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 ${\mathcal{A}}_{exp}(G)$. It depends on the large-scale geometry of $G$. We also consider some interesting examples including complex-analytic analogues of classical $\mathrm{\hslash }$-adic quantum groups.