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

Abstract

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 $\tilde{U}(\mathfrak{s}{\mathfrak{l}}_2{)}_{\hslash }$ (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 $\hslash$, 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.

version 4: minor changes, version 3: the proofs of auxilary assertions for Theorem 1 are corrected

References in corpus (3)

• Arens-Michael envelopes of nilpotent Lie algebras, holomorphic functions of exponential type, and homological epimorphisms
• On analytic analogues of quantum groups
• The relation "commutator equals function'' in Banach algebras