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Let $k$ be a field of odd prime characteristic $p$. We calculate the Liealgebra structure of the first Hochschild cohomology of a class of quantumcomplete intersections over $k$. As a consequence, we prove that if $B$ is adefect $2$-block of a finite group algebra $kG$ whose Brauer correspondent $C$has a unique isomorphism class of simple modules, then a basic algebra of $B$is a local algebra which can be generated by at most $2\sqrt I$ elements, where$I$ is the inertial index of $B$, and where we assume that $k$ is a splittingfield for $B$ and $C$.

On blocks of defect two and one simple module, and Lie algebra structure of $HH^1$