Leutwyler, H. ; Sudarshan, E. C. G. (1967) Dynamical symmetries and symmetry algebras Physical Review, 156 (5). pp. 16371643. ISSN 0031899X

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Official URL: http://prola.aps.org/abstract/PR/v156/i5/p1637_1
Related URL: http://dx.doi.org/10.1103/PhysRev.156.1637
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 selfconsistency 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 selfconsistency 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=Nn1) 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 threeparameter Abelian group, the orthogonal group in three dimensions, and the 24element permutation group S_{4}, respectively.
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