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^[1]) and German mathematician Heinz Hopf. The concept was introduce by Doi in his 1992 paper "unifying Hopf modules^[2]".

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

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

The generalization of Doi-Hopf module in weak Hopf algebra case is given by Gabriella Böhm in 2000.^[3]