Codimension Growth for Weak Polynomial Identities, and Non-integrality of the PI Exponent

Abstract

Let K be a field of characteristic zero. We study the asymptotic behavior of the codimensions for polynomial identities of representations of Lie algebras, also called weak identities. These identities are related to pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. First we obtain a characterization of ideals of weak identities with polynomial growth of the codimensions in terms of their cocharacter sequence. Moreover we obtain examples of pairs that generate varieties of pairs of almost polynomial growth. Second we show that any variety of pairs of associative type is generated by the Grassmann envelope of a finitely generated superpair. As a corollary we obtain that any special variety of pairs which does not contain pairs of type (R, sl2), consists of pairs with a solvable Lie algebra. Here sl2 denotes the Lie algebra of the 2 × 2 traceless matrices. Finally we give an example of a pair that contradicts a conjecture due to Amitsur.