nLab Hopf module




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.


Hopf modules

Given a commutative ring kk and a kk-bialgebra (H,m H,η,Δ,ϵ)(H,m_H,\eta,\Delta,\epsilon), a left-right Hopf module of HH is a kk-module MM with the structure of left HH-module and right HH-comodule, where the action ν:HMM\nu: H\otimes M\to M and right HH-coaction ρ:MMH\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 MHM\otimes H is the standard tensor product of modules over Hopf algebras, with the action given by (νm H)(HτH)(ΔMH)(\nu\otimes m_H)\circ(H\otimes \tau\otimes H)\circ(\Delta \otimes M\otimes H) as kk-linear map H(MH)MHH\otimes(M\otimes H)\to M\otimes H where τ=τ H,M:HMMH\tau=\tau_{H,M}:H\otimes M\to M\otimes H is the standard flip of tensor factors in the symmetric monoidal category of kk-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,ρ E)(E,\rho_E) is a right HH-comodule algebra (a monoid in the category of HH-comodules); then one can define the category E H{}_E\mathcal{M}^H of left EE- right HH- relative Hopf modules (less precisely, (E,H)(E,H)-relative Hopf modules, or simply (relative) Hopf modules), which are left EE-modules that are right HH-comodules with a natural compatibility condition. In Sweedler notation for comodules. where ρ(m)=m (0)m (1)\rho(m) = \sum m_{(0)}\otimes m_{(1)}, ρ E(e)=e (0)e (1)\rho_E(e) = \sum e_{(0)}\otimes e_{(1)}, the compatibility condition for the left-right relative Hopf modules is ρ(em)=e (0)m (0)e (1)m (1)\rho (e m) = \sum e_{(0)} m_{(0)} \otimes e_{(1)} m_{(1)} for all mMm\in M and eEe\in E. A morphism of relative Hopf modules is a morphism of modules which is also a morphism of comodules. If the category of kk-modules is denoted by \mathcal{M} then the left EE- right HH-relative Hopf modules and their morphisms form a category in Hopf algebraic literature denoted by E H{}_E\mathcal{M}^H.

Given a left HH-comodule algebras EE and FF, one considers also the category E F H{}_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 EF op H{}_{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 HH-bimodule is left and right HH-comodule and left and right HH-bimodule, where all four structure are compatible in standard way.
The category of Hopf bimodules, H H H H{}_H^H\mathcal{M}^H_H is monoidally equivalent to the category of Yetter-Drinfeld modules.


There are further generalizations where instead of a bialgebra HH and a HH-comodule algebra EE one replaces EE by an arbitrary algebra AA, and HH by a coalgebra CC 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.


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 HH is a Hopf algebra and UEU\hookrightarrow E a faithfully flat HH-Hopf-Galois extension then the natural adjunction between the categories of relative (E,H)(E,H)-Hopf modules and left UU-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 HH is a Hopf algebra over a field kk, then the category of the ordinary Hopf modules H H{}_H^H\mathcal{M} is equivalent to the category of kk-vector spaces. See Section 1 of Montgomery 1993 for more.

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

ρ:hhvhhv \rho: h'\otimes h\otimes v\mapsto h'h\otimes v

and the coaction as

σ:hvΔ(h)v \sigma: h\otimes v\mapsto \Delta(h)\otimes v

for Δ:HHH\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 MM arises precisely in this way, as one shows that

MHM coH M\cong H \otimes M^{\text{co} H}

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

VHV V \mapsto H\otimes V
MM coH M\mapsto M^{\text{co} H}

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


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

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category: algebra

Last revised on May 16, 2024 at 15:39:51. See the history of this page for a list of all contributions to it.