sl(3) link homology
We define a bigraded homology theory whose Euler characteristic is the quantum sl(3) link invariant.
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arXiv subjects
Mikhail Khovanov
Publications and source records attributed to Mikhail Khovanov.
Matrix factorizations and link homology
For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.
math.QA↗
Homological realization of Nakajima varieties and Weyl group actions
We describe points on Nakajima varieties and Weyl group actions on them via complexes of semisimple and projective modules over certain finite-dimensional algebras.
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Categorifications of the colored Jones polynomial
The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.
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A functor-valued invariant of tangles
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.
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An invariant of tangle cobordisms
We prove that the construction of our previous paper math.QA/0103190 yields an invariant of tangle cobordisms.
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Categorification of some level two representations of sl(n)
We categorify representations of quantum sl(n) whose highest weight is twice a fundamental weight.
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Crossingless matchings and the cohomology of (n,n) Springer varieties
The sequence of rings $H^n, n\ge 0,$ introduced in math.QA/0103190, controls categorification of the quantum sl(2) invariant of tangles. We prove that the center of $H^n$ is isomorphic to the cohomology ring of the (n,n) Springer variety and show that the braid group action in the derived category of $H^n$-modules descends to the Springer action of the symmetric group.
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Patterns in knot cohomology I
Cohomology theory of links, introduced by the author, is combinatorial. Dror Bar-Natan recently wrote a program that found ranks of cohomology groups of all prime knots with up to 11 crossings. His surprising experimental data is discussed in this note.
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A category for the adjoint representation
We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of C. The category C is the direct sum of a semisimple category and the category of modules over a certain algebra A, associated to a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root systems.
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Quivers, Floer cohomology, and braid group actions
We consider the derived categories of modules over a certain family A_m of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type A_m. We show that each of these two rather different objects encodes the topology of curves on an (m+1)-punctured disc. We prove that the braid group B_{m+1} acts faithfully on the derived category of A_m-modules and that it injects into the symplectic mapping class group of Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of A_m-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.
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A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the wall functors act as generators of the Temperley-Lieb algebra while Zuckerman functors act as generators of U(sl(2)).
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A categorification of the Jones polynomial
We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
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NilCoxeter algebras categorify the Weyl algebra
We show that induction and restriction functors for inclusions of nilCoxeter algebras provide a categorical realization of the algebra of polynomial differential operators in one variable.
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Web bases for sl(3) are not dual canonical
We compare two natural bases for the invariant space of a tensor product of irreducible representations of A_2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A_2 spider. The other basis is the dual canonical basis, the dual of the basis defined by Lusztig and Kashiwara. For sl(2) or A_1, the web bases have been discovered many times and were recently shown to be dual canonical by Frenkel and Khovanov. We prove that for sl(3), the two bases eventually diverge even though they agree in many small cases. The first disagreement comes in the invariant space Inv((V^+ tensor V^+ tensor V^- tensor V^-)^{tensor 3}), where V^+ and V^- are the two 3-dimensional representations of sl(3). If the tensor factors are listed in the indicated order, only 511 of the 512 invariant basis vectors coincide.
q-alg↗
Kazhdan-Lusztig polynomials and canonical basis
In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of the quantum group $U_q sl_k$. We also use known results about canonical bases for $U_q sl_2$ to get a new proof of recurrent formulas for KL polynomials for maximal parabolic subgroups (geometrically, this case corresponds to Grassmanians), due to Lascoux-Schutzenberger and Zelevinsky.
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Representations of tensor categories and Dynkin diagrams
In this note we illustrate by a few examples the general principle: interesting algebras and representations defined over Z_+ come from category theory, and are best understood when their categorical origination has been discovered. We show that indecomposable Z_+-representations of the character ring of SU(2) satisfying certain conditions correspond to affine and infinite Dynkin diagrams with loops. We also show that irreducible Z_+-representations of the Verlinde algebra (the character ring of the quantum group SU(2)_q, where q is a root of unity), satisfying similar conditions correspond to usual (non-affine) Dynkin diagrams with loops. Conjecturedly, the last result is related to the ADE classification of conformal field theories with the chiral algebra \hat{sl(2)}.
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