## A Note on Jordan Algebras, Three Generations and Exceptional Periodicity

**Authors:** Carlos Castro

It is shown that the algebra $ {{\bf J } }_3 [ { \bf C \otimes O } ] \otimes {\bf Cl(4,C) } $ based on the complexified Exceptional Jordan, and the complex Clifford algebra in $ {\bf 4D}$,
is rich enough to describe all the spinorial degrees of freedom of three generations of fermions in ${\bf 4D}$, and include additional fermionic dark matter candidates. Furthermore, the model described in this letter 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 $ {\bf 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 ${\bf 4}$ generations of fermions in $ {\bf 4D}$ with the off-diagonal entries of the spinorial elements of the $pair$ $ {\bf T}_3^{ 8, n}, ( {\bf {\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 $ {\bf 4 } $.
Three $pairs$ of ${\bf T}$ algebras and their conjugates $ {\bf {\bar T} }$ were essential in the Magic Star construction of Exceptional Periodicity \cite{Alessio1} that extends
the $ {\bf e}_8$ algebra to $ {\bf e}_8^{ (n) } $ with $ n $ integer.

**Comments:** 13 Pages.

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### Submission history

[v1] 2019-07-17 05:59:16

[v2] 2019-11-17 13:47:02

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