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.


Given 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.

An immediate generalization of Hopf modules is for the case where (E,eho E)(E,\eho_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.

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 [BW 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].

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.

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.

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

  • BW2003: T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

  • Škoda 2008: 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

  • A. Borowiec, G. A. Vazquez Coutino, Hopf modules and their duals, math.QA/0007151

  • H-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 167–195 MR92a:16047 doi

  • Francesco d’Andrea, Alessandro de Paris, On noncommutative equivariant bundles, arXiv:1606.09130

  • Joost Vercruysse. Hopf algebras—Variant notions and reconstruction theorems. (2012). (arXiv:1202.3613)

category: algebra

Last revised on February 14, 2024 at 21:52:29. See the history of this page for a list of all contributions to it.