Poisson bracket realizations of Lie algebras and subrepresentations of (ad⊗k)s

Abstract

A procedure which associates Poisson bracket realizations of a Lie algebra L to subrepresentations of the extension (ad⊗k)s of the adjoint action to the algebra of polynomials defined on the dual space L* is pointed out. The procedure is applied, for k=2, to the real forms of the semisimple Lie algebras of types D3 and B2∼C2, in particular to the algebras so(4,2), so(4,1), and so(3,2)∼sp(4,R). The results obtained for the algebra sp(4,R) have led to an algebraic foundation for the constraints satisfied by the dynamical variables for the classical limit of the generalized helium problem.