Hopf-cyclic Cohomology of Quantum Enveloping Algebras

TL;DR

This paper computes the Hopf-cyclic cohomology of quantum enveloping algebras associated with semi-simple Lie algebras, revealing that Hochschild cohomology is concentrated in a specific degree.

Contribution

It provides explicit calculations of Hopf-cyclic cohomology for quantum enveloping algebras of arbitrary semi-simple Lie algebras, a novel extension in the field.

Findings

Hochschild cohomology is concentrated in a degree determined by the Lie algebra's rank

Explicit formulas for periodic and non-periodic Hopf-cyclic cohomology are derived

Results apply to quantum groups with coefficients in a modular pair in involution

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 show that its Hochschild cohomology is concentrated in a single degree determined by the rank of the Lie algebra $\mathfrak{g}$.