On C $$\varvec{\otimes }$$ H $$\varvec{\otimes }$$ O-Valued Gravity, Sedenions, Hermitian Matrix Geometry and Nonsymmetric Kaluza–Klein Theory

Sedenions, the Clifford algebra $\mathbb{C}l(8)$, and three fermion generations

“…The left adjoint algebra in this case is (C ⊗ H ⊗ O) L = C (8), the same as the left adjoint algebra of C ⊗ S. In [11] on the other hand, all the degrees of freedom are included by considering all eight minimal left ideals of C (6), instead of just two as in [2,10]. An alternative way of describing the full degrees of freedom is to therefore consider all 16 minimal left ideals of C (8).…”

“…It was mentioned earlier that (C ⊗ H ⊗ O) L ∼ = (C ⊗ S) L ∼ = C (8). Unlike the smaller algebra C (6) associated with C ⊗ O, this larger algebra admits a triality automorphism associated with Spin (8). Triality is a non-linear outer automorphism of Spin(8) of order three.…”

“…Finding a theoretical basis for the mathematical structure that underlies the SM remains a prominent challenge in physics. Instead of the common approach of embedding the SM gauge group into some larger group, recently there has been an interest in using the fundamental and generative structures of the four normed division algebras as a simple mathematical framework for particle physics [1,2,3,4,5,6,7,8].…”

The Standard Model particle content with complete gauge symmetries from the minimal ideals of two Clifford algebras

“…In [2], the basis states of two minimal left ideals of the Clifford algebra C (6) were shown to transform precisely as a single generation of leptons and quarks under the electrocolor group SU (3) C ⊗ U (1) E M . The Clifford algebra C (6) there is generated from the left adjoint actions of the complex octonions C ⊗ O on itself. 1 A Witt decomposition of C (6) splits the algebra into a basis of nilpotent ladder operators, and the unitary symmetries that preserve this Witt decomposition are SU (3) and U (1).…”

“…The Clifford algebra C (6) there is generated from the left adjoint actions of the complex octonions C ⊗ O on itself. 1 A Witt decomposition of C (6) splits the algebra into a basis of nilpotent ladder operators, and the unitary symmetries that preserve this Witt decomposition are SU (3) and U (1). In other words, the minimal ideals are closed under the action of these unitary symmetry, and one finds that each basis state of the ideal transforms like a specific lepton or quark.…”