1 results
Classifying the Minimal Varieties of Polynomial Growth
-
- Journal:
- Canadian Journal of Mathematics / Volume 66 / Issue 3 / 01 June 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 625-640
- Print publication:
- 01 June 2014
-
- Article
-
- You have access
- Export citation
-
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.
