On the Property M Conjecture for the Heisenberg Lie Algebra

We prove a fundamental case of a conjecture of the first author which expresses the homology of the extension of the Heisenberg Lie algebra by C[t]/(tk+1) in terms of the homology of the Heisenberg Lie algebra itself. More specifically, we show that both the 0th and (k+1)th x-graded components of homology of this extension of the three-dimensional Heisenberg Lie algebra have dimension 3k+1 by constructing a simple basis for cohomology.