On classification of Lie algebra realizations

TL;DR

This paper develops a generalized framework for classifying Lie algebra realizations by vector fields, extending existing methods to regular local realizations and proposing an algorithm for their systematic classification.

Contribution

It introduces a generalized classification approach for Lie algebra realizations, including an algorithm for constructing classifications of regular local realizations.

Findings

Established a correspondence between transitive local realizations and subalgebra classifications.

Formulated a rigorous classification problem for general realizations.

Presented an algorithm for constructing classifications of Lie algebra realizations.

Abstract

We study realizations of Lie algebras by vector fields. A correspondence between classification of transitive local realizations and classification of subalgebras is generalized to the case of regular local realizations. A reasonable classification problem for general realizations is rigorously formulated and an algorithm for construction of such classification is presented.