On blocks of defect two and one simple module, and Lie algebra structure of HH1

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 2I elements, where I is the inertial index of B, and where we assume that k is a splitting field for B and C.