nLab entwined module

Idea

Given an entwining structure between a $k$-algebra $A$ and a $k$-coalgebra $C$ one defines the corresponding analogue of Hopf modules: they are $A$-modules with structure of $C$-comodules with a compatibility dictated by the entwining structure. If an algebra and a coalgebra are generalized to a monad and a comonad, then entwining structure generalizes to a mixed distributive law. Every entwining structure defines the composed coring (as observed by Takeuchi); the entwined modules are then precisely the objects in the category of comodules (Eilenberg-Moore category) of the composed coring.

Literature

• T. Brzeziński, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 (arXiv version).
• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
• Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770 MR2011b:14004

Historically, the entwined modules are first introduced under the name “bialgebras” by van Osdol for the mixed distributive laws between monads and comonads instead of $k$-algebras and $k$-coalgebras. The same terminology is used by Power and Watanabe.

• Donovan van Osdol, Bicohomology Theory, Transactions of the American Mathematical Society 183 (1973) 449-476 [jstor:1996479]

• John Power, Hiroshi Watanabe, Combining a monad and a comonad, Theoretical Computer Science 280 1–2 (2002) 137–162 [doi:10.1016/S0304-3975(01)00024-X]