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Odd Khovanov's arc algebra
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Abstract
We construct an odd version of Khovanov's arc algebra . Extending the center to elements that anticommute, we get a subalgebra that is isomorphic to the oddification of the cohomology of the (n,n)-Springer varieties. We also prove that the odd arc algebra can be twisted into an associative algebra.
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Blanchet introduced certain singular cobordisms to fix the functoriality of Khovanov homology. In this paper we introduce graded algebras consisting of such singular cobordisms à la Blanchet. As the main result we explicitly describe these algebras in algebraic terms using the combinatorics of arc diagrams.
We define parameter dependent
gl2-foams and their associated web
and arc algebras and verify that they specialize to several known
sl2 or gl2 constructions related to higher link and tangle invariants. Moreover, we show
that all these specializations are equivalent, and we deduce several applications,
e.g. for the associated link and tangle invariants, and their functoriality.
For each integer we describe diagrammatically a positively graded
Koszul algebra such that the category of finite dimensional
-modules is equivalent to the category of perverse sheaves on the
isotropic Grassmannian of type or , constructible
with respect to the Schubert stratification. The algebra is obtained by a
(non-trivial) ``folding'' procedure from a generalized Khovanov arc algebra.
Properties like graded cellularity and explicit closed formulas for graded
decomposition numbers are established by elementary tools.
In this master thesis we construct an oddification of the rings from
arXiv:math/0103190 using the functor from arXiv:0710.4300 . This leads to a
collection of non-associative rings where C represent some choices
of signs. Extending the center up to anti-commutative elements, we get a ring
which is isomorphic to the oddification of the ring cohomology of
the (n,n)-Springer variety from arXiv:1203.0797 .
This paper is concerned with the modular representation theory of the affine Hecke-Clifford superalgebra, the cyclotomic Hecke-Clifford superalgebras, and projective representations of the symmetric group. Our approach exploits crystal graphs of affine Kac-Moody algebras.
In this paper, we define an explicit basis for the 𝔤𝔩n-web algebra Hn(k) (the 𝔤𝔩n generalization of Khovanov’s arc algebra) using categorified q-skew Howe duality.
Our construction is a 𝔤𝔩n-web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and Hn(k), and it gives an explicit graded cellular basis of the 2-hom space between two 𝔤𝔩n-webs. We use this to give a (in principle) computable version of colored Khovanov–Rozansky 𝔤𝔩n-link homology, obtained from a complex defined purely combinatorially via the (thick cyclotomic) KLR algebra and needs only F.
In this paper we use Kuperberg’s
-webs and Khovanov’s
-foams to define a new algebra
, which we call the
-web algebra. It is the
analogue of Khovanov’s arc algebra. We prove that
is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of
q
-skew Howe duality, which allows us to prove that
is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group
, to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that
is a graded cellular algebra.
We give an explicit graded cellular basis of the 𝔰𝔩3-web algebra KS.
In order to do this, we identify Kuperberg's basis for the 𝔰𝔩3-web space WS with a version of Leclerc-Toffin's intermediate crystal basis and we identify Brundan, Kleshchev and Wang's degree of tableaux with the weight of flows on webs and the q-degree of foams.
We use these observations to give a "foamy" version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the 𝔰𝔩3-web algebra.
We restrict ourselves to the 𝔰𝔩3 case over ℂ here, but our approach should, up to the combinatorics of 𝔰𝔩N-webs, work for all N>1 or over ℤ.
We create a framework for odd Khovanov homology in the spirit of Bar-Natan's
construction for the ordinary Khovanov homology. Namely, we express the cube of
resolutions of a link diagram as a diagram in a certain 2-category of
chronological cobordisms and show that it is 2-commutative: the composition of
2-morphisms along any 3-dimensional subcube is trivial. This allows us to
create a chain complex, whose homotopy type modulo certain relations is a link
invariant. Both the original and the odd Khovanov homology can be recovered
from this construction by applying certain strict 2-functors. We describe other
possible choices of functors, including the one that covers both homology
theories and another generalizing dotted cobordisms to the odd setting. Our
construction works as well for tangles and is functorial with respect to tangle
cobordisms up to sign.
In this paper, which is a follow-up to my paper with Yonezawa "The sl(N)-web
categories and categorified skew Howe duality", I define and study sl(N)-web
algebras for any N greater than one. For N=2 these algebras are isomorphic to
Khovanov's arc algebras and for N=3 they are Morita equivalent to the sl(3)-web
algebras which I defined and studied with Pan and Tubbenhauer.
The main result of this paper is that the sl(N)-web algebras are Morita
equivalent to blocks of certain level-N cyclotomic KLR algebras, for which I
use the categorified quantum skew Howe duality from the aforementioned paper
with Yonezawa.
Using this Morita equivalence and Brundan and Kleshchev's work on cyclotomic
KLR algebras, I show that there exists an isomorphism between a certain space
of sl(N)-webs and the split Grothendieck group of the corresponding sl(N)-web
algebra, which maps the dual canonical basis elements to the Grothendieck
classes of the indecomposable projective modules (with a certain normalization
of their grading).
We construct a bigraded cohomology theory which depends on one parameter a, and whose graded Euler characteristic is the quantum sl.2/ link invariant. We follow Bar-Natan's approach to tangles on one side, and Khovanov's sl.3/ theory for foams on the other side. Our theory is properly functorial under tangle cobordisms, and a version of the Khovanov sl.2/ invariant (or Lee's modification of it) corresponds to a D 0 (or a D 1/: In particular, the construction naturally resolves the sign ambiguity in the functoriality of Khovanov's sl.2/ theory.
We give an alternative presentation of Khovanov homology of links. The original construction rests on the Kauffman bracket model for the Jones polynomial, and the generators for the complex are enhanced Kauffman states. Here we use an oriented sl(2) state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces. Functoriality in original Khovanov homology holds up to sign. Variants of Khovanov homology fixing functoriality were obtained by Clark–Morrison–Walker [7] and also by Caprau [6]. Our construction is similar to those variants. Here we work over integers, while the previous constructions were over gaussian integers.
We identify the ring of odd symmetric functions introduced by Ellis and
Khovanov as the space of skew polynomials fixed by a natural action of the
Hecke algebra at q=-1. This allows us to define graded modules over the Hecke
algebra at q=-1 that are `odd' analogs of the cohomology of type A Springer
varieties. The graded module associated to the full flag variety corresponds to
the quotient of the skew polynomial ring by the left ideal of nonconstant odd
symmetric functions. The top degree component of the odd cohomology of Springer
varieties is identified with the corresponding Specht module of the Hecke
algebra at q=-1.
We introduce some graded versions of the walled Brauer algebra, working over
a field of characteristic zero. This allows us to prove that the walled Brauer
algebra is Morita equivalent to an idempotent truncation of a certain infinite
dimensional version of Khovanov's arc algebra, as suggested by recent work of
Cox and De Visscher. We deduce that the walled Brauer algebra is Koszul
whenever its defining parameter is non-zero.
We investigate properties of the odd Khovanov homology, compare and contrast them with those of the original (even) Khovanov homology, and discuss applications of the odd Khovanov homology to other areas of knot theory and low-dimensional topology.
We prove that blocks of the general linear supergroup are Morita equivalent
to a limiting version of Khovanov's diagram algebra. We deduce that blocks of
the general linear supergroup are Koszul.
We prove that integral blocks of parabolic category O associated to the
subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary
covers of generalised Khovanov algebras. Although this result is in principle
known, the existing proof is quite indirect, going via perverse sheaves on
Grassmannians. Our new approach is completely algebraic, exploiting Schur-Weyl
duality for higher levels. As a by-product we get a concrete combinatorial
construction of 2-Kac-Moody representations in the sense of Rouquier
corresponding to level two weights in finite type A.
We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
We describe a modification of Khovanov homology (math.QA/9908171), in the spirit of Bar-Natan (math.GT/0410495), which makes the theory properly functorial with respect to link cobordisms. This requires introducing `disorientations' in the category of smoothings and abstract cobordisms between them used in Bar-Natan's definition. Disorientations have `seams' separating oppositely oriented regions, coming with a preferred normal direction. The seams satisfy certain relations (just as the underlying cobordisms satisfy relations such as the neck cutting relation). We construct explicit chain maps for the various Reidemeister moves, then prove that the compositions of chain maps associated to each side of each of Carter and Saito's movie moves (MR1238875, MR1445361) always agree. These calculations are greatly simplified by following arguments due to Bar-Natan and Khovanov, which ensure that the two compositions must agree, up to a sign. We set up this argument in our context by proving a result about duality in Khovanov homology, generalising previous results about mirror images of knots to a `local' result about tangles. Along the way, we reproduce Jacobsson's sign table (math.GT/0206303) for the original `unoriented theory', with a few disagreements. Comment: 91 pages. Added David Clark as co-author. Further detail on variations of third Reidemeister moves, to allow treatment of previously missing cases of movie move six. See changelog section for more detail
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle. When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.
We categorify one-half of the quantum group associated to an arbitrary Cartan
datum.
We define a 2-category that categorifies the covering Kac–Moody algebra for introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a -grading giving its Grothendieck group the structure of a free module over the group algebra of . By specializing the -action to +1 or to −1, the construction specializes to an “odd” categorification of and to a supercategorification of , respectively.
We consider the monoidal subcategory of finite dimensional representations of
U_q(gl(1|1)) generated by the vector representation, and we provide a diagram
calculus for the intertwining operators. We construct a categorification using
subquotient categories of the BGG category O(gl_n). Using Soergel's functor V
we compute the Soergel modules corresponding to the indecomposable projective
modules in these subcategories and the homomorphism spaces between them; hence
we describe the regular blocks of these categories as modules over some
explicit diagram algebras. We construct diagrammatically standard and proper
standard modules for the proper stratified structure of these algebras.
Finally, we prove that they have an interesting connection with the geometry of
Springer fibres associated with hook partitions.
We study in detail two row Springer fibres of even orthogonal type from an
algebraic as well as topological point of view. We show that the irreducible
components and their pairwise intersections are iterated P^1-bundles. Using
results of Kumar and Procesi we compute the cohomology ring with its action of
the Weyl group. The main tool is a type D diagram calculus labelling the
irreducible components in an convenient way which relates to a diagrammatical
algebra describing the category of perverse sheaves on isotropic Grassmannians
based on work of Braden. The diagram calculus generalizes Khovanov's arc
algebra to the type D setting and should be seen as setting the framework for
generalizing well-known connections of these algebras in type A to other types.
The results will be connected to Brauer algebras at non-generic parameters in a
subsequent paper.
We show that the octonions are a twisting of the group algebra of 2 × 2 × 2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. In particular, we show that they are quasialgebras associative up to a 3-cocycle isomorphism. We show that one may make general constructions for quasialgebras exactly along the lines of the associative theory, including quasilinear algebra, representation theory, and an automorphism quasi-Hopf algebra. We study the algebraic properties of quasialgebras of the type which includes the octonions. Further examples include the higher 2n-onion Cayley algebras and examples associated to Hadamard matrices.
We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations.
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a `TQFT') to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.
This is the second of a series of four articles studying various generalisations of Khovanov's diagram algebra. In this article we develop the general theory of Khovanov's diagrammatically defined "projective functors" in our setting. As an application, we give a direct proof of the fact that the quasi-hereditary covers of generalised Khovanov algebras are Koszul. Comment: Minor changes, extra sections on Kostant modules and rigidity of cell modules added
This is the first of four articles studying some slight generalisations
H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m)
of these algebras in the sense of Rouquier, and certain infinite dimensional
limiting versions. In this article we prove that H(n,m) is a cellular symmetric
algebra and that K(n,m) is a cellular quasi-hereditary algebra. In subsequent
articles, we relate these algebras to level two blocks of degenerate cyclotomic
Hecke algebras, parabolic category O and the general linear supergroup,
respectively.
- P Ozsvath
- J Rasmussen
- Z Szabo
P. Ozsvath, J. Rasmussen, and Z. Szabo. Odd Khovanov homology. Alg. Geom. Topol.,
13(3):1465-1488, 2013, math.QA:0710.4300.
On quasiassociative algebras and monoidal categories of set graded modules
- K Putyra
K. Putyra. On quasiassociative algebras and monoidal categories of set graded modules.
Preprint.
Towards odd Khovanov homology for tangles
- K Putyra
- A Shumakovitch
K. Putyra and A. Shumakovitch.
Towards odd Khovanov homology
for tangles.
Topology Seminar, University of California, Berkeley, CA,
http://www.math.columbia.edu/~putyra/talks/2013-Berkeley/handout.pdf,
2013.