  Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie
algebra structure of the first Hochschild cohomology of a class of quantum
complete intersections over $k$. As a consequence, we prove that if $B$ is a
defect $2$-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 splitting
field for $B$ and $C$.
