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Piecewise periodicity structure estimates in Shirshov’s height theorem

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Abstract

The Gelfand-Kirillov dimension of l-generated general matrices is equal to (l − 1)n 2 + 1. Due to the Amitzur-Levitsky theorem, the minimal degree of the identity of this algebra is 2n. That is why the essential height of A being an l-generated PI-algebra of degree n over every set of words is greater than (l − 1)n 2/4 + 1. In this paper we prove that if A has a finite Gelfand-Kirillov dimension, then the number of lexicographically comparable subwords with the period (n − 1) in each monoid of A is not greater than (l − 2)(n − 1). The case of subwords with the period 2 can be generalized to the proof of Shirshov’s height theorem.

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Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia

    M. I. Kharitonov

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  1. M. I. Kharitonov
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Original Russian Text © M.I. Kharitonov, 2013, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2013, Vol. 68, No. 1, pp. 10–16.

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Kharitonov, M.I. Piecewise periodicity structure estimates in Shirshov’s height theorem. Moscow Univ. Math. Bull. 68, 26–31 (2013). https://doi.org/10.3103/S0027132213010051

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