  Let k be a field, q in k. We derive a cup product formula on the Hochschild
cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we
determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N,
where N is the ideal generated by homogeneous nilpotent elements. We explicitly
construct non-nilpotent Hochschild cocycles which cannot be generated by lower
homological degree elements, thus disproving the Snashall-Solberg finite
generation conjecture.
