Invariant metrics on central extensions of Quadratic Lie algebras

TL;DR

This paper characterizes the structure of central extensions of quadratic Lie algebras that admit invariant metrics, linking algebraic and geometric descriptions of these extensions.

Contribution

It provides a comprehensive description of how central extensions of quadratic Lie algebras can be characterized algebraically and geometrically.

Findings

Central extensions can be described algebraically in terms of the original quadratic Lie algebra.

Invariant metrics induce specific direct sum decompositions.

The structure of these extensions is characterized by the properties of the invariant metric.

Abstract

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and non degenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariantmetrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involvedgive rise to.