Concept in Hopf algebra and weak Hopf algebra
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 formula_1 with formula_2 a Hopf algebra, formula_3 a left formula_2-comodule algebra, and formula_5 a right formula_2-module coalgebra. A left-right Doi-Hopf formula_1-module formula_8 is a left formula_3-module and a right formula_5-comodule via formula_11 such that formula_12 for all formula_13. The subscript is the Sweedler notation.
A left Doi-Hopf datum is a triple formula_1 with formula_2 a Hopf algebra, formula_3 a right formula_2-comodule algebra, and formula_5 a left formula_2-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.
References.
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