Research ArticleNo Access

A note on Jordan algebras, three generations and exceptional periodicity

Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, GA 30314, USA

Ronin Institute, 127 Haddon Pl, NJ 07043, USA

https://doi.org/10.1142/S0219887820500711Cited by:1 (Source: Crossref)

Abstract

It is shown that the algebra $J_{3} [C \otimes O] \otimes C l (4, C)$ $J_{3} [C \otimes O] \otimes C l (4, C)$ based on the complexified Exceptional Jordan, and the complex Clifford algebra in 4D, is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in 4D, and include additional fermionic dark matter candidates. Furthermore, the model described in this paper can account also for the Standard Model gauge symmetries. We extend these results to the Magic Star algebras of Exceptional Periodicity developed by Marrani–Rios–Truini and based on the Vinberg cubic $T$ algebras which are generalizations of exceptional Jordan algebras. It is found that there is a one-to-one correspondence among the real spinorial degrees of freedom of four generations of fermions in 4D with the off-diagonal entries of the spinorial elements of the $p a i r$ $T_{3}^{8, n}, ({\bar{T}}_{3}^{8, n})$ of Vinberg matrices at level $n = 2$ . These results can be generalized to higher levels $n > 2$ leading to a higher number of generations beyond 4. Three $p a i r s$ of $T$ algebras and their conjugates $\bar{T}$ were essential in the Magic Star construction of Exceptional Periodicity that extends the $e_{8}$ algebra to $e_{8}^{(n)}$ with $n$ integer.

Dedicated to the loving memory of Chris Baker

Keywords:

AMSC: 15A66, 17C40, 17C60, 17C90

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