Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras.
Let \(H\) be a Hopf algebra. An associative right \(H\)-comodule algebra \(A\) finitely generated over a field \(k\) belongs to the class \(\mathcal A\) if it has an ideal \(I_A\) such that \(A/I_A\) is a commutative domain and \(I_A\) has no nonzero ideals of \(A\) which are \(H\)-stable. Suppose that \(H\) is cosemisimple. Then the subalgebra of coinvariants \(A_H\) is finitely generated and \(A\) is finitely generated as a comodule over \(A_H\). Suppose that the field \(k\) has characteristic zero and \(H\) is the four-dimensional Taft algebra. Then there exists an \(H\)-comodule algebra \(A\) which is not finitely generated as \(A_H\)-comodule and \(A_H\) is no finitely generated as \(k\)-algebra. Related problems were considered by \textit{S. Skryabin} [Trans. Am. Math. Soc. 359, No. 6, 2597-2623 (2007; Zbl 1123.16032)], by \textit{S. Zhu}, [J. Algebra 180, No. 1, 187-205 (1996; Zbl 0845.16039)], \textit{A. A. Totok}, [Mosc. Univ. Math. Bull. 52, No. 3, 33-36 (1997); translation from Vestn. Mosk. Univ., Ser. I 1997, No. 3, 32-35 (1997; Zbl 0914.16018)], and the reviewer [Mosc. Univ. Math. Bull. 51, No. 4, 41-44 (1996); translation from Vestn. Mosk. Univ., Ser. I 1996, No. 4, 45-49 (1996; Zbl 0913.16018), and Errata 2, 64 (1997)].
finite-dimensional semisimple Hopf algebras
scientific article; zbMATH DE number 5975551