Hochschild cohomology of $U(\mathfrak{sl}_2(k))$

TL;DR

This paper computes the Hochschild cohomology of the universal enveloping algebra of sl_2 over an algebraically closed field of characteristic p>2, revealing new invariants under the adjoint action.

Contribution

It provides the first explicit determination of Hochschild cohomology for U(sl_2) in positive characteristic and analyzes invariants under the adjoint action.

Findings

Hochschild cohomology groups of U(sl_2) are explicitly computed.

Invariants of sl_2 and SL_2 in the divided power algebra are characterized.

New algebraic structures related to sl_2 in characteristic p>2 are identified.

Abstract

Let $k$ be an algebraically closed field of characteristic $p>2$. We determine the Hochschild cohomology of $U(\mathfrak{sl}_2(k))$ and the invariants of $\mathfrak{sl}_2(k)$ and $\mathsf{SL}_2(k)$ in the adjoint action on the divided power algebra $D(\mathfrak{sl}_2(k))$.