Minimal ⁎-varieties and minimal supervarieties of polynomial growth

Abstract By a φ-variety V , we mean a supervariety or a ⁎-variety generated by an associative algebra over a field F of characteristic zero. In this case, we consider its sequence of φ-codimensions c n φ ( V ) and say that V is minimal of polynomial growth n k if c n φ ( V ) grows like n k , but any proper φ-subvariety grows like n t with t k . In this paper, we deal with minimal φ-varieties generated by unitary algebras and prove that for k ≤ 2 there is only a finite number of them. We also explicit a list of finite dimensional algebras generating such minimal φ-varieties. For k ≥ 3 , we show that the number of minimal φ-varieties can be infinity and we classify all minimal φ-varieties of polynomial growth n k by giving a recipe for the construction of their T φ -ideals.