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arXiv · 2006.05237

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

Viktor Abramov
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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.

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Viktor Abramov. 2020-06-09. Ternary algebras associated with irreducible tensor representations of SO(3) and quark model. https://arxiv.org/abs/2006.05237

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