6533b7dbfe1ef96bd127003c

RESEARCH PRODUCT

Minimal star-varieties of polynomial growth and bounded colength

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Abstract 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.