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

We prove that every nondegenerate Banach space representation of the Drinfeld– Jimbo algebra U_q.g/ of a semisimple complex Lie algebra g is finite dimensional when jqj¤1. As a corollary, we find an explicit form of the Arens–Michael envelope of U_q.g/, which is similar to that of U.g/ obtained by Joseph Taylor in 1970s. In the case when g D sl₂, we also consider the representation theory of the corresponding analytic form, the Arens–Michael algebra e_U.sl2/_„ (with e^„ D q) and show that it is simpler than for U_q.sl₂/. For example, all irreducible continuous representations of e_U.sl2/_„ are finite dimensional for every admissible value of the complex parameter „, while U_q.sl₂/ has a topologically irreducible infinite-dimensional representation when jqjD1 and q is not a root of unity.