Nonunital decompositions of the matrix algebra of order three

TL;DR

This paper classifies all ways to decompose the 3x3 complex matrix algebra into two subalgebras that do not contain the identity, completing the understanding of such decompositions and related Rota-Baxter operators.

Contribution

It provides a complete classification of nonunital decompositions of M_3(C) into two subalgebras and describes associated Rota-Baxter operators of nonzero weight.

Findings

All decompositions of M_3(C) into two subalgebras without the identity are classified.

The classification of Rota-Baxter operators of nonzero weight on M_3(C) is completed.

New structural insights into the algebraic decompositions of 3x3 complex matrices.

Abstract

All decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two subalgebras such that none of the subalgebras contains the identity matrix are classified. Thus, the classification of all decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two subalgebras as well as description of Rota-Baxter operators of nonzero weight on $M_3(\mathbb{C})$ is finished.