P. I. algebras with Hopf algebra actions (Q1291084)

All algebras considered are over a field \(K\) of characteristic \(0\). Let \(H\) be a finite dimensional Hopf algebra acting on a PI algebra \(A\). The authors give a rather tight classification of such algebras. Namely, they prove that in this case there exists a nilpotent \(H\)-ideal \(I\) of \(A\) with the property that \(A/I\) is \(H\)-verbally semiprime. Further they prove that every \(H\)-verbally semiprime PI algebra is \(H\)-PI equivalent to direct sum (possibly infinite) of \(H\)-verbally prime algebras. Finally they describe the \(H\)-verbally prime algebras. They prove that these are either of the type \(B\otimes E\) for \(E\) being the infinite dimensional Grassmann algebra or of the type \(B_0\otimes E_0\oplus B_1\otimes E_1\). Here \(B\) is an \(H\)-prime algebra, \(E=E_0\oplus E_1\) is the natural \(\mathbb{Z}_2\) grading of \(E\), and in the second case \(B=B_0\oplus B_1\) is also \(\mathbb{Z}_2\) graded. The classification made in the paper follows the spirit of the celebrated work of \textit{A. R. Kemer} [Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 5, 1042-1059 (1984; Zbl 0586.16010)]. It is a serious contribution to understanding the nature of PI algebras with additional properties.