# Cohomology of some families of Lie algebras and quadratic Lie algebras

## TL;DR

This paper investigates the cohomology of specific Lie algebra classes, including Heisenberg and quadratic Lie algebras, providing new calculations of their cohomological properties and Betti numbers.

## Contribution

It offers a complete description of the cohomology for the $MD(n,1)$-class and computes the second Betti number for generalized complex diamond Lie algebras.

## Key findings

- Complete cohomology description for $MD(n,1)$-class
- Second Betti number for generalized complex diamond Lie algebras computed
- Extension of known cohomology results to new classes of Lie algebras

## Abstract

The paper studies the cohomology of Lie algebras and quadratic Lie algebras. Firstly, we propose to describe the cohomology of $MD(n,1)$-class which was introduced in \cite{LHNCN16}. This class contains Heisenberg Lie algebras. In 1983, L. J. Santharoubane \cite{San83} computed the cohomology of Heisenberg Lie algebras. In this paper, we will completely describe the cohomology of the other ones of $MD(n, 1)$-class. Finally, we will be concerned about the cohomology of quadratic Lie algebras. In 1985, A. Medina and P. Revoy \cite{MR85} computed the second Betti number of the generalized real diamond Lie algebras. We will compute in this paper the second Betti number of the generalized complex diamond Lie algebras by using the super-Poisson bracket.