Study of formality for the Heisenberg algebra

TL;DR

This paper investigates the formality of the three-dimensional Heisenberg algebra by computing its Chevalley-Eilenberg cohomology and analyzing the L-infinity structure, revealing no formality and complex higher-degree terms.

Contribution

It provides the first detailed cohomology computation for this algebra and demonstrates the non-existence of formality, highlighting the complexity of its L-infinity structure.

Findings

No formality for the Heisenberg algebra.

Non-trivial higher-degree terms in the L-infinity structure.

Explicit cohomology computations for the algebra.

Abstract

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