Holomorphically finitely generated Hopf algebras and quantum Lie groups

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 applying the analytization functor. We show that a commutative HFG Hopf algebra is always an algebra of holomorphic functions on a complex Lie group (actually a Stein group), and prove that the corresponding categories are equivalent. With a compactly generated complex Lie group~$G$, Akbarov associated a cocommutative topological Hopf algebra, the 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, using previous results of the author we establish a theorem on the analytic structure of ${\mathscr A}_{exp}(G)$. It depends on the large-scale geometry of $G$. We also consider some interesting examples including complex-analytic analogues of classical $\hbar$-adic quantum groups.