Английская Википедия: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]".
Doi-Hopf module
A right Doi-Hopf datum is a triple <math>(H,A,C)</math> with <math>H</math> a Hopf algebra, <math>A</math> a left <math>H</math>-comodule algebra, and <math>C</math> a right <math>H</math>-module coalgebra. A left-right Doi-Hopf <math>(H,A,C)</math>-module <math>M</math> is a left <math>A</math>-module and a right <math>C</math>-comodule via <math>\beta: M\to M\otimes C</math> such that <math>\beta(am)=\sum a_{(0)}m_{[0]}\otimes a_{(1)}\rightharpoonup m_{[1]}</math> for all <math>a\in A,m\in M</math>. The subscript is the Sweedler notation.
A left Doi-Hopf datum is a triple <math>(H,A,C)</math> with <math>H</math> a Hopf algebra, <math>A</math> a right <math>H</math>-comodule algebra, and <math>C</math> a left <math>H</math>-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.[3]
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