Biunits of ternary algebra of hypermatrices

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.