arXiv subjects

Giovanni Busalacchi

Publications and source records attributed to Giovanni Busalacchi.

2 recordsLinked to original sources

arXiv2025

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 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.

math.RA↗

arXiv2025

Cocharacters of generalized polynomial identities

In this paper we extend the cocharacter theory to generalized identities of $W$-algebras. We prove that the Hilbert series of the relatively free $W$-algebra admits an expansion in terms of Schur functions whose coefficients coincide with generalized cocharacter multiplicities. Moreover, we prove analogues of the Hook and Strip theorems for $W$-algebras and we derive growth bounds for generalized codimension and colenght sequences. Finally, we establish that every variety $\mathcal{V}$ of $W$-algebras is generated by the Grassmann envelope of a finitely generated $W$-superalgebra, and if $\mathcal{V}$ satisfies a generalized Capelli set, then it is generated by a finitely generated $W$-algebra.

math.RA↗