Multipliers, W‐algebras, and the growth of generalized polynomial identities

Let 𝐴 be a 𝑊-algebra over a field 𝐹 of characteristic zero, where 𝑊 is any 𝐹-algebra. We first develop a comprehensive theory of generalized identities independent of the algebraic structure of 𝑊, using the multiplier algebra of 𝐴. Then, we investigate the generalized variety generated by the 𝑘×𝑘 matrix algebra with a suitable action, proving that it exhibits almost polynomial growth of the generalized codimensions. Furthermore, we characterize the generalized varieties of almost polynomial growth generated by finite-dimensional 𝑊-algebras. Finally, we provide a counterexample to the Specht property of generalized 𝑇_𝑊-ideals in characteristic zero.