Holomorphically finitely generated Hopf algebras and quantum Lie groups

Abstract

We study topological Hopf algebras that are holomorphically finitely generated (HFG) as Fr\'echet Arens-Micheal algebras in the sense of Pirkovskii. Some of them but not all can be obtained from affine Hopf algebras by an application of the analytization functor. We find that a commutative HFG Hopf algebra is always an algebra of holomorphic functions on a complex Lie (in fact, Stein) group and prove that the corresponding categories are equivalent. Akbarov associated with a compactly generated complex Lie group $G$ the cocommutative topological Hopf algebra ${\mathscr A}_{exp}(G)$ of exponential analytic functionals. We show that it is HFG but not every cocommutative HFG Hopf algebra is of this form. In the case when $G$ is connected, a structure theorem based on previous results of the author is established and the analytic structure of ${\mathscr A}_{exp}(G)$ (that depends on the large-scale geometry of $G$) is described. We also consider some interesting examples including complex-analytic analogues of classical $\hbar$-adic quantum groups.Comment: 38 pages, Version 3: Corrections in Question 2 and Example 4.11; Version 2: the notation \widehat U_\hbar(sl_2) is changed by \widetilde U(sl_2)_\hba