Alexander Varchenko

Multidimensional hypergeometric functions and representation theory of lie algebras and quantum groups

Construction of complexes calculating homology of the complement of a configuration construction of homology complexes for a discriminantal configuration algebraic interpretation of chain complexes of a discriminantal configuration quasi-isomorphism of two-sided Hochschild complexes to suitable one-sided Hochschild complexes bundle properties of a discriminantal configuration R-matrix for the two-sided complexes monodromy R-matrix operator as the canonical element, quantum doubles hypergeometric integrals KacMoody Lie algebras without Serres relations and their doubles hypergeometric integrals of a discriminantal configuration resonances at infinity degenerations of discriminantal configurations remarks on homology groups of a configuration with coefficients in local systems.

Geometry and Classificatin of Solutions of the Classical Dynamical Yang–Baxter Equation

Abstract:The classical Yang–Baxter equation(CYBE) is an algebraic equation central in the theory of integrable systems. Its nondegenerate solutions were classified by Belavin and Drinfeld. Quantization of CYBE led to the theory of quantum groups. A geometric interpretation of CYBE was given by Drinfeld and gave rise to the theory of Poisson–Lie groups. The classical dynamical Yang–Baxter equation (CDYBE) is an important differential equation analogous to CYBE and introduced by Felder as the consistency condition for the differential Knizhnik–Zamolodchikov–Bernard equations for correlation functions in conformal field theory on tori. Quantization of CDYBE allowed Felder to introduce an interesting elliptic analog of quantum groups. It becomes clear that numerous important notions and results connected with CYBE have dynamical analogs. In this paper we classify solutions to CDYBE and give geometric interpretation to CDYBE. The classification and interpretation are remarkably analogous to the Belavin–Drinfeld picture.

Solutions of the Quantum Dynamical Yang–Baxter Equation and Dynamical Quantum Groups

Abstract:The quantum dynamical Yang–Baxter (QDYB) equation is a useful generalization of the quantum Yang–Baxter (QYB) equation. This generalization was introduced by Gervais, Neveu, and Felder. Unlike the QYB equation, the QDYB equation is not an algebraic but a difference equation, with respect to a matrix function rather than a matrix. The QDYB equation and its quasiclassical analogue (the classical dynamical Yang–Baxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder. In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical Yang–Baxter equation, obtained in our previous paper. All solutions we found can be obtained from Felders elliptic solution by a limiting process and gauge transformations. Fifteen years ago the quantum Yang–Baxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum Yang–Baxter equation. In this paper we propose a similar language, originating from Felders ideas, which we found to be adequate for the dynamical Yang–Baxter equation. This is the language of dynamical quantum groups (or ?-Hopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper.

Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors

In this paper we strenghten a theorem by Esnault-Schechtman-Viehweg, [3], which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with coefficients in a local system using only logarithmic global differential forms, provided certain ”Aomoto non-resonance conditions” for monodromies are fulfilled at some ”edges” (intersections of hyperplanes). We prove that it is enough to check these conditions on a smaller subset of edges, see Theorem 4.1. We show that for certain known one dimensional local systems over configuration spaces of points in a projective line defined by a root system and a finite set of affine weights (these local systems arise in the geometric study of Knizhnik-Zamolodchikov differential equations, cf. [8]), the Aomoto resonance conditions at non-diagonal edges coincide with Kac-Kazhdan conditions of reducibility of Verma modules over affine Lie algebras, see Theorem 7.1.

Bethe eigenvectors of higher transfer matrices

We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra . The models are defined on a tensor product of irreducible -modules. For each model, there exist N one-parameter families of commuting operators on , called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra

Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations

gl_N

Schubert calculus and representations of the general linear group

. The models are defined on a tensor product irreducible

EXCHANGE DYNAMICAL QUANTUM GROUPS

gl_N

Geometry of q-hypergeometric functions as a bridge between Yangians and quantum affine algebras

-modules. For each model, there exist

Differential Equations Compatible with KZ Equations

N