Comment on "Generators of matrix algebras in dimension 2 and 3"

TL;DR

This paper critically examines a theorem claiming to identify generators of 3x3 matrix algebras, revealing that the theorem's core identity is invalid, thus challenging previous assumptions about algebra generation.

Contribution

The authors demonstrate that a previously published theorem on generating 3x3 matrix algebras contains a fundamental error, invalidating its conclusions.

Findings

Theorem 7 in the referenced paper is incorrect.

The identity used in the theorem does not hold in general.

The paper clarifies the limitations of previous generator criteria.

Abstract

Theorem 7 in Ref. [Linear Algebra Appl., 430, 1-6, (2009)] states sufficient conditions to determine whether a pair generates the algebra of 3x3 matrices over an algebraically closed field of characteristic zero. In that case, an explicit basis for the full algebra is provided, which is composed of words of small length on such pair. However, we show that this theorem is wrong since it is based on the validity of an identity which is not true in general.