Algebras of cubic matrices

TL;DR

This paper studies algebraic structures of cubic matrices with various multiplications, focusing on associative Maksimov's multiplication, and explores their properties, isomorphisms, and subalgebras.

Contribution

It introduces a framework for algebras of cubic matrices based on Maksimov's associative multiplication and analyzes their structural properties and homomorphisms.

Findings

ACMs are not baric.

ACMs are commutative only when m=1.

Homomorphisms exist from ACMs to accompanying algebras.

Abstract

We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size $m$. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative iff $m=1$. We introduce a notion of accompanying algebra (which is $m^2$-dimensional) and show that there is a homomorphism from any ACM to the accompanying algebra. We describe (left and right) symmetric operations and give left and right zero divisors of the corresponding ACMs. Moreover several subalgebras and ideals of an ACM are constructed.