(FUNDAMENTAL AND APPLIED MATHEMATICS)

2011/2012, VOLUME 17, NUMBER 1, PAGES 169-188

## Classification of matrix subalgebras of length $1$

O. V. Markova

Abstract

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We define the length of a finite system of generators of a given algebra $\mathcal A$ as the smallest number $k$ such that words of length not greater than $k$ generate $\mathcal A$ as a vector space, and the length of the algebra is the maximum of the lengths of its systems of generators. In this paper, we obtain a classification of matrix subalgebras of length $1$ up to conjugation. In particular, we describe arbitrary commutative matrix subalgebras of length $1$, as well as those that are maximal with respect to inclusion.

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