\(X\)-inner objects for Hopf crossed products (Q1125906)

Let \(H\) be a pointed Hopf algebra over a field \(k\), \(R\) a prime \(k\)-algebra with Martindale ring of quotients \(Q\) and extended center \(K\). Assume \(R\) is an \(H\)-module algebra, let \(\sigma\) be convolution invertible in \(\Hom(H\otimes H,R)\) and form the crossed product \(R\#_\sigma H\). There exist greatest \(X\)-inner Hopf subalgebras \(H_{X\text{-inn}}\) and \((K\otimes H)_{X\text{-inn}}\) of \(H\) and \(K\otimes H\) respectively. \((K\otimes H)_{X\text{-inn}}\) turns out not to be useful in the non-cocommutative case to extend results on \(X\)-outer group actions to Hopf crossed products. What is needed is an \(X\)-inner subobject of \(K\otimes H\) which generalizes \(X\)-inner automorphisms for group actions and \(X\)-inner derivations for Lie algebra actions. The author proves the existence of a one-sided coideal of \(K\otimes H\) with these properties. It is of the form \(\widehat u(E)\), \(E\) the centralizer of \(R\) in \(Q\#_\sigma H\), where \(\widehat u:E\to K\otimes H\) depends on a certain convolution invertible \(u\) in \(\Hom (H,Q)\). The desired one-sided coideal is \(\widehat u(E)\) (sometimes described as left coideal, sometimes as right coideal). Neither \(u\) nor \(\widehat u(E)\) is unique, but the author says it should be viewed as a maximal \(X\)-inner object. It always contains \((K\otimes H)_{X\text{-inn}}\), and equals it if \(H\) is cocommutative. An example shows it need not be a \(K\)-subalgebra of \(K\otimes H\). He then uses it to characterize \(X\)-outer actions, and finally also uses it to relate \(E\), \(H_{X\text{-inn}}\) and \((K\otimes H)_{X\text{-inn}}\).