Biunits of ternary algebra of hypermatrices

TL;DR

This paper investigates the structure of ternary algebras formed by third-order hypermatrices, exploring their geometric invariants, algebraic properties, and potential physical implications like a generalized Pauli principle.

Contribution

It introduces a new ternary algebra framework for hypermatrices, analyzes SO(3)-invariants, and identifies biunits related to the representation theory of rotation groups.

Findings

Identification of two SO(3)-invariants of hypermatrices.

Definition of a Hermitian metric via one invariant.

Characterization of biunits in the algebra as elements satisfying regularity conditions.

Abstract

In this paper we study ternary algebras of third-order hypermatrices. By hypermatrix we mean a complex-valued variable with three indices, which is also called a three-dimensional matrix or spatial matrix. We assume that a hypermatrix is defined in three-dimensional Euclidean space and when this space is rotated, it transforms as a SO(3)-tensor. We consider two ternary multiplications of hypermatrices, which have the property of generalized associativity. We explore the geometric meaning of two independent SO(3)-invariants of hypermatrices and show that one of them defines a Hermitian metric. We study the 10-dimensional subspace of hypermatrices, known in the theory of representations of the rotation group, as the space of the weight 2 tensor representation of the rotation group. It is proved that the elements of this subspace that satisfy the regularity condition are right biunits of the ternary algebra of hypermatrices. The motivation for studying biunits of ternary algebra of hypermatrices was a ternary generalization of the Pauli exclusion principle.