Classification of Realizations of Lie Algebras of Vector Fields on a Circle

TL;DR

This paper classifies finite-dimensional Lie algebra realizations of vector fields on a circle, providing canonical forms, reduction methods, and counting formulas for inequivalent realizations.

Contribution

It introduces canonical forms for 2- and 3-dimensional noncommutative Lie algebra realizations on a circle and shows all others reduce to these forms via global transformations.

Findings

Canonical forms for 2- and 3-dimensional realizations are established.

All other realizations can be reduced to these canonical forms.

Formulas for counting inequivalent realizations are derived.

Abstract

Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical forms of realizations of two- and three-dimensional noncommutative algebras are obtained. It is shown that all other realizations of smooth vector fields are reduced to this form using global transformations. Some combinatorial formulas for the number of inequivalent realizations of these algebras are obtained.