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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