nLab Hopf module

Idea

The Hopf modules over bimonoids are modules in the category of comodules or viceversa. This notion has many generalizations and variants. Relative Hopf modules are an algebraic and possibly noncommutative analogue of a notion of an equivariant sheaf.

Definitions

Hopf modules

Given a commutative ring $k$ and a $k$-bialgebra $(H,m_H,\eta,\Delta,\epsilon)$, a left-right Hopf module of $H$ is a $k$-module $M$ with the structure of left $H$-module and right $H$-comodule, where the action $\nu: H\otimes M\to M$ and right $H$-coaction $\rho : M\to M\otimes H$ are compatible in the sense that the coaction is a morphism of left modules. In this, the structure of left module on $M\otimes H$ is the standard tensor product of modules over Hopf algebras, with the action given by $(\nu\otimes m_H)\circ(H\otimes \tau\otimes H)\circ(\Delta \otimes M\otimes H)$ as $k$-linear map $H\otimes(M\otimes H)\to M\otimes H$ where $\tau=\tau_{H,M}:H\otimes M\to M\otimes H$ is the standard flip of tensor factors in the symmetric monoidal category of $k$-modules. morphism of relative Hopf modules is a morphism of modules which is also a morphism of comodules.

Relative Hopf modules

An immediate generalization of Hopf modules is for the case where $(E,\rho_E)$ is a right $H$-comodule algebra (a monoid in the category of $H$-comodules); then one can define the category ${}_E\mathcal{M}^H$ of left $E$- right $H$- relative Hopf modules (less precisely, $(E,H)$-relative Hopf modules, or simply (relative) Hopf modules), which are left $E$-modules that are right $H$-comodules with a natural compatibility condition. In Sweedler notation for comodules. where $\rho(m) = \sum m_{(0)}\otimes m_{(1)}$, $\rho_E(e) = \sum e_{(0)}\otimes e_{(1)}$, the compatibility condition for the left-right relative Hopf modules is $\rho (e m) = \sum e_{(0)} m_{(0)} \otimes e_{(1)} m_{(1)}$ for all $m\in M$ and $e\in E$. A morphism of relative Hopf modules is a morphism of modules which is also a morphism of comodules. If the category of $k$-modules is denoted by $\mathcal{M}$ then the left $E$- right $H$-relative Hopf modules and their morphisms form a category in Hopf algebraic literature denoted by ${}_E\mathcal{M}^H$.

Given a left $H$-comodule algebras $E$ and $F$, one considers also the category ${}_E\mathcal{M}_F^H$ of two-sided relative Hopf modules (sometimes also called the category of relative Hopf bimodules. It is equivalent to the category ${}_{E\otimes F^{op}}\mathcal{M}^H$. See Caen. et al 2007.

Hopf bimodules

One can also consider Hopf bimodules, and similar categories. A Hopf $H$-bimodule is left and right $H$-comodule and left and right $H$-bimodule, where all four structure are compatible in standard way.
The category of Hopf bimodules, ${}_H^H\mathcal{M}^H_H$ is monoidally equivalent to the category of Yetter-Drinfeld modules.

Generalizations

There are further generalizations where instead of a bialgebra $H$ and a $H$-comodule algebra $E$ one replaces $E$ by an arbitrary algebra $A$, and $H$ by a coalgebra $C$ and introduces a compatibility in the sense of a mixed distributive law or entwining (structure). Then the relative Hopf modules become a special case of so-called entwined modules, see the monograph Brzezinski, Wisbauer 2003.

Properties

Geometrically, relative Hopf modules are instances of equivariant objects (equivariant quasicoherent sheaves) in noncommutative algebraic geometry, the statement of which can be made precise, cf. Škoda 2008. For another approach/candidate for noncommutative equivariant objects, see d’Andrea, Paris 2019 and its MR review by Gabriella Böhm for an excellent comparison.

Furthermore, in the context of relative Hopf modules there is an analogue of the faithfully flat descent along torsors from commutative algebraic geometry, and the Galois descent theorems in algebra. Its main instance is Schneider's theorem, asserting that if $H$ is a Hopf algebra and $U\hookrightarrow E$ a faithfully flat $H$-Hopf-Galois extension then the natural adjunction between the categories of relative $(E,H)$-Hopf modules and left $U$-modules is an equivalence of categories. This corresponds to the classical theorem saying that the category of equivariant quasicoherent sheaves over the total space of a torsor is equivalent to the category of the quasicoherent sheaves over the base of the torsor.

Fundamental theorem on Hopf modules

If $H$ is a Hopf algebra over a field $k$, then the category of the ordinary Hopf modules ${}_H^H\mathcal{M}$ is equivalent to the category of $k$-vector spaces. See Section 1 of Montgomery 1993 for more.

The equivalence may be seen as follows. Any vector space $V$ can be endowed with a (left-) Hopf module structure, for $H$ a Hopf algebra, simply by tensoring with $H$. The action of $H$ is given as

$$\rho: h'\otimes h\otimes v\mapsto h'h\otimes v$$

and the coaction as

$$\sigma: h\otimes v\mapsto \Delta(h)\otimes v$$

for $\Delta:H\to H\otimes H$ the comultiplication. This is known as a trivial Hopf module.

The fundamental theorem of Hopf modules states that any Hopf module $M$ arises precisely in this way, as one shows that

$$M\cong H \otimes M^{\text{co} H}$$

where $M^{\text{co} H}:= \{m\in M \vert \sigma(m)= 1_H \otimes m\}$ is the space of coinvariant of $M$ under the coaction $\sigma$ of $H$. In fact, the operations

$$V \mapsto H\otimes V$$

$$M\mapsto M^{\text{co} H}$$

come as functors realizing an equivalence of categories between vector spaces, and $H$-Hopf modules (see Vercruysse 2012 for more on this).

References

Related entries include comodule algebra, Schneider's descent theorem, Yetter-Drinfeld module, entwined module

• 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
• Susan Montgomery, Hopf algebras and their actions on rings, CBMS Lecture Notes 82, AMS 1993, 240p.
• Peter Schauenburg, Hopf modules and Yetter - Drinfel′d modules, J. Algebra 169:3 (1994) 874-890 doi; Hopf modules and the double of a quasi-Hopf algebra, Trans. Amer. Math. Soc. 354 (2002), 3349-3378 doi pdf; Actions of monoidal categories, and generalized Hopf smash products, Journal of Algebra 270 (2003) 521-563, doi ps; Hopf bimodules over Hopf-Galois extensions, Miyashita-Ulbrich actions, and monoidal center constructions, Comm. Algebra 24 (1996) 143–163 doi
• A. Borowiec, G. A. Vazquez Coutino, Hopf modules and their duals, math.QA/0007151
• Hans-Jürgen Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195 MR92a:16047 doi
• Stefaan Caenepeel, Septimiu Crivei, Andrei Marcus, Mitsuhiro Takeuchi, Morita equivalences induced by bimodules over Hopf–Galois extensions, J. Alg. 314:1, (2007) 267–302 doi
• Francesco d'Andrea, Alessandro de Paris, On noncommutative equivariant bundles, Commun. Alg. 47:12 (2019) 5443–5461 arXiv:1606.09130 doi
• Joost Vercruysse. Hopf algebras—Variant notions and reconstruction theorems. (2012). (arXiv:1202.3613)

category: algebra