Derivations of the two-parameter quantized enveloping algebra $\U$

TL;DR

This paper computes derivations and automorphisms of a two-parameter quantized enveloping algebra, providing insights into its Hochschild cohomology and symmetry structure, which are crucial for understanding its algebraic properties.

Contribution

It explicitly determines the derivations, Hochschild cohomology, and automorphism groups of the two-parameter quantized enveloping algebra and its Hopf algebra structure, a novel comprehensive analysis.

Findings

Computed the derivations of the algebra.

Calculated the first Hochschild cohomology group.

Determined the automorphism groups of the algebra and Hopf algebra.

Abstract

Let $r,s$ be two parameters chosen from $\C^{\ast}$ such that $r^{m}s^{n}=1$ implies $m=n=0$. We compute the derivations of the two-parameter quantized enveloping algebra $\U$ and calculate its first degree Hochschild cohomology group. We further determine the group of algebra automorphisms for the two-parameter Hopf algebra $\V$. As a result, we determine the group of Hopf algebra automorphisms for $\V$.