CLASSIFICATION OF THE REALIZATIONS OF LIE ALGEBRAS FOR THE VECTOR FIELDS ON A CIRCLE

摘要

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.