On dynamical groups: Classification of Lie algebras with Galilei subalgebras

Abstract

Because of the pure group theoretical approach to the free nonrelativistic particle through an integrable irreducible representation of the quantum mechanical Galilei (Lie) algebra G, it is reasonable to construct so-called dynamical (Lie) algebras D which possess an integrable representation describing interacting (nonrelativistic) systems. Such dynamical algebras D should contain the geometrical subalgebra G0 of G, spanned by the mass operator, the momentum, the angular-momentum and the position operators. Furthermore, the relation between the free and the interacting system is simplified if the one-dimensional Lie algebra T generating free time translations is a subalgebra of D. Hence preferred candidates for D are those Lie algebras L which possess a subalgebra isomorphic to G or to G0, i.e., those L for which an injective homomorphism ε:G→ε(G)⊂L or G0→ε(G0)⊂L exists. ε is called an embedding of G or of G0 in L. Our main result is a complete classification of (i) all nonsemisimple L with Levi decomposition L=S[addition in left half circle]F with G or G0 embedding ε such that S⊂ε(G) or S⊂ε(G0) (ii) all complex semisimple L̄ with an embedding of the complex extension of G, (iii) all real simple L being real forms or realifications of the lowest dimensional L̄ (i.e., A5, B3, C4) with G-embedding. The result gives a fairly complete list of all candidates for dynamical nonrelativistic algebras. The physical aspects of two of them, the conformal Galilei algebra Gc and a limitable dynamical algebra Dt, are discussed. A method for the construction of physically useful integrable representations for Gc and Dt is given. Some general properties of nonrelativistic and of relativistic dynamical algebras with G-embeddings are considered.