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

Let be a ‐algebra over a field of characteristic zero, where is any ‐algebra. We first develop a comprehensive theory of generalized identities independent of the algebraic structure of , using the multiplier algebra of . Then, we investigate the generalized variety generated by the 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 ‐algebras. Finally, we provide a counterexample to the Specht property of generalized ‐ideals in characteristic zero.