Study of formality for the Heisenberg algebra

In this paper, we compute the Chevalley-Eilenberg cohomology of the three-dimensional Heisenberg Lie algebra with values in its universal enveloping algebra. We also compute the Schouten brackets on cochains and cohomology level in order to write the formality equations. It turns out that there is no formality, and that the perturbed L∞ structure on the cohomology has non-trivial terms in infinitely many degrees.