We define and study a class of entwined modules (stable anti-Yetter-Drinfeld
modules) that serve as coefficients for the Hopf-cyclic homology and
cohomology. In particular, we explain their relationship with Yetter-Drinfeld
modules and Drinfeld doubles. Among sources of examples of stable
anti-Yetter-Drinfeld modules, we find Hopf-Galois extensions with a flipped
version of the Miyashita-Ulbrich action

Hajac, Piotr M.

Khalkhali, Masoud

Rangipour, Bahram

Sommerhaeuser, Yorck

English

arXiv

Piotr M. Hajac

Masoud Khalkhali

Bahram Rangipour

Yorck Sommerhäuser

Comptes Rendus Mathématique

s for therinfeldflippedYetter–liquonsmodulesaction des. Theyn ofthed nottive. WeC. R. Acad. Sci. Paris, Ser. I 338 (2004) 587–590Algebra/TopologyStable anti-Yetter–Drinfeld modulesPiotr M. Hajaca,b, Masoud Khalkhalic, Bahram Rangipourc, Yorck Sommerhäuserda Instytut Matematyczny, Polska Akademia Nauk, ul.Śniadeckich 8, Warszawa, 00-956 Polandb Katedra Metod Matematycznych Fizyki, Uniwersytet Warszawski, ul. Ho˙za 74, Warszawa, 00-682 Polandc Department of Mathematics, University of Western Ontario, London ON, Canadad Mathematisches Institut, Universität München, Theresienstr. 39, 80333 München, GermanyReceived 31 July 2003; accepted 12 November 2003Presented by Alain ConnesAbstractWe define and study a class of entwined modules (stable anti-Yetter–Drinfeld modules) that serve as coefficientHopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter–Drinfeld modules and Ddoubles. Among sources of examples of stable anti-Yetter–Drinfeld modules, we find Hopf–Galois extensions with aversion of the Miyashita–Ulbrich action.To cite this article: P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméModules anti-Yetter–Drinfeld stables.Nous définissons et étudions une classe de modules enlacés (modules anti-Drinfeld stables) qui servent de coefficients pour l’homologie et la cohomologie Hopf-cyclique. En particulier, nous expleurs liens avec les modules de Yetter–Drinfeld et les doublets de Drinfeld. Parmi les sources d’exemples deanti-Yetter–Drinfeld stables, nous trouvons des extensions de Hopf–Galois munies d’une version transposée de l’Miyashita–Ulbrich.Pour citer cet article : P.M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. IntroductionThe aim of this paper is to define and provide sources of examples of stable anti-Yetter–Drinfeld moduleplay the role of coefficients for Hopf-cyclic theory [7]. In particular, we claim that modular pairs in involutioConnes and Moscovici are precisely 1-dimensional stable anti-Yetter–Drinfeld modules.Throughout the paper we assume thatH is a Hopf algebra with a bijective antipode. On the one hand,bijectivity of the antipode is implied by the existence of a modular pair in involution, so that then it neebe assumed. On the other hand, some parts of arguments might work even if the antipode is not bijecavoid such discussions. The coproduct, counit and antipode ofH are denoted by∆, ε andS, respectively. ForE-mail addresses:p.m.hajac@impan.gov.pl (P.M. Hajac), masoud@uwo.ca (M. Khalkhali), brangipo@uwo.ca (B. Rangipour),sommerh@mathematik.uni-muenchen.de (Y. Sommerhäuser).URL: http://www.fuw.edu.pl/~pmh (P.M. Hajac).1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2003.11.037588 P.M. Hajac et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 587–590e allfYetter–tweenatibilitydingrinfelda Hopfitrphismar pairse ample.the coproduct we use the notation∆(h)= h(1) ⊗ h(2), for a left coaction onM we writeM∆(m)=m(−1) ⊗m(0),and for a right coaction∆M(m)=m(0) ⊗m(1). The summation symbol is suppressed everywhere. We assumalgebras to be associative, unital and over the same ground fieldk. The symbolO(X) stands for the algebra opolynomial functions onX.2. The transformation of Yetter–Drinfeld modulesIt turns out that, in order to incorporate coefficients into cyclic theory, we need to alter the concept of aDrinfeld module by replacing the antipode by its inverse in the Yetter–Drinfeld compatibility condition beactions and coactions. We call the modules-comodules satisfying the thus modified Yetter–Drinfeld compconditionanti-Yetter–Drinfeld modules.1 Just as Yetter–Drinfeld modules come in 4 different versions depenon the side of actions and coactions (see [3, p. 181] for a general formulation), so do the anti-Yetter–Dmodules. All versions are completely equivalent and can be derived from one another by replacingalgebraH byH cop, H op, orH op,cop, respectively.Definition 2.1. Let H be a Hopf algebra with a bijective antipodeS, andM a module and comodule overH . WecallM an anti-Yetter–Drinfeld module iff the action and coaction are compatible in the following sense:M∆(hm)= h(1)m(−1)S−1(h(3)) ⊗ h(2)m(0) if M is a left module and a left comodule, (1)∆M(hm)= h(2)m(0) ⊗ h(3)m(1)S(h(1))if M is a left module and a right comodule, (2)M∆(mh)= S(h(3))m(−1)h(1) ⊗m(0)h(2) if M is a right module and a left comodule, (3)∆M(mh)=m(0)h(2) ⊗ S−1(h(1))m(1)h(3) if M is a right module and a right comodule. (4)To make cyclic theory work, we also need to assume that the action splits coaction, i.e., for allm ∈ M,m(−1)m(0) = m, m(1)m(0) = m, m(0)m(−1) = m, m(0)m(1) = m, for the left–left, left–right, right–left, andright–right versions, respectively. We call modules satisfying this conditionstable. Let us emphasize thatis the anti-Yetter–Drinfeld condition rather than the Yetter–Drinfeld condition that makes the homomoaction◦ coactionH -linear andH -colinear. Therefore the stability conditionaction◦ coaction= id suits the formerand not the latter. The first class of examples of stable anti-Yetter–Drinfeld modules is provided by modulin involution [4, p. 8]. Since such pairs occur naturally in different contexts, Lemmas 2.2 and 2.3 guaranteamount of examples of anti-Yetter–Drinfeld modules.Lemma 2.2.Let the ground fieldk be a right module overH via a characterδ and a left comodule overH viaa group-likeσ . Thenk =σkδ is a stable anti-Yetter–Drinfeld moduleif and only if (δ, σ ) is a modular pair ininvolution.The anti-Yetter–Drinfeld modules do not form a monoidal category themselves, but rather a so-calledC-cat goryover the category of Yetter–Drinfeld modules (see [11, p. 351] for details). More precisely:Lemma 2.3.LetN be a Yetter–Drinfeld module andM an anti-Yetter–Drinfeld module. ThenN ⊗M is an anti-Yetter–Drinfeld module viah(n⊗m)= h(1)n⊗ h(2)m, N⊗M∆(n⊗m)= n(−1)m(−1) ⊗ n(0) ⊗m(0), for the left–left case, and viah(n ⊗ m) = h(2)n ⊗ h(1)m, ∆N⊗M(n ⊗ m) = n(0) ⊗ m(0) ⊗ n(1)m(1), for the left–right caseSimilarly,M⊗N is an anti-Yetter–Drinfeld module via(m⊗n)h=mh(2)⊗nh(1), M⊗N∆(n⊗m)=m(−1)n(−1)⊗m(0) ⊗ n(0), for the right–left case, and via(m⊗ n)h=mh(1) ⊗ nh(2), ∆M⊗N(m⊗ n)=m(0) ⊗ n(0) ⊗m(1)n(1),for the left–right case.1 This concept was devised independently by Ch. Voigt and, also independently, by P. Jara and D. ¸Stefan.P.M. Hajac et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 587–590 589iningt togivenbetween--withy.sin theion andebrapthewnditionetter–[2].groundtensionsver therstoodjectivityNote that, just as the right–right Yetter–Drinfeld modules are entwined modules [1] for the entwψ(h′ ⊗ h) = h(2) ⊗ S(h(1))h′h(3), the right–right anti-Yetter–Drinfeld modules are entwined with respecψ(h′ ⊗ h)= h(2) ⊗ S−1(h(1))h′h(3). (Other cases are completely analogous.)An intermediate step between modular pairs in involution and stable anti-Yetter–Drinfeld modules isby matched and comatched pairs of [10]. Whenever the antipode is equal to its inverse, the differencethe Yetter–Drinfeld and anti-Yetter–Drinfeld conditions disappears. For a group ring Hopf algebrakG, a leftH -comodule is simply aG-graded vector spaceM = ⊕g∈GMg , where the coaction is defined byMg m → g ⊗m.An action ofG onM defines an (anti-)Yetter–Drinfeld module if and only if for allg,h ∈G andm ∈Mg we havehm ∈ Mhgh−1. The stability condition means simply thatgm = m for all g ∈ G, m ∈Mg . A very concrete classical example of a stable (anti-)Yetter–Drinfeld module is provided by the Hopf fibration. ThenH =O(SU(2)) andM = O(S2). SinceS2 ∼= SU(2)/U(1), we have a natural left action of SU(2) onS2. Its pull-back makesM a leftH -comodule. On the other hand, one can viewS2 as the set of all traceless matrices of SU(2). The pull-back of this embeddingj :S2 ↪→ SU(2) together with the multiplication inO(S2) defines a left action ofH onM. It turns out thatthe equivariance propertyj (gx)= gj (x)g−1 guarantees the anti-Yetter–Drinfeld condition, and this combinedthe injectivity ofj ensures the stability ofM. This stability mechanism can be generalized in the following waLemma 2.4.LetM be an algebra and a leftH -comodule. Assume thatπ :H →M is an epimorphism of algebraand the actionhm = π(h)m makesM an anti-Yetter–Drinfeld module. Assume also thatπ(1(−1))1(0) = 1. ThenM is a stable module.3. Hopf–Galois extensions and the opposite Miyashita–Ulbrich actionAnother source of examples is provided by Hopf–Galois theory. These examples are purely quantumsense that the employed actions are automatically trivial for commutative algebras. To fix the notatterminology, recall that an algebra and anH -comodule is called a comodule algebra if the coaction is an alghomomorphism. AnH -extensionB := {p ∈ P |∆P (p)= p⊗ 1} ⊆ P is called Hopf–Galois iff the canonical macan:P ⊗B P → P ⊗H, can(p ⊗ p′) = p∆(p′), is bijective. The bijectivity assumption allows us to definetranslation mapT :H → P ⊗B P , T (h) := can−1(1⊗h)=: h[1] ⊗B h[2] (summation suppressed). It can be shothat when everything is over a field (our standing assumption), the centralizerZB(P) := {p ∈ P | bp = pb,∀b ∈B}of B in P is a subcomodule ofP . On the other hand, the formulaph = h[1]ph[2] defines a right action onZB(P)called the Miyashita–Ulbrich action. This action and coaction satisfy the Yetter–Drinfeld compatibility con[6, (3.11)]. The following proposition modifies the Miyashita–Ulbrich action so as to obtain stable anti-YDrinfeld modules.Proposition 3.1.LetB ⊆ P be a Hopf–GaloisH -extension such thatB is central inP . ThenP is a right–rightstable anti-Yetter–Drinfeld module via the actionph= (S−1(h))[2]p(S−1(h))[1] and the right coaction onP .The simplest examples are obtained forP = H . A broader class is given by the so-called Galois objectsThen quantum-group coverings at roots of unity provide examples with central coinvariants bigger than thefield (see [5] and examples therein). Finally, one can generalize Proposition 3.1 to arbitrary Hopf–Galois exby replacingP by P/[B,P ] [8, Remark 4.2].4. The Drinfeld double comodule algebraFor finite-dimensional Hopf algebras, the Yetter–Drinfeld modules can be understood as modules oDrinfeld double [9, p. 220]. Much in the same way, the anti-Yetter–Drinfeld modules can also be undeas modules over a certain algebra. This makes the usual notions and operations for modules, like pro590 P.M. Hajac et al. / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 587–590n anti-choice,nld4. Allench98.ations, in:in:mutative1 (1989)d. Sci.or induction, directly available for anti-Yetter–Drinfeld modules. To this end, the comodule structure of aYetter–Drinfeld module has to be converted into a module structure over the dual Hopf algebraH ∗, so that fromnow on we assume that the Hopf algebraH is finite-dimensional.Proposition 4.1.LetH be a finite-dimensional Hopf algebra. The formula(ϕ ⊗ h)(ϕ′ ⊗ h′)= ϕ′(1)(S−1(h(3)))ϕ′(3)(S2(h(1)))ϕϕ′(2) ⊗ h(2)h′ (5)turns the vector spaceA(H) :=H ∗ ⊗H into an associative algebra with the unitε⊗ 1.Note that the above product differs from the product in the Drinfeld double ofH [9, p. 214] only by theadditional squared antipode in the second factor. To relate the modules overA(H) with anti-Yetter–Drinfeldmodules, recall first that every rightH -comoduleM becomes a leftH ∗-module via ϕm := ϕ(m(1))m(0).Conversely, any leftH ∗-module yields a rightH -comodule via∆M(m)= ∑ni=1h∗i m⊗ hi . Here{h1, . . . , hn} is abasis ofH and{h∗1, . . . , h∗n} is the dual basis. (Of course, this comodule structure does not depend on theof a basis.) Using this, we get the following connection between the modules overA(H) and anti-Yetter–Drinfeldmodules:Proposition 4.2.Let H be a finite-dimensional Hopf algebra. IfM is a left–right anti-Yetter–Drinfeld moduleit becomes a leftA(H)-module by(ϕ ⊗ h)m := ϕ((hm)(1))(hm)(0). Conversely, ifM is a leftA(H)-module, itbecomes a left–right anti-Yetter–Drinfeld module byhm := (ε ⊗ h)m, ∆M(m) := ∑ni=1(h∗i ⊗ 1)m ⊗ hi . Here{h1, . . . , hn} is a basis ofH and{h∗1, . . . , h∗n} its dual basis.The claim of Lemma 2.3 is reflected in the fact that althoughA(H) is not a Hopf algebra itself, it can be showthat the formula(ϕ ⊗ h) → (ϕ(2) ⊗ h(1))⊗ (ϕ(1) ⊗ h(2)) makesA(H) a right comodule algebra over the DrinfedoubleD(H).AcknowledgementsP.M.H. acknowledges the Marie Curie Fellowship HPMF-CT-2000-00523 and KBN grant 2 P03A 013 2authors are grateful to T. Brzeziński and M. Furuta for their help and comments, and to D. Perrot for the Frtranslation.References[1] T. Brzezínski, On modules associated to coalgebra Galois extensions, J. Algebra 215 (1999) 290–317.[2] S. Caenepeel, Brauer Groups, Hopf Algebras and Galois Theory, in:K-Monographs in Math., vol. 4, Kluwer Academic, Dordrecht, 19[3] S. Caenepeel, G. Militaru, S. Zhu, Frobenius and Separable Functors for Generalized Module Categories and Nonlinear EquLecture Notes in Math., vol. 1787, Springer-Verlag, Berlin, 2002.[4] A. Connes, H. Moscovici, Cyclic cohomology and Hopf algebra symmetry, Lett. Math. Phys. 52 (2000) 1–28.[5] L. Da̧browski, P.M. Hajac, P. Siniscalco, Explicit Hopf–Galois description of SLe2iπ/3(2)-induced Frobenius homomorphisms,D. Kastler, M. Rosso, T. Schucker (Eds.), Enlarged Proceedings of the ISI GUCCIA Workshop on Quantum Groups, NoncomGeometry and Fundamental Physical Interactions, Nova Science, Commack–New York, 1999, pp. 279–298.[6] Y. Doi, M. Takeuchi, Hopf–Galois extensions of algebras, the Miyashita–Ulbrich action, and Azumaya algebras, J. Algebra 12488–516.[7] P.M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhäuser, Hopf-cyclic homology and cohomology with coefficients, C. R. AcaParis, Ser. I 338 (2004).[8] P. Jara, D. ¸Stefan, Cyclic homology of Hopf Galois extensions and Hopf algebras, Preprint, math.KT/0307099.[9] C. Kassel, Quantum Groups, in: Graduate Texts in Math., vol. 155, Springer-Verlag, Berlin, 1995.[10] M. Khalkhali, B. 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Stable anti-Yetter-Drinfeld modules

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