Lie Algebras : Classifications, Deformations, Rigidity and Differential Geometry

TL;DR

This paper reviews classical results on finite-dimensional complex Lie algebras, including classifications, deformations, and rigidity, and explores their applications in differential geometry such as structures on Lie groups and symmetric spaces.

Contribution

It provides a concise overview of Lie algebra classifications, deformations, and their geometric applications, highlighting recent insights and classical results.

Findings

Classification of nilpotent Lie algebras

Applications to contact and symplectic structures

Discussion of G-symmetric spaces

Abstract

This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to Differential Geometry considering some left invariant structures on Lie groups : contact and symplectic structures, Generalized complex structures on real Lie Group. We present also the notion of riemannian and pseudo-riemannian G-symmetric spaces.