Three generations of colored fermions with $$S_3$$ family symmetry from Cayley–Dickson sedenions

Abstract

An algebraic representation of three generations of fermions with $$SU(3)_C$$ S U ( 3 ) C color symmetry based on the Cayley–Dickson algebra of sedenions $${\mathbb {S}}$$ S is constructed. Recent constructions based on division algebras convincingly describe a single generation of leptons and quarks with Standard Model gauge symmetries. Nonetheless, an algebraic origin for the existence of exactly three generations has proven difficult to substantiate. We motivate $$\mathbb {S}$$ S as a natural algebraic candidate to describe three generations with $$SU(3)_C$$ S U ( 3 ) C gauge symmetry. We initially represent one generation of leptons and quarks in terms of two minimal left ideals of $$\mathbb {C}\ell (6)$$ C ℓ ( 6 ) , generated from a subset of all left actions of the complex sedenions on themselves. Subsequently we employ the $$S_3$$ S 3 automorphism of order three, which is an automorphism of $$\mathbb {S}$$ S but not of $$\mathbb {O}$$ O , to generate two additional generations. Given the relative obscurity of sedenions, efforts have been made to present the material in a self-contained manner.

Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems

“…Note that the complex Clifford algebra Cl(6, C) [57][58][59][60][61][62][63][64][65][66][67][68][69][70][71] and its isomorphic equivalences such as Cl(7, 0) 16 and the complex 8 × 8 matrix M (8, C) algebra, 72 either as standalone algebra or in association with octonions or sedenions, have been investigated in connection with the color SU (3) c symmetry and the other standard model symmetries. Nevertheless, in our approach 27,29,30 we choose to stick with the real Cl(0, 6) which is based on two reasons: The first reason is that an algebraic spinor of the real Cl(0, 6) with 64 real basis elements is a perfect match with one generation of 16 Weyl fermions in the standard model with 32 complex degrees of freedom (i.e.…”

Towards a Generalized Cayley–Dickson Construction through Involutive Dimagmas

“…These algebras extend the real numbers to more complex structures, incorporating properties such as addition, multiplication, and conjugation. Originally introduced in 1845 by the mathematician Arthur Cayley [1], and later analyzed by Leonard Eugene Dickson in 1919 [2], the Cayley-Dickson construction has found significant applications in various branches of mathematics, including algebra, analysis, and geometry [3][4][5][6][7][8][9][10][11][12][13][14] along with mathematical physical applications [15][16][17][18][19][20][21][22][23][24][25][26][27][28].…”