Pavel Etingof

Quantization of Lie bialgebras, III

In the paper [Dr3] V. Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the problem of quantization of Lie bialgebras. This paper gives a positive answer to a number of Drinfelds questions, using the methods and ideas of [KL]. In particular, we show the existence of a quantization for Lie bialgebras. The universality and functoriality properties of this quantization will be discussed in the second paper of this series. We plan to provide positive answers to most of the remaining questions in [Dr3] in the following papers of this series.

Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations

Introduction Representations of finite-dimensional and affine Lie algebras Knizhnik-Zamolodchikov equations Solutions of the Knizhnik-Zamolodchikov equations Free field realization Quantum groups Local systems and configuration spaces Monodromy of the Knizhnik-Zamolodchikov equations Quantum affine algebras Quantum Knizhnik-Zamolodchikov equations Solutions of the quantum Knizhnik-Zamolodchikov equations for

Finite-dimensional representations of rational Cherednik algebras

\mathfrak {sl}_2

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

Connection matrices for the quantum Knizhnik-Zamolodchikov equations and elliptic functions Current developments and future perspectives References Index.

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

A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed.

Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential

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.

Quantum Groups and Lie Theory: Lectures on the dynamical Yang-Baxter Equations

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.

On finite-dimensional semisimple and cosemisimple Hopf algebras in positive characteristic

Let G be a simple simply connected algebraic group over ℂ with Lie algebra \( \mathfrak{g} \) . Given a parabolic subgroup P ⊂ G, in tikya[1] the first author introduced a certain generating function Z G,P aff . Roughly speaking, these functions count (in a certain sense) framed G-bundles on ℙ2 together with a P-structure on a fixed (horizontal) line in ℙ2. When P = B is a Borel subgroup, the function Z G,B aff was identified in tikya[1] with the Whittaker matrix coefficient in the universal Verma module over the affine Lie algebra \( \overset{\lower0.5em\hbox{

Fusion categories and homotopy theory

\smash{\scriptscriptstyle\smile}

EXCHANGE DYNAMICAL QUANTUM GROUPS

}}{\mathfrak{g}} _{aff} \) (here we denote by \( \mathfrak{g}_{aff} \) the affinization of \( \mathfrak{g} \) and by \( \overset{\lower0.5em\hbox{