Unitary Doi-Koppinen modules

A Doi-Koppinen datum consists of a bialgebra A together with a right A-comodule algebra B and a left A-module coalgebra C. A Doi-Koppinen module is then an A-module which is at the same time a C-comodule, such that the module and comodule structure are compatible in a natural way. Doi-Koppinen data can be constructed from right coideal subalgebras in bialgebras. In this talk, we will revisit the theory of Doi-Koppinen modules in the setting of right coideal *-subalgebras of compact quantum group Hopf *-algebras. We introduce the notion of a unitary Doi-Koppinen module, and explain how the collection of unitary Doi-Koppinen modules forms a tensor C*-category. We then  explain how a natural regular unitary Doi-Koppinen module can be constructed in the presence of a relatively invariant functional on the orthogonal left coideal. This is based on joint work with J.R. Dzokou Talla.