Doi–Koppinen modules for quantum groupoids

A definition of a Doi–Koppinen datum over a noncommutative algebra is proposed. The idea is to replace a bialgebra in a standard Doi–Koppinen datum with a bialgebroid. The corresponding category of Doi–Koppinen modules over a noncommutative algebra is introduced. A weak Doi–Koppinen datum and module of Böhm (Comm. Algebra 28 (2000) 4687) are shown to be examples of a Doi–Koppinen datum and module over an algebra. A coring associated to a Doi–Koppinen datum over an algebra is constructed and various properties of induction and forgetful functors for Doi–Koppinen modules over an algebra are deduced from the properties of corresponding functors in the category of comodules of a coring.