\title[On blocks of defect two and one simple module]
{On blocks of defect two and one simple module, and Lie algebra 
structure of $\HH^1$} 
\author{D. J. Benson, Radha Kessar, and Markus Linckelmann} 


\begin{abstract}    
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 $C$.
\end{abstract}