Codimension growth for polynomial identities of representations of Lie algebras

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)$(A,L)$ where A is an associative enveloping algebra for the Lie algebra L. We obtain a characterization of ideals of weak identities with polynomial growth of the codimensions in terms of their cocharacter sequence. Recall that such a characterization was obtained by Kemer in [12] for associative algebras and by Benediktovich and Zalesskii in [2] for Lie algebras. We prove that the pairs (UT2,UT2(−))$\Big (UT_2,UT_2^{(-)}\Big )$, (E,E(−))$\big (E,E^{(-)}\big )$ and (M2,sl2)$\big (M_2,sl_2\big )$ generate varieties of pairs of almost polynomial growth. Here E denotes the infinite dimensional Grassmann algebra with 1. Also UT2$UT_2$ is the associative subalgebra of M2 (the 2 × 2 matrices over the field K) consisting of upper triangular matrices and sl2$sl_2$ is the Lie subalgebra of M2(−)$M_2^{(-)}$ of the traceless matrices.