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 $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth 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 $|F|$, the cardinality of the base field and we give a recipe of how to construct them.

Giambruno, A., La Mattina, D., Zaicev, M. (2014). Classifying the Minimal Varieties of Polynomial Growth. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 66, no. 3, 625-640.

### Classifying the Minimal Varieties of Polynomial Growth

#### Abstract

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 $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth like $n^k$ but any proper subvariety grows like $n^t$ with $t 4$, the number of minimal varieties is at least $|F|$, the cardinality of the base field and we give a recipe of how to construct them.
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Giambruno, A., La Mattina, D., Zaicev, M. (2014). Classifying the Minimal Varieties of Polynomial Growth. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 66, no. 3, 625-640.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10447/97275