Multipliers, $W$-algebras and the growth of generalized polynomial identities
Fabrizio Martino, Carla Rizzo
TL;DR
The paper develops a general framework for generalized polynomial identities in $W$-algebras using multiplier algebras to separate the action from the algebraic structure, and proves a WM-type decomposition for finite-dimensional $W$-algebras. It then classifies $W$-varieties by growth, showing that, when $W$ acts in a finite-dimensional way, the generalized codimensions either grow polynomially or exponentially, with almost polynomial growth occurring only for the $UT_2^F$ and $UT_2^D$ cases. Through a detailed analysis of $UT_2$ under various $W$-actions, it establishes explicit generators for $\mathrm{Id}^W$ and formulates a growth dichotomy tied to the presence of these UT$_2$-varieties. Finally, it provides a counterexample to the Specht property in characteristic zero for generalized $T_W$-ideals via Grassmann algebras with non-finitely generated $W$-action, highlighting the central role of the finiteness of the action in the generalized setting.
Abstract
Let $A$ be a $W$-algebra over a field $F$ of characteristic zero, where $W$ is any $F$-algebra. We develop a comprehensive theory of generalized identities independently on the algebraic structure of $W$, using the multiplier algebra of $A.$ We also characterize the generalized varieties of almost polynomial growth generated by finite dimensional $W$-algebras. Finally, we provide a counter-example to the Specht property of generalized $T_W$-ideals in characteristic zero.