International Centre for Theoretical Physics a Non-lie Algebraic Framework and Its Possible Merits for Symmetry Descriptions

A non-associative algebraic construction is introduced vbich "bears a relation to a Lie algebra SL paralleling the relation between an associative enveloping algebra and oC , The key ingredient of this algebraic construction is the presence of two parameters which relate it to the enveloping algebra of jf . The analogue of the Poincare-Birkhoff-Witt theorem is proved for the new algebra. Possibilities of physical relevance are also considered. It is noted that, if fully developed, the mathematical framework suggested by this nev algebra should be non-Lie. Subsequently, a certain scheme resulting from specific considerations connected with this (non-Lie) algebraic structure is found to bear striking resemblance to a recent phenomenological theory proposed for explaining CP violation by the K system. Some relevant speculations are also made in viev of certain recent trends of thought in elementary particle physics. Finally, in an appendix, a Gell-Mann-Okubo-like mass formula for the new algebra is derived for on SU{3) octet. MIEAMARE TRIESTE October 19T* * To be submitted for publication. I. The effectiveness of Lie algebraB in theoretical physics must be considered, by now, as established. nevertheless, with the deepening of our understanding of problems like symmetry breaking, discrete space-time symmetry violations etc., the need for a more effective algebraic structure emerges as a plausible alternative. Of course,if one were to adopt such an attitude one should not lose sight of the vast and "numerous successes of Lie algebras (and groups) in connection with elementary particles. In this regard, one may recall the socalled Jordan algebras introduced as far back aa 1334 by Jordan, von Neumann and Wigner defined by the identities