Inner and outer actions of pointed Hopf algebras (Q2708923)

Let \(H\) be a finite-dimensional pointed Hopf algebra over an algebraically closed field of characteristic zero, acting on a prime algebra \(A\). The action is called \(X\)-inner if it becomes inner when extended to the symmetric Martindale ring of quotients \(Q\) of \(A\). A notion of \(X\)-outer action was given by \textit{A. Milinski} [Commun. Algebra 23, No.~1, 313-333 (1995; Zbl 0827.16027)], which is called \(M\)-outer in the paper under review. The author has previously given an Galois type correspondence theory for \(M\)-outer actions [J. Algebra 219, No.~2, 606-624 (1999; Zbl 0936.16039)]. In that paper, she gave an extended definition of \(X\)-inner action. In the paper under review, she slightly modifies that definition, calling it an \(R\)-inner action, whose negation is called an \(R\)-outer action. For group actions, \(M\)-outer and \(R\)-outer coincide. She shows that \(M\)-outer actions are \(R\)-outer. The main theorems of the paper show that under certain circumstances, \(M\)-outer and \(R\)-outer are equivalent. For example, this is true for the Taft algebras and for the Frobenius-Lusztig kernels acting on \(A\), and is true when \(A\) is semisimple. A major role in her approach to \(R\)-outer actions is played by the centralizer \(H'\) in \(H\) of the center of \(Q\). She shows that if the action is \(R\)-outer, then \(H'\) is a normal Hopf subalgebra of \(H\), and \(H\) is isomorphic to a crossed product of \(H'\) with the group algebra of \(G(H)/G(H')\), \(G(H)\) the group-like elements of \(H\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].