Varieties of graded $W$-algebras and asymptotic behavior of codimension growth
Giovanni Busalacchi, Fabrizio Martino, Carla Rizzo
TL;DR
This work develops a framework for generalized W-graded polynomial identities in finite-dimensional G-graded W-algebras using the multiplier algebra, proving the generalized W-exponent exists and equals the ordinary graded PI-exponent. It provides explicit calculations of graded identities, codimensions, and cocharacters for UT2 under all possible graded W-actions, and identifies almost polynomial growth varieties among these cases. The results combine multiplier-algebra methods, generic algebras, and representation-theoretic analysis to map the growth landscape of graded identities. Overall, the paper clarifies when graded W-actions preserve exponent behavior and when they yield APG varieties, with UT2 serving as a concrete testbed across four action types.
Abstract
Let $W$ be a $G$-graded algebra over a field of characteristic zero, where $G$ is a finite group. We develope a theory of generalized $G$-graded polynomial identities satisfied by any finite-dimensional $W$-algebra $A$, by mean 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 $UT_2,$ the $2 \times 2$ upper triangular matrix algebra equipped with its canonical $\mathbb{Z}_2$-grading, under all the possible graded $W$-actions. Finally, we exhibit examples of varieties of graded $W$-algebras with almost polynomial growth of the codimensions.