Varieties of graded W-algebras and asymptotic behavior of codimension growth

Let W be a G-graded algebra over a field of characteristic zero, where G is a finite group. We develop a theory of generalized G-graded polynomial identities satisfied by any finite-dimensional W-algebra A, by means of the graded multiplier algebra of A. In particular, we first prove that the graded generalized exponent exists and equals the ordinary one. Then, we explicitly compute the G-graded generalized identities of UT2, the 2×2 upper triangular matrix algebra equipped with its canonical Z2-grading, under all possible graded W-actions. Finally, we exhibit examples of varieties of graded W-algebras with almost polynomial growth of the codimensions.