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Classifying the Minimal Varieties of Polynomial Growth

Published online by Cambridge University Press:  20 November 2018

Antonio Giambruno ,
Daniela La Mattina  and
Mikhail Zaicev
Antonio Giambruno
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, 90123 Palermo, Italy. e-mail: antonio.giambruno@unipa.it, daniela.lamattina@unipa.it
Daniela La Mattina
Affiliation:
Dipartimento di Matematica e Informatica, Università di Palermo, 90123 Palermo, Italy. e-mail: antonio.giambruno@unipa.it, daniela.lamattina@unipa.it
Mikhail Zaicev
Affiliation:
Department of Algebra, Faculty of Mathematics and Mechanics, Moscow State University, 119992 Moscow Russian State University of Trade and Economics, Smol'naya 36, 125993 Moscow, Russia. e-mail: zaicevmv@mail.ru
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Abstract

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Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with 1 over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ that are minimal of polynomial growth (i.e., their sequence of codimensions grows like ${{n}^{k}}$, but any proper subvariety grows like ${{n}^{t}}$ with $t\,<\,k$). These varieties are the building blocks of general varieties of polynomial growth.

It turns out that for $k\,\le \,4$ there are only a finite number of varieties of polynomial growth ${{n}^{k}}$, but for each $k\,>\,4$, the number of minimal varieties is at least $\left| F \right|$, the cardinality of the base field, and we give a recipe for their construction.

Keywords

16R1016P90polynomial identitycodimensionpolynomial growthT-ideal
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 625 - 640
Copyright
Copyright © Canadian Mathematical Society 2014

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