Hopf-cyclic cohomology of quantum enveloping algebras

Abstract

In this paper we calculate both the periodic and non-periodic Hopf-cyclic cohomology of Drinfeld–Jimbo quantum enveloping algebra $U_q(\mathfrak{g})$ for an arbitrary semi-simple Lie algebra $\mathfrak{g}$ with coefficients in a modular pair in involution. We obtain this result by showing that the coalgebra Hochschild cohomology of these Hopf algebras are concentrated in a single degree determined by the rank of the Lie algebra $\mathfrak{g}$.