Minimal star-varieties of polynomial growth and bounded colength

Let V be a variety of associative algebras with involution ⁎ over a field F of characteristic zero. Giambruno and Mishchenko proved in [6] that the ⁎-codimension sequence of V is polynomially bounded if and only if V does not contain the commutative algebra D=F⊕F, endowed with the exchange involution, and M, a suitable 4-dimensional subalgebra of the algebra of 4×4 upper triangular matrices, endowed with the reflection involution. As a consequence the algebras D and M generate the only varieties of almost polynomial growth. In [20] the authors completely classify all subvarieties and all minimal subvarieties of the varieties var⁎(D) and var⁎(M). In this paper we exhibit the decompositions of the ⁎-cocharacters of all minimal subvarieties of var⁎(D) and var⁎(M) and compute their ⁎-colengths. Finally we relate the polynomial growth of a variety to the ⁎-colengths and classify the varieties such that their sequence of ⁎-colengths is bounded by three.