A class of solvable Lie algebras and their Casimir Invariants

TL;DR

This paper classifies a specific class of solvable Lie algebras with a nilpotent structure, computes their Casimir invariants, and explores their applications in geometry and Einstein spaces.

Contribution

It provides a complete classification of solvable Lie algebras with a given nilradical and calculates their Casimir invariants, extending understanding of their structure and applications.

Findings

Classified all indecomposable solvable Lie algebras with nilradical n_{n,1}

Computed generalized Casimir invariants for these algebras

Connected these algebras to Einstein spaces and Riemannian manifold classification

Abstract

A nilpotent Lie algebra n_{n,1} with an (n-1) dimensional Abelian ideal is studied. All indecomposable solvable Lie algebras with n_{n,1} as their nilradical are obtained. Their dimension is at most n+2. The generalized Casimir invariants of n_{n,1} and of its solvable extensions are calculated. For n=4 these algebras figure in the Petrov classification of Einstein spaces. For larger values of n they can be used in a more general classification of Riemannian manifolds.