Relative Hopf modules for (dual) quasi-Hopf algebras
The notions of quasi-bialgebras and quasi-Hopf algebras are due to Drinfel'd as generalizations of bialgebras and Hopf algebras by relaxing the coassociativity of the comultiplication up to conjugation. For a dual quasi-bialgebra \(H\) over \(k\), the authors introduce the notions of an \(H\)-comodule algebra \(A\) as an algebra in the tensor category of right \(H\)-comodules (note that this algebra is not necessarily associative), and the relative right Hopf modules as \(k\)-vector spaces which are right \(H\)-comodules and right modules in the tensor category of right \(H\)-comodules. If \(H\) is finite dimensional, the category of relative Hopf modules is isomorphic to the category of modules over the smash product. The authors extend the structure theorem of relative Hopf modules of \textit{Y. Doi} [Commun. Algebra 11, 243-255 (1983; Zbl 0502.16009)] for these \(H\)-comodule algebras. In the last section the dual case of \([C,H]\)-Hopf modules for a coalgebra \(C\) in the tensor category of right \(H\)-modules is considered.
scientific article; zbMATH DE number 1502168