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

Abstract

We describe the realizations of finite-dimensional Lie algebras of smooth tangential vector fields on a circle and construct “canonical” realizations of the two-dimensional noncommutative algebra, as well as the algebra 𝔰𝔩(2,ℝ) . It is shown that any realization of these algebras by smooth vector fields can be reduced to one of “canonical” realizations with the help of piecewise-smooth global transformations of a circle onto itself. We also deduce the formulas for the number of nonequivalent realizations.