Contractions of Low-Dimensional Lie Algebras

TL;DR

This paper develops a rigorous framework for continuous contractions of low-dimensional Lie algebras, introduces new criteria, and systematically classifies all one-parametric contractions for algebras up to dimension four.

Contribution

It formulates new criteria for contractions, presents an algorithm for handling one-parametric contractions, and classifies all such contractions for low-dimensional Lie algebras.

Findings

All one-parametric contractions for Lie algebras of dimension ≤4 are constructed.

New criteria for contractions are proposed and validated.

Properties of multi-parametric and repeated contractions are analyzed.

Abstract

Theoretical background of continuous contractions of finite-dimensional Lie algebras is rigorously formulated and developed. In particular, known necessary criteria of contractions are collected and new criteria are proposed. A number of requisite invariant and semi-invariant quantities are calculated for wide classes of Lie algebras including all low-dimensional Lie algebras. An algorithm that allows one to handle one-parametric contractions is presented and applied to low-dimensional Lie algebras. As a result, all one-parametric continuous contractions for the both complex and real Lie algebras of dimensions not greater than four are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases. Levels and co-levels of low-dimensional Lie algebras are discussed in detail. Properties of multi-parametric and repeated contractions are also investigated.