Doi-Hopf module

In quantum group, Hopf algebra and weak Hopf algebra, the Doi-Hopf module is a crucial construction that has many applications. It's named after Japanese mathematician Yukio Doi (土井 幸雄 ) and German  mathematician Heinz Hopf. The concept was introduce by Doi in his 1992 paper "unifying Hopf modules ".

Doi-Hopf module
A right Doi-Hopf datum is a triple $$(H,A,C)$$ with $$H$$ a Hopf algebra, $$A$$ a left $$H$$-comodule algebra, and $$C$$ a right $$H$$-module coalgebra. A left-right Doi-Hopf $$(H,A,C)$$-module $$M$$ is a left $$A$$-module and a right $$C$$-comodule via $$\beta: M\to M\otimes C$$ such that $$\beta(am)=\sum a_{(0)}m_{[0]}\otimes a_{(1)}\rightharpoonup m_{[1]}$$ for all $$a\in A,m\in M$$. The subscript is the Sweedler notation.

A left Doi-Hopf datum is a triple $$(H,A,C)$$ with $$H$$ a Hopf algebra, $$A$$ a right $$H$$-comodule algebra, and $$C$$ a left $$H$$-module coalgebra. A Doi-Hopf module can be defined similarly.

Doi-Hopf module in weak Hopf algebra
The generalization of Doi-Hopf module in weak Hopf algebra case is given by Gabriella Böhm in 2000.