A new look at Lie algebras

TL;DR

This paper introduces a novel framework for describing real finite-dimensional Lie algebras using pairs of linear maps and eigenvectors, revealing their solvable structure and foundational role in constructing all Lie algebras.

Contribution

It presents a new approach to Lie algebra classification based on pairs (F, v), linking algebra structure to geometric solutions of invariant equations.

Findings

The basic building blocks are solvable Lie algebras constructed from pairs (F, v).

When F is nilpotent, the resulting Lie algebra is also nilpotent.

The approach connects Lie algebra invariants to geometric equations involving Casimir functions.

Abstract

We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on a dual space to a linear space $V$. This algebra is solvable. In particular, when $F$ is nilpotent, the Lie algebra is also nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. %Which allows, having a collection of pairs $(F_i,v_i)$, $i=1, \dots, n$, to construct any Lie algebra. Using relations between the Lie algebra, the Lie--Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has a geometric sense. Several examples illustrate the importance of these constructions.