## 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 Exceptional Jordan algebra of the complexified octonions, 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.
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{Alessio} that extends
the $ {\bf e}_8$ algebra to $ {\bf e}_8^{ (n) } $ with $ n $ integer.

**Comments:** 11 Pages. Submitted to Advances in Applied Clifford Algebras

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### Submission history

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

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