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Commuting Clifford actions
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It shown that if a vector space carries commuting actions of two Clifford algebras, then the quadratic monomials using generators from either Clifford algebra determine a spinor representation of an orthogonal Lie algebra. Examples of this construction have applications to high energy physics, particularly to the standard model and unification. It is shown how to use Clifford data to construct spectral triples for the Pati-Salam model that admit an action of Spin(10).
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Standard Model Symmetries and the Nested Embeddings of $\mathbb{R}\subset\mathbb{C}\subset\mathbb{H}\subset\mathbb{O}$
The Standard Model gauge group and its charge operators emerge from O⊕H⊕C⊕R by stabilizing top-graded elements and imposing an equal-trace condition.
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