Examples and patterns on quadratic Lie algebras

TL;DR

This paper explores the structure and construction of quadratic Lie algebras, emphasizing classical features, necessary conditions, and the existence of local quadratic forms within this broad class.

Contribution

It provides an overview of classical features and introduces methods for constructing local quadratic Lie algebras, clarifying structural conditions for their existence.

Findings

Quadratic Lie algebras include semisimple and orthogonal subspace examples.

Existence of invariant forms imposes structural conditions on the algebra.

Methods for constructing local quadratic Lie algebras are discussed.

Abstract

A Lie algebra is said to be quadratic if it admits a symmetric invariant and non-degenerated bilinear form. Semisimple algebras with the Killing form are examples of these algebras, while orthogonal subspaces provide abelian quadatric algebras. The class of quadratic algebras is outsize, but at first sight it is not clear weather an algebra is quadratic. Some necessary structural conditions appear due to the existence of an invariant form forces elemental patterns. Along the paper we overview classical features and constructions on this topic and focus on the existence and constructions of local quadratic.