Derivations and Hochschild cohomology of quantum nilpotent algebras

TL;DR

This paper computes derivations of quantum nilpotent algebras and explicitly describes their Hochschild cohomology, providing new insights into the structure of quantized enveloping algebras of Lie algebras.

Contribution

It offers the first explicit description of the Hochschild cohomology of the positive part of quantum groups, using cluster algebra techniques.

Findings

Derived the derivations of quantum nilpotent algebras.

Explicitly described the first Hochschild cohomology group of $U_q^+(rak{g})$.

Leveraged cluster algebra structures for the computations.

Abstract

We compute the derivations of Quantum Nilpotent Algebras under a technical (but necessary) assumption on the center. As a consequence, we give an explicit description of the first Hochschild cohomology group of $U_q^+(\mathfrak{g})$, the positive part of the quantized enveloping algebra of a finite-dimensional complex simple Lie algebra $\mathfrak{g}$. Our results are obtained leveraging an initial cluster constructed by Goodearl and Yakimov.