Realization of Lie Algebras by Analytic Functions of Generators of a Given Lie Algebra

Abstract

In this paper we discuss the problem of the Poisson bracket realization of various Lie algebras in terms of analytic functions of the generators of a given Lie algebra. We pose and solve the problem of realizing the general O(4), O(3, 1), and E(3) algebras in terms of analytic functions of the generators of a prescribed realization of an E(3) algebra. A similar problem is solved for the symmetric tensor realizations of SU(3) and SL(3, R). Related questions are discussed for O(n + 1), O(n, 1), E(n), SU(n), and SL(n, R). We study in some detail the finite canonical transformations realized by the generators of the various groups. The relation of these results to the reconstruction problem is briefly discussed.