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