2020/06/09 by Viktor Abramov, Abramov, Viktor
Mathematics · Physics and Astronomy · #17B10 #20C35 #Advanced Topics in Algebra #Black Holes and Theoretical Physics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2006.05237
openalex publication_date 2020/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that each irreducible tensor representation of weight 2 of the\nrotation group of three-dimensional space in the space of rank 3 covariant\ntensors gives rise to an associative algebra with unity. We find the algebraic\nrelations that the generators of these algebras must satisfy. Part of these\nrelations has a form of binary relations and another part has a form of ternary\nrelations. The structure of ternary relations is based on the cyclic group Z3\nand the primitive cubic root of unity q=\exp(2\π i/3). The subspace of each\nalgebra spanned by the triple products of generators is 5-dimensional and it is\nthe space of an irreducible tensor representation of weight 2 of the rotation\ngroup SO(3). We define a Hermitian scalar product in this 5-dimensional\nsubspace and construct an orthonormal basis for it. Then we find the\nrepresentation matrix of an infinitesimal rotation. We show that constructed\nalgebras with binary and ternary relations can have applications in the quark\nmodel and Grand Unification Theories.\n