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ON POSSIBLE GENERALIZATION OF THE CONCEPT OF REPRESENTATION OF GROUPS
Kh. Yiglane
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Abstract
A generalized representation of groups is defined which, in the case of nonsingunlar representatlon matrices submitted to commutation relations, can be reduced to a projective representation. It was proved that the generalized representation of the reflection group in two-dimensional (or three-dimensional) space provides three-row spin matrices, whereas the projeetive representation of the same group yields two-row spin matrices. A similar generalization of representation of the reflection group in four dimensions is described by five- or ten-row KemmerDuffin matrices P. The corresponding projective representation is characterized by Dirac matrices gamma v. Neither the generalized nor the projective representations of the reflection group space-time describe spins higher than 1. The non-associative Cayley algebra is shown to be a nonassociative representation of the reflection group in three dimensions. (auth)