Doi-Hopf module explained

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

(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 ⊗ C such that \beta(am) = \sum a _{(0)} m _{[0]} ⊗ 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.^[3]

Notes and References

1. Web site: 土井 幸雄 (Yukio Doi) - マイポータル - researchmap . 2022-12-30 . researchmap.jp.
2. Doi . Yukio . 1992-12-15 . Unifying Hopf modules . Journal of Algebra . en . 153 . 2 . 373–385 . 10.1016/0021-8693(92)90160-N . 0021-8693. free .
3. Böhm . Gabriella . 2000-01-01 . Doi-hopf modules over weak hopf algebras . Communications in Algebra . 28 . 10 . 4687–4698 . 10.1080/00927870008827113 . math/9905027 . 123012465 . 0092-7872.