We show that three generations of leptons and quarks with unbroken Standard Model gauge symmetry SU (3) c × U (1) em can be described using the algebra of complexified sedenions C ⊗ S. A primitive idempotent is constructed by selecting a special direction, and the action of this projector on the basis of C ⊗ S can be used to uniquely split the algebra into three complex octonion subalgebras C ⊗ O. These subalgebras all share a common quaternionic subalgebra. The left adjoint actions of the 8 C-dimensional C ⊗ O subalgebras on themselves generates three copies of the Clifford algebra C (6). It was previously shown that the minimal left ideals of C (6) describe a single generation of fermions with unbroken SU (3) c × U (1) em gauge symmetry. Extending this construction from C ⊗ O to C ⊗ S naturally leads to a description of exactly three generations.

This paper represents a first attempt at unifying two promising attempts to understand the origin of the internal symmetries of leptons and quarks. It is shown that each of the four normed division algebras over the reals admits a representation of a circular braid group. For the complex numbers and the quaternions, the represented circular braid groups are B 2 and B c 3 , precisely those used to construct leptons and quarks as framed braids in the Helon model of Bilson-Thompson. It is then shown that these framed braids coincide with the states that span the minimal left ideals of the complex (chained) octonions, shown by Furey to describe one generation of leptons and quarks with unbroken SU (3) c and U (1) em symmetry.The identification of basis states of minimal ideals with certain framed braids is possible because the braiding in B 2 and B c 3 in the Helon model are interchangeable. It is shown that the framed braids in the Helon model can be written as pure braid words in B c 3 with trivial braiding in B 2 , something which is not possible for framed braids in general.

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We argue that quantum gravity theories that carry a Lie-algebraic modification of the Poincaré and Heisenberg algebras inevitably provide inhomogeneities that may serve as seeds for cosmological structure formation. Furthermore, in this class of theories one must expect a strong polarization and spin dependence of various quantum gravity effects.

Building upon previous works, it is shown that two minimal left ideals of the complex Clifford algebra C (6) and two minimal right ideals of C (4) transform as one generation of leptons and quarks under the gauge symmetry SU (3) C × U (1) E M and SU (2) L respectively. The SU (2) L weak symmetries are naturally chiral. Combining the C (6) and C (4) ideals, all the gauge symmetries of the Standard Model, together with its lepton and quark content for a single generation are represented. The combined ideals can be written as minimal left ideals of C (6) ⊗ C (4) ∼ = C (10) in a way that preserves individually the C (6) structure and C (4) structure of physical states. This resulting model captures many of the attractive features of the Georgi and Glashow SU (5) Grand Unified Theory without introducing proton decay or other unobserved processes. Such processes are naturally excluded because they do not preserve the underlying algebraic structure.