Algebras of cubic matrices
M. Ladra, U.A. Rozikov

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 -cubic matrices (with ). 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 . 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 . We introduce a notion of accompanying algebra (which is -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…
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Taxonomy
TopicsAdvanced Topics in Algebra · Rings, Modules, and Algebras · Matrix Theory and Algorithms
