Classifying the Minimal Varieties of Polynomial Growth

Giambruno, Antonio ; Mattina, Daniela La ; Zaicev, Mikhail

Canadian journal of mathematics, Tome 66 (2014) no. 3, pp. 625-640

Résumé

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.

Détail

DOI : 10.4153/CJM-2013-016-5

Mots-clés : 16R10, 16P90, polynomial identity, codimension, polynomial growth, T-ideal

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