Cohomology rings for quantized enveloping algebras

TL;DR

This paper computes the cohomology ring structure of quantum groups associated with simple Lie algebras, revealing it is generated by elements in degrees matching classical Lie algebra cohomology.

Contribution

It provides the first detailed description of the cohomology ring for quantized enveloping algebras, showing it is an exterior algebra generated in specific degrees.

Findings

Cohomology ring is generated by homogeneous elements in odd degrees.

Structure is similar to classical Lie algebra cohomology.

Partial results for specialized quantum groups at non-zero parameters.

Abstract

We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak{g}$. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as generate the cohomology ring for the Lie algebra $\mathfrak{g}$. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from $U_q$ by specializing the parameter $q$ to a non-zero value $\epsilon \in \mathbb{C}$.