Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Alina Dobrogowska,Karolina Wojciechowicz

doi:10.3390/sym13081455

Abstract

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

Highlights

To begin, we recall the definition of a Lie bundle
We proposed a new approach to the classification of low-dimensional Lie algebras using concepts well known in the theory of integrable systems, such as the bi-Hamiltonian structure
In the case of the four-dimensional Lie algebras, we described them in terms of three such families

Summary

Introduction

We recall the definition of a Lie bundle. Let V, W be finite dimensional vector spaces. Lie bundle, which corresponds to the compatible Poisson structure on the dual space (the same for all of these algebras). We consider Lie bundles generating real four-dimensional Lie algebras. Let g be a real n-dimensional Lie algebra It is a well-known fact that on the dual space g∗ , there exists a canonical Poisson structure. The Lie bundle A a1 ,...,an−1 (n), S a1 ,...,an−1 (n) introduces a family of Lie–Poisson brackets on the dual space L+ (n):. The Lie bundle A a1 ,...,an−1 (n), S a1 ,...,an−1 (n) can be generalized to the case when some parameters ai are equal to zero. ...,an ( n ) is a linear bundle of Lie algebras In this case, we will allow that matrix elements are complex numbers.

Three-Dimensional Lie Algebras and Lie Bundles

Four-Dimensional Lie Algebras and Lie Bundles

Conclusions