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 Uq(g) a semisimple complex Lie g is finite dimensional when |q|≠1. As corollary, we find an explicit form Arens–Michael envelope Uq(g), which similar to U(g) obtained by Joseph Taylor in 1970s. In case g=sl2, also consider theory corresponding analytic form, U˜(sl2)ℏ (with eℏ=q) and show it simpler than for Uq(sl2). For example, all irreducible continuous representations are admissible value parameter ℏ, while Uq(sl2) has topologically infinite-dimensional |q|=1 q not root unity.