The effect of assuming the identity as a generator on the length of the matrix algebra

2015/01/23 by Thomas Laffey, Laffey, Thomas, Olga Markova +3
Mathematics · #15A30 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:15A30

paper · pdf · doi:10.48550/arxiv.1501.05806

15 pages

arxiv created 2015/01/23 · arxiv updated 2015/01/26

Abstract

Let M_n(\mathbbF) be the algebra of n × n matrices and let \mathcal S be a generating set of M_n(\mathbbF) as an \mathbbF-algebra. The length of a finite generating set \mathcal S of M_n(\mathbbF) is the smallest number k such that words of length not greater than k generate M_n(\mathbbF) as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets \mathcal S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed.

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