The cubic chessboard

TL;DR

This paper surveys recent developments in cubic (ternary) algebraic structures, exploring their symmetries, non-associative forms, graded algebras, and potential applications to gauge theories and quark field descriptions.

Contribution

It introduces new concepts like Z_3-graded matrix algebras, a generalized exterior calculus, and a ternary Clifford algebra framework, connecting them to physical theories.

Findings

Z_3-graded matrix algebras and their properties

A new gauge theory based on ternary differential calculus

A ternary Dirac equation and its potential link to quark fields

Abstract

We present a survey of recent results, scattered in a series of papers that appeared during past five years, whose common denominator is the use of cubic relations in various algebraic structures. Cubic (or ternary) relations can represent different symmetries with respect to the permutation group S_3, or its cyclic subgroup Z_3. Also ordinary or ternary algebras can be divided in different classes with respect to their symmetry properties. We pay special attention to the non-associative ternary algebra of 3-forms (or ``cubic matrices''), and Z_3-graded matrix algebras. We also discuss the Z_3-graded generalization of Grassmann algebras and their realization in generalized exterior differential forms. A new type of gauge theory based on this differential calculus is presented. Finally, a ternary generalization of Clifford algebras is introduced, and an analog of Dirac's equation is discussed, which can be diagonalized only after taking the cube of the Z_3-graded generalization of Dirac's operator. A possibility of using these ideas for the description of quark fields is suggested and discussed in the last Section.