  We show that the Hochschild cohomology of a monomial algebra over a field of
characteristic zero vanishes from degree two if the first Hochschild cohomology
is semisimple as a Lie algebra. We also prove that first Hochschild cohomology
of a radical square zero algebra is reductive as a Lie algebra. In the case of
the multiple loops quiver, we obtain the Lie algebra of square matrices of size
equal to the number of loops.
