Coactions, smash products, and Hopf modules (Q1330020)

Blattner and Montgomery (1985) have shown that the smash product \(H \# H^*\) of a finite dimensional Hopf algebra \(H\) with its dual \(H^*\) is naturally isomorphic to the algebra of linear transformations on the underlying vector space of \(H\). In the present paper the following generalization of this result for infinite dimensional Hopf algebras is proven: if \(H^{* \text{rat}}\neq 0\), then \(H\# H^{* \text{rat}}\) is a central simple algebra and also a dense ring of linear transformations with finite ranks on the vector space \(H\). Here \(H^{* \text{rat}}\) is the maximal rational submodule on the left (or, equivalently, right) regular module \(H^*\), and \(H^{* \text{rat}}\neq 0\) iff there exists a left (right) integral in \(H^*\). The proof uses correspondences between \(H\)-comodules and rational modules over a dense subalgebra of \(H^*\), and between relative Hopf modules and modules over certain smash products.