Banach space representations of Drinfeld–Jimbo algebras and their complex-analytic forms

We prove that every non-degenerate Banach space representation of the Drinfeld-Jimbo algebra $U_q(\mathfrak{g})$ of a semisimple complex Lie algebra $\mathfrak{g}$ is finite dimensional when $|q|\ne 1$. As a corollary, we find an explicit form of the Arens-Michael envelope of $U_q(\mathfrak{g})$, which is similar to that of $U(\mathfrak{g})$ obtained by Joseph Taylor in 70s. In the case when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_2$, we also consider the representation theory of the corresponding analytic form $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ (with $e^\hbar=q$) and show that it is simpler than for $U_q(\mathfrak{s}\mathfrak{l}_2)$. For example, all irreducible continuous representations of $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ are finite dimensional for every admissible value of the complex parameter $\hbar$, while $U_q(\mathfrak{s}\mathfrak{l}_2)$ has a topologically irreducible infinite-dimensional representation when $|q|= 1$ and $q$ is not a root of unity.