Dynamical symmetries and symmetry algebras

Abstract

We consider the interaction of three multiplets of particles under the assumption that the members of each one of these multiplets have the same mass and the same spin. The vertex self-consistency conditions lead to an algebraic structure involving the coupling matrices. This structure, referred to as a symmetry algebra and denoted by the symbol (N,n,ν), is characterized essentially by the numbers of particles N, n, and ν belonging to each one of the three multiplets and is independent of the particular underlying dynamics. A question of particular interest is whether dynamical self-consistency implies the existence of a symmetry group that leaves the interaction invariant. We analyze this problem in detail for three particularly simple but instructive symmetry algebras. It is shown that the algebra (N,n,ν=Nn) corresponds to the case of maximal symmetry, the interaction being invariant under the group U(N)×U(n). The algebra (N,n,ν=Nn-1) is shown to have a solution only if n=N, in which case it corresponds to symmetry under the group U(N). Finally we consider the particularly instructive algebra (N=3,n=3,ν=3) which is shown to admit of three physically distinct solutions, which correspond to invariance of the interaction under a three-parameter Abelian group, the orthogonal group in three dimensions, and the 24-element permutation group S₄, respectively.