A duality theorem for Hopf module algebras
Let \(H\) be a Hopf algebra over a field \(F\), \(A\) an \(H\)-module algebra. The smash product algebra \(A\#H\) is an \(H^ o\)-module algebra for the continuous dual \(H^ o\) of \(H\). Under certain technical conditions, the authors prove that \((A\#H)\#U\cong A\otimes (H\#U)\) for \(U\) a Hopf subalgebra of \(H^ o\). This had been proved previously by \textit{M. Cohen} and \textit{S. Montgomery} [Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001)] for \(H=FG\), the group algebra of a finite group \(G\), or \(H=(FG)^*\), \(U=H^*\). In both of these cases, \((A\#H)\#H^*\) and \(A\otimes (H\#H^*)\) are isomorphic to the matrix algebra \(M_ n(A)\), \(n=|G|\), and the authors' theorem yields \((A\#H)\#H^*\cong M_ n(A)\) for \(H\) finite-dimensional. The technical conditions in the authors' theorem require \(H\) and \(U\) to have bijective antipodes, that the action of \(H\) on \(A\) be \(U\)-locally finite and that \(U\) satisfy a certain symmetry condition with respect to \(H\): An example is given to show that the local finiteness condition is needed. The generality of the theorem allows the authors to treat universal enveloping algebras of Lie algebras and group algebras. A sharpened version of the theorem for residually linear \(H\), and \(U\) dense in \(H^*\), yields a dense embedding of \(H\#U\) in \(\text{End\,}H\), and has some interesting applications to \(U(L)\) for \(L\) a nilpotent finite-dimensional Lie algebra over \(F\) of characteristic zero, and to algebras \(A\) graded by a group \(G\) which is residually \(F\)-linear with finite conjugacy classes.
scientific article; zbMATH DE number 3946379
scientific article