Complex-analytic approach to quantum groups

Abstract: We discuss quantum analogues of complex Lie groups. Our approach is closer to classical quantum group theory than to C*-algebraic one (no multipliers and no invariant weights). I propose to consider a topological Hopf algebra with a finiteness condition (holomorphically finitely generated or HFG for short). This topic seems to offer a wide range of research opportunities.

Our focus is on examples, such as analytic forms of some classical quantum groups (a deformation of a solvable Lie group and Drinfeld-Jimbo algebras). I also present some general results: (1) the category of Stein groups is anti-equivalent to the category of commutative Hopf HFG algebras; (2) If G is a compactly generated Lie group, the associated convolution cocommutative topological Hopf algebra (introduced by Akbarov) is HFG. When, in addition, G is connected and linear, the structure of this cocommutative algebra can be described explicitly. I also plan to discuss briefly holomorphic duality (which is parallel to Pontryagin duality).

operator algebrasquantum algebra

Audience: researchers in the topic

Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

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