Cohomology for quantum groups via the geometry of the nullcone

TL;DR

This paper computes the cohomology algebra of small quantum groups at roots of unity using complex geometry of the nullcone, providing new insights especially for cases where the order is less than the Coxeter number.

Contribution

It offers a uniform calculation of the cohomology algebra of small quantum groups for all roots of unity, utilizing geometric methods related to the nullcone of Lie algebras.

Findings

Cohomology algebra computed for small quantum groups at roots of unity.

Established finite generation of cohomology modules over the algebra.

Provided new results on support varieties for quantum group modules.

Abstract

Let $\zeta$ be a complex $\ell$th root of unity for an odd integer $\ell>1$. For any complex simple Lie algebra $\mathfrak g$, let $u_\zeta=u_\zeta({\mathfrak g})$ be the associated "small" quantum enveloping algebra. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the Coxeter number $h$ of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible $G$-modules stipulates that $p \geq h$. The main result in this paper provides a surprisingly uniform answer for the cohomology algebra $\opH^\bullet(u_\zeta,{\mathbb C})$ of the small quantum group. When $\ell>h$, this cohomology algebra has been calculated by Ginzburg and Kumar \cite{GK}. Our result requires powerful tools from complex geometry and a detailed knowledge of the geometry of the nullcone of $\mathfrak g$. In this way, the methods point out difficulties present in obtaining similar results for the restricted enveloping algebra $u$ in small characteristics, though they do provide some clarification of known results there also. Finally, we establish that if $M$ is a finite dimensional $u_\zeta$-module, then $\opH^\bullet(u_\zeta,M)$ is a finitely generated $\opH^\bullet(u_\zeta,\mathbb C)$-module, and we obtain new results on the theory of support varieties for $u_\zeta$.