Unital 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, with one containing the identity, providing a comprehensive structural understanding.

Contribution

It provides a complete classification of unital decompositions of the 3x3 matrix algebra into two subalgebras, a novel structural result.

Findings

All such decompositions are explicitly characterized.

The classification reveals the possible algebraic structures involved.

The results contribute to understanding algebraic decompositions in matrix algebras.

Abstract

We classify all decompositions of $M_3(\mathbb{C})$ into a direct vector-space sum of two subalgebras such that one of the subalgebras contains the identity matrix.