Ternary algebras associated with irreducible tensor representations of SO(3) and quark model

TL;DR

This paper constructs associative algebras from irreducible tensor representations of SO(3), revealing binary and ternary relations, with potential applications in quark models and Grand Unification Theories.

Contribution

It introduces a new class of associative algebras derived from tensor representations, detailing their algebraic relations and potential physical applications.

Findings

Algebras have binary and ternary relations based on Z_3 and cubic roots of unity.

The 5-dimensional subspace forms an irreducible tensor representation of SO(3).

Constructed algebras can be applied in quark models and Grand Unification Theories.

Abstract

We show that each irreducible tensor representation of weight 2 of the rotation group of three-dimensional space in the space of rank 3 covariant tensors gives rise to an associative algebra with unity. We find the algebraic relations that the generators of these algebras must satisfy. Part of these relations has a form of binary relations and another part has a form of ternary relations. The structure of ternary relations is based on the cyclic group Z_3 and the primitive cubic root of unity q=\exp(2\pi i/3). The subspace of each algebra spanned by the triple products of generators is 5-dimensional and it is the space of an irreducible tensor representation of weight 2 of the rotation group SO(3). We define a Hermitian scalar product in this 5-dimensional subspace and construct an orthonormal basis for it. Then we find the representation matrix of an infinitesimal rotation. We show that constructed algebras with binary and ternary relations can have applications in the quark model and Grand Unification Theories.