Finite-dimensional representability for finitely generated W-(super)algebras

Establish that for every finitely generated W-(super)algebra A there exists a finite-dimensional W-(super)algebra B such that their generalized polynomial identity ideals coincide, i.e., gid(A) = gid(B).

Background

The paper develops a cocharacter theory for generalized polynomial identities of W‑algebras, proving analogues of the Hook and Strip theorems and deriving bounds on generalized codimension and colength sequences. It further shows that every variety of W‑algebras is generated by the Grassmann envelope of a finitely generated W‑superalgebra; if the variety satisfies a generalized Capelli set, it is generated by a finitely generated W‑algebra.

In the classical setting, Kemer’s representability theorem states that every ordinary PI‑algebra is PI‑equivalent to the Grassmann envelope of a finite-dimensional superalgebra, and in the presence of a Capelli identity, to a finite-dimensional algebra. The authors indicate that an analogous representability result for W‑(super)algebras would follow from proving the stated conjecture about finite-dimensional representability of finitely generated W‑(super)algebras with respect to generalized polynomial identities.