Cocharacter sequences of algebras with Hopf algebra actions
The author develops the theory of cocharacters of polynomial identities of an algebra \(A\) with action of a finite dimensional Hopf algebra \(H\) over a field of characteristic 0. He proves analogues of the basic quantitative theorems for PI-algebra cocharacters. The results are in the spirit of the ordinary polynomial identities, *-polynomial identities for algebras with involution * and \(\mathbb{Z}/2\mathbb{Z}\)-graded polynomial identities. First the author defines the \(S_n\)- and the \(\text{GL}(k)\)-cocharacter sequences for the \(H\)-identities of \(A\) and the Poincaré (or Hilbert) series of the relatively free algebra with \(H\)-action. He establishes that these three invariants are equivalent. He compares them with their origins from the ordinary PI-theory and proves various bounds on the cocharacters. In particular, several quantitative characterizations of the property that an algebra with an \(H\)-identity is PI are given. Then the author introduces refinements of the above invariants involving representations of wreath products of \(H\). He extends his main theorems to this case when \(H\) is semisimple as an algebra and the field is algebraically closed. It turns out that the methods developed in the paper work in a more general setup. As a byproduct the author establishes some new results of independent interest concerning *-polynomial identities.
actions of finite dimensional Hopf algebras
scientific article; zbMATH DE number 954786