Contact and 2-compatible Lie algebras

TL;DR

This paper introduces the concept of 2-compatible Lie algebras, which are quadratic deformations of a given Lie algebra, and explores their role in classifying contact Lie algebras.

Contribution

It defines 2-compatible Lie algebras via quadratic deformations and links this notion to the classification of contact Lie algebras, especially those related to the Heisenberg algebra.

Findings

2-compatible Lie algebras are quadratic deformations of original Lie algebras.

Contact Lie algebras of odd dimension are quadratic deformations of Heisenberg algebras.

The notion aids in the classification of contact Lie algebras.

Abstract

A $n$-dimensional Lie algebra $g=(V,\mu)$ is called $2$-compatible if it is isomorphic to a quadratic deformation of a Lie algebra $g_0=(V,\mu_0)$. By quadratic deformation we means a formal deformation $\mu_t=\mu_0+t\varphi_1+t^2\varphi_2$ where $\mu_t$ is a Lie algebra on $V \otimes K[[t]]$. It is equivalent to say that we have the following system $\sum_{i+j \leq 4} \varphi_i \circ \varphi_j= 0$. This notion naturally appears in the theory of classification of contact Lie algebras because any $(2p+1)$-dimensional contact Lie algebra is isomorphic to a quadratic deformation of the Heisenberg algebra $\mathcal{H}_{2p+1}$.