Cohomology of quantum groups: An analog of Kostant's Theorem

TL;DR

This paper extends Kostant's Lie algebra cohomology theorem to quantum groups at generic parameters and roots of unity, providing new formulas and conditions for cohomology calculations.

Contribution

It proves Kostant's cohomology formula for quantum groups at generic q and roots of unity, generalizing previous results to new algebraic settings.

Findings

Kostant's formula holds for quantum groups at generic q

Kostant's formula applies at roots of unity with specific weight conditions

Extends classical Lie algebra cohomology results to quantum group context

Abstract

We prove the analog of Kostant's Theorem on Lie algebra cohomology in the context of quantum groups. We prove that Kostant's cohomology formula holds for quantum groups at a generic parameter $q$, recovering an earlier result of Malikov in the case where the underlying semisimple Lie algebra $\mathfrak{g} = \mathfrak{sl}(n)$. We also show that Kostant's formula holds when $q$ is specialized to an $\ell$-th root of unity for odd $\ell \ge h-1$ (where $h$ is the Coxeter number of $\mathfrak{g}$) when the highest weight of the coefficient module lies in the lowest alcove. This can be regarded as an extension of results of Friedlander-Parshall and Polo-Tilouine on the cohomology of Lie algebras of reductive algebraic groups in prime characteristic.