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arXiv2000

Semi-infinite cohomology and Hecke algebras

This paper provides a homological algebraic foundation for generalizations of classical Hecke algebras introduced in math.QA/9805134. These new Hecke algebras are associated to triples of the form (A,B,e), where A is an associative algebra containing subalgebra B with character e. These algebras are connected with cohomology of associative algebras in the sense that for every left A-module V and right A-module W the Hecke algebra associated to triple (A,B,e) naturally acts in the B-cohomology and B-homology spaces of V and W, respectively. We also introduce the semi-infinite cohomology functor for associative algebras and define modifications of Hecke algebras acting in semi-infinite cohomology spaces. We call these algebras semi-infinite Hecke algebras. As an example we realize the W-algebra W(g) associated to a complex semisimple Lie algebra g as a semi-infinite Hecke algebra. Using this realization we explicitly calculate the algebra W(g) avoiding the bosonization technique used by Feigin and Frenkel.

math.RA↗
arXiv2004

The Chevalley Involution and a Duality of Weight Varieties

We study the action of the Chevalley involution of a simple complex Lie group G on the set of its weight varieties (i.e. torus quotients of its flag manifolds). We find for torus quotients of Grassmannians (for GL(n)) that we obtain the classical notion of "association of projective point sets" (see Ch. III of Dolgachev-Ortland, Asterisque 165). We say a weight variety is "self-dual" if it is carried into itself by the action of the Chevalley involution. Our main theorem is a classification of those self-dual weight varieties for which the induced self-map is the identity. This problem is the "classical problem" associated to the "quantum problem" of classifying the self-dual representations V of G for which the induced action of the Chevalley involution on the zero weight space V[0] is a scalar. This latter problem is important in the study of the representation of the Artin group associated to G on V[0] obtained as the monodromy representation of the Casimir connection - see Millson - Toledano Laredo arXiv:math.QA/0305062.

math.SG↗
arXiv2005

Recurrences for elliptic hypergeometric integrals

In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group E_7. In the present note, we consider the action of the affine Weyl group, or more precisely, the recurrences satisfied by special cases of the integral. These are of two flavors: linear recurrences that hold only up to dimension 6, and three families of bilinear recurrences that hold in arbitrary dimension, subject to a condition on the parameters. As a corollary, we find that a codimension one special case of the integral is a tau function for the elliptic Painlevé equation.

math.CA↗
arXiv2005

Solutions of the classical Yang-Baxter equation and noncommutative deformations

In this paper, I will show that, if a Lie algebra $\G$ acts on a manifold $P$, any solution of the classical Yang-Baxter equation on $\G$ gives arise to a Poisson tensor on $P$ and a torsion-free and flat contravariant connection (with respect to the Poisson tensor). Moreover, if the action is locally free, the matacurvature of the above contravariant connection vanishes. This will permit to get a large class of manifolds which satisfy the necessary conditions, presented by Hawkins in math.QA/0504232, to the existence of a noncommutative deformation.

math.SG↗
arXiv2006

A dual point of view on the ribbon graph decomposition of moduli space

In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in math.AG/0402015, and used in math.QA/0412149 to construct open-closed topological conformal field theories. This dual version of the ribbon graph decomposition is a compact orbi-cell complex with a natural weak homotopy equivalence to the moduli space.

math.GT↗
arXiv2006

Elliptic gamma functions, triptic curves and SL_3(Z)

This is a condensed exposition of the results of math.QA/0601337, based on a talk of the first author at the Oberwolfach workshop "Deformations and Contractions in Mathematics and Physics", 15-21 January 2006.

math.CV↗
arXiv2006

Quadratic transformations of Macdonald and Koornwinder polynomials

When one expands a Schur function in terms of the irreducible characters of the symplectic (or orthogonal) group, the coefficient of the trivial character is 0 unless the indexing partition has an appropriate form. A number of q-analogues of this fact were conjectured in math.QA/0112035; the present paper proves most of those conjectures, as well as some new identities suggested by the proof technique. The proof involves showing that a nonsymmetric version of the relevant integral is annihilated by a suitable ideal of the affine Hecke algebra, and that any such annihilated functional satisfies the desired vanishing property. This does not, however, give rise to vanishing identities for the standard nonsymmetric Macdonald and Koornwinder polynomials; we discuss the required modification to these polynomials to support such results.

math.RT↗
arXiv2007

Limits of elliptic hypergeometric integrals

In math.QA/0309252, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level.

math.CA↗
arXiv2006

Full heaps and representations of affine Weyl groups

We use the author's combinatorial theory of full heaps (defined in math.QA/0605768) to categorify the action of a large class of Weyl groups on their root systems, and thus to give an elementary and uniform construction of a family of faithful permutation representations of Weyl groups. Examples include the standard representations of affine Weyl groups as permutations of ${\Bbb Z}$ and geometrical examples such as the realization of the Weyl group of type $E_6$ as permutations of 27 lines on a cubic surface; in the latter case, we also show how to recover the incidence relations between the lines from the structure of the heap. Another class of examples involves the action of certain Weyl groups on sets of pairs $(t, f)$, where $t \in {\Bbb Z}$ and $f$ is a function from a suitably chosen set to the two-element set $\{+, -\}$. Each of the permutation representations corresponds to a module for a Kac--Moody algebra, and gives an explicit basis for it.

math.CO↗
arXiv2007

The small quantum group and the Springer resolution

In math.RT/0304173 the derived category of the principal block in modules over the Lusztig quantum algebra at a root of unity is related to the derived category of equivariant coherent sheaves on the Springer resolution. In the present paper we deduce a similar relation between the derived category of the principal block for the {\em small (reduced) quantum algebra} and the derived category of (non-equivariant) coherent sheaves on the Springer resolution. As an application we get a geometric description of Hochschild cohomology (in particular, the center) of the regular block for the small quantum group, and use it to give an explicit description of a certain subalgebra in the center (obtained previously by another method and under more restrictive assumptions in math.QA/0107098. We also briefly explain the relation of our result to the geometric description math.RT/0407048 of the derived category of modules over the De Concini -- Kac quantum algebra.

math.RT↗
arXiv2006

Cylindrical Combinatorics and Representations of Cherednik Algebras of type A

We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions on periodic diagrams, and in particular, compute some generating functions concerning plane partitions, where Kostka polynomials and their level restricted generalization appear. On representation side, we study representations of Cherednik algebras which admit weight decomposition with respect to a certain commutative subalgebra. All the irreducible representations of this class are constructed combinatorially through standard tableaux on periodic diagrams, and this realization as "tableaux representations" provides a new combinatorial approach to the investigation of these representations. As consequences, we describe the decomposition of a tableaux representation as a representation of the degenerate affine Hecke algebra, which is a subalgebra of the Cherednik algebra, and also describe the spectral decomposition of the spherical subspace (the invariant subspace under the action of the Weyl group) of a tableaux representation with respect to the center of the degenerate affine Hecke algebra, In particular, the computation of the character of the spherical subspace is reduced to the computation of the generating function for the set of column strict plane partitions, and we obtain an expression of the characters in terms of Kostka polynomials as announced in math.QA/0508274.

math.RT↗
arXiv2013

Coisotropic submanifolds and dual pairs

The Poisson sigma model is a widely studied two-dimensional topological field theory. This note shows that boundary conditions for the Poisson sigma model are related to coisotropic submanifolds (a result announced in [math.QA/0309180]) and that the corresponding reduced phase space is a (possibly singular) dual pair between the reduced spaces of the given two coisotropic submanifolds. In addition the generalization to a more general tensor field is considered and it is shown that the theory produces Lagrangian evolution relations if and only if the tensor field is Poisson.

math.SG↗
arXiv2007

Integrating L-infinity algebras

Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group. In the case of the string Lie 2-algebra of Baez and Crans, this recovers the model of the string group introduced in math.QA/0504123 .

math.AT↗
arXiv2006

On Khovanov's cobordism theory for su(3) knot homology

We reconsider the su(3) link homology theory defined by Khovanov in math.QA/0304375 and generalized by Mackaay and Vaz in math.GT/0603307. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of (complexes of) `cobordisms with seams' (actually, a `canopolis'), making it local in the sense of Bar-Natan's local su(2) theory of math.GT/0410495. We show that this `seamed cobordism canopolis' decategorifies to give precisely what you'd both hope for and expect: Kuperberg's su(3) spider defined in q-alg/9712003. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory. Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the su(3) homology of the (2,n) torus knots.

math.GT↗
arXiv2008

Fixing the functoriality of Khovanov homology

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.

math.GT↗
arXiv2017

Quantum Hyperdeterminants and Hyper-Pfaffians

The notion of generalized quantum monoids is introduced. It is proved that the quantum coordinate ring of the monoid can be lifted to a quantum hyper-algebra, in which the quantum determinant and quantum Pfaffian are sent to the quantum hyperdeterminant and quantum hyper-Pfaffian respectively. The quantum hyperdeterminant in even dimension is shown to be a $q$-analog of Cayley's first hyperdeterminant.

math.QA↗
arXiv2021

Representations of the orthosymplectic Yangian

We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $\frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one variable, they are classified in terms of highest weights. We give explicit constructions of a family of elementary modules of the Yangian and show that a wide class of irreducible representations can be produced by taking tensor products of the elementary modules.

math.RT↗