## On the property M conjecture for the Heisenberg Lie algebra

**
Phil Hanlon and Michelle L. Wachs**
**Abstract:**
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
$\Bbb C[t]/(t^{k+1})$ in terms of the homology of the Heisenberg Lie algebra itself. More
specifically, we show that both the $0^{th}$ and $k+1^{st}$ $x$-graded components of
homology of this extension of the
$3$-dimensional Heisenberg Lie algebra have dimension $3^{k+1}$ by constructing a
simple basis for cohomology.

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