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

Abstract

We prove that every nondegenerate 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 1970s. In the case when $\mathfrak{g}=\mathfrak{s}{\mathfrak{l}}_2$, we also consider the representation theory of the corresponding analytic form, the Arens–Michael algebra $\tilde{U}{(\mathfrak{s}{\mathfrak{l}}_2)}_{\hslash }$ (with $e^{\hslash }=q$) and show that it is simpler than for $U_q(\mathfrak{s}{\mathfrak{l}}_2)$. For example, all irreducible continuous representations of $\tilde{U}{(\mathfrak{s}{\mathfrak{l}}_2)}_{\hslash }$ are finite dimensional for every admissible value of the complex parameter ℏ, 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.