  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 $D(\mathfrak{sl}_2(k))$.
