
B
q – IsopRepp
A
,
B
qq,
where Iso means the set of objects up to isomorphism.
Remark A.17. Note that whenever kis a field, all dg-categories are k-flat.
We refer to elements in Repp
A
,
B
qas quasi-functors. Note that a quasi-functor F:
A
Ñ
B
induces a functor
rFs:H0p
A
q Ñ H0p
B
q.
Thus, we can think of Repp
A
,
B
qas the category of ‘representations up to homotopy’ of
A
in
B
.
A.5.1. Closed monoidal structure. If
A
is cofibrant, then ´b
A
preserves quasi-equivalences
and one can define the bifunctor
´ bL´: Hqe ˆHqe ÑHqe,
A
bL
B
:“Q
A
bQ
B
,
where Q
A
and Q
B
are cofibrant replacements. Then, as proven by Toen [41], there exists
an internal hom-functor
RH
omHqep´,´q such that
HomHqep
A
bL
B
,
C
q – HomHqep
A
,
RH
omHqep
B
,
C
qq.
Therefore, Hqe is a symmetric closed monoidal category.
Remark A.18. Note that the internal hom can not simply be the derived hom functor
(because tensor product of cofibrant dg-categories is not cofibrant in general).
Define the dg-category of quasi-functors Repdgp
A
,
B
qas
‚the objects in Repp
A
,
B
q X p
A
op b
B
-modqcf ;
‚the dg-homs
H
ompX, Y qof Intp
A
op b
B
-modq.
In other words, Repdgp
A
,
B
qis the full subcategory of quasi-functors in
D
dgp
A
op b
B
q,
thus of cofibrant dg-bimodules that preserves quasi-representable modules. It is a dg-
enhancement of Repp
A
,
B
q.
If
A
is k-flat, then
RH
omHqep
A
,
B
q –Hqe Repdgp
A
,
B
q.
Thus H0p
RH
omHqep
A
,
B
qq – HomHqep
A
,
B
q.
Remark A.19. If kis a field of characteristic 0, then the dg-category
RH
omHqep
A
,
B
q
is equivalent to the A8-category of strictly unital A8-functors [13].
Example A.20. We have Repdgp
A
, IntpCpkqqq – Intp
A
op -modq –
D
dgp
A
q.

2-VERMA MODULES 61
Recall that classical Morita theory says that for Aand Bbeing k-algebras, there is an
equivalence
HomcoppA-mod, B -modq – Aop bkB-mod,
where Homcop is given by the functors that preserve coproducts.
Similarly, we put Repcop
dg p
D
dgp
A
q,
D
dgp
B
qq for the subcategory of Repdg p
D
dgp
A
q,
D
dgp
B
qq
where FPRepcop
dg p
D
dgp
A
q,
D
dgp
B
qq if and only if rFs:
D
p
A
q Ñ
D
p
B
qpreserves coprod-
ucts.
Theorem A.21. If
A
is k-flat, then we have
RH
omcop
Hqep
D
dgp
A
q,
D
dgp
B
qq :“Repcop
dg p
D
dgp
A
q,
D
dgp
B
qq –Hqe
D
dgp
A
op b
B
q.
Under the hypothesis of Theorem A.21, the internal composition of dg-quasifunctors
preserving coproducts is given by taking a cofibrant replacement of the derived tensor
product over
A
.
A.6. Pretriangulated dg-categories. Basically, a triangulated dg-category is a dg-cate-
gory such that its homotopy category is canonically triangulated. But before being able to
give a precise definition, we need to do a detour through Quillen exact categories, Frobenius
categories and stable categories.
A.6.1. Frobenius structure on Cp
A
q.Recall a Quillen exact category [36] is an additive
category with a class of short exact sequences
0ÑXf
ÝÑ Yg
ÝÑ ZÑ0,
called conflations, which are pairs of ker-coker, where fis called an inflation and ga
deflation, respecting some axioms:
‚the identity is a deflation;
‚the composition of deflations is a deflation;
‚deflations (resp. inflations) are stable under base (resp. cobase) change.
AFrobenius category is a Quillen exact category having enough injectives and projectives,
and where injectives coincide with projectives. The stable category
C
of a Frobenius cat-
egory
C
is given by modding out the maps that factor through an injective/projective
object. It carries a canonical triangulated structure where:
‚the suspension functor Sis obtained by taking the target of a conflation
0ÑXÑIX ÑSX Ñ0,
where IA is an injective hull of X, for all XP
C
;
‚the distinguished triangles are equivalent to standard triangles
Xf
ÝÑ Yg
ÝÑ Zh
ÝÑ SX,

62 GR ´
EGOIRE NAISSE AND PEDRO VAZ
obtained from conflations by the following commutative diagram:
0X Y Z 0
0X IX SX 0.
f
Id
g
h
Example A.22. Let
A
be a small dg-category. One can put a Frobenius structure on
Cp
A
qp:“Z0p
A
op -modqq by using split short exact sequences as class of conflations. Then
there is an equivalence Cp
A
q – H0p
A
op -modq, and the suspension functor coincides with
the usual homological shift. Moreover,
D
p
A
qinherits the triangulated structure from
H0p
A
-modq, where distinguished triangles are equivalent to distinguished triangles ob-
tained from all short exact sequences in Cp
A
q.
A.6.2. Pretriangulated dg-categories. Remark for any dg-category
A
there is a Yoneda
functor
Z0p
A
q Ñ Cp
A
q, X ÞÑ Hom
A
p´, Xq.
Definition A.23. A dg-category
T
is pretriangulated if the image of the Yoneda functor is
stable under translations and extensions (for the Quillen exact structure on Cp
T
qdescribed
in Example A.22).
This definition implies that
‚Z0p
T
qis a Frobenius subcategory of Cp
T
q;
‚H0p
T
qinherits a triangulated structure, called canonical triangulated structure,
from H0p
T
-modq.
Example A.24. Let
A
be a dg-category. We have that
D
dgp
A
qis pretriangulated with
Z0p
D
dgp
A
qq – Cp
A
qcf . Moreover, the canonical triangulated structure of H0p
D
dgp
A
qq
coindices with the usual on
D
p
A
q.
Then, it is possible to show that
‚any dg-category
A
admits a pretriangulated hull pretrp
A
qsuch that
RH
omHqep
A
,
T
q»
ÝÑ
RH
omHqeppretrp
A
q,
T
q,
for all pretriangulated dg-category
T
;
‚
RH
omHqep
A
,
T
qis pretriangulated whenever
T
is pretriangulated;
‚any dg-functor F:
T
Ñ
T
1between pretriangulated dg-categories induces a trian-
gulated functor rFs:H0p
T
q Ñ H0p
T
1q.
Note that for
A
being k-flat, the pretriangulated structure of
RH
omHqep
D
dgp
A
q,
D
dgp
B
qq
restricts to the one of
D
dgp
A
op b
B
q(viewed as sub-dg-category). In particular, we obtain
distinguished triangles of quasi-functors from short exact sequences of dg-bimodules.
Definition A.25. For a morphism f:XÑYPZ0p
T
qin the underlying category of
pretriangulated dg-category
T
, one calls mapping cone an object Conepfq P
T
such that
Conepfq^–ConepX^¯
f
ÝÑ Y^q P H0p
T
-modq.

2-VERMA MODULES 63
A.6.3. Dg-Morita equivalences.
Definition A.26. A dg-functor F:
A
Ñ
B
is a dg-Morita equivalence if it induces an
equivalence
LF:
D
p
A
q»
ÝÑ
D
p
B
q:XÞÑ FpQXq,
where QX is a cofibrant replacement of X.
Example A.27. In particular, a quasi-equivalence is a dg-Morita equivalence and the
functor that sends dg-categories to their pretriangulated hull
A
ÞÑ pretrp
A
qis a dg-Morita
equivalence.
Theorem A.28 ([?]).There is a model structure dg-catmor on dg-cat where the weak-
equivalences are the dg-Morita equivalences and the fibrations are the same as before.
Definition A.29. We say that
T
is triangulated if it is fibrant in dg-catmor.
Equivalently,
T
is triangulated if and only if the Yoneda functor induces an equivalence
H0p
T
-modq»
ÝÑ
D
cp
T
q(i.e. every compact object is quasi-representable). Also equiva-
lently,
T
is triangulated if and only if
T
is pretriangulated and H0p
T
-modqis idempotent
complete.
In particular, any category admits a triangulated hull trp
A
q(i.e. fibrant replacement).
It is given by trp
A
q:“
D
c
dgp
A
q, the dg-category of compact objects in
D
dgp
A
q.
Example A.30. Let Rbe a k-algebra viewed as a dg-category. Then
D
c
dgpRqis the
dg-category of perfect complexes, i.e. bounded complexes of finitely generated projective
R-modules.
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Institut de Recherche en Math´
ematique et Physique, Universit´
e Catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
E-mail address:gregoire.naisse@uclouvain.be
Institut de Recherche en Math´
ematique et Physique, Universit´
e Catholique de Louvain,
Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
E-mail address:pedro.vaz@uclouvain.be