The method of virtual copies and contractions of simple Lie algebras

Abstract

Contractions of simple Lie algebras onto semidirect sums are analyzed in the context of the method of virtual copies, and various existence conditions are obtained in terms of the branching rules of representations and the properties of the characteristic representation associated to an embedding of semisimple algebras. Contractions of the Lie algebra sp(2N, R) with respect to nonmaximal symplectic subalgebras are studied in some detail. In particular, a realization of the subalgebra in terms of cubic polynomial operators is constructed.