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

TL;DR

This paper proves that non-degenerate Banach space representations of Drinfeld-Jimbo algebras are finite dimensional when |q|≠ 1, and explores the structure and representation theory of their complex-analytic forms, especially for sl_2.

Contribution

It establishes the finite dimensionality of Banach space representations for these algebras when |q|≠ 1 and explicitly describes their Arens-Michael envelopes, also analyzing the simpler representation theory of their analytic forms for sl_2.

Findings

All non-degenerate Banach space representations are finite dimensional for |q|≠ 1.

Explicit form of the Arens-Michael envelope of U_q(g) is obtained.

Representation theory of the analytic form for sl_2 is simpler, with all irreducible representations finite dimensional.

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 $\widetilde U(\mathfrak{s}\mathfrak{l}_2)_\hbar$ (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.