Table Of ContentMatilde Marcolli | Deepak Parashar (Eds.)
Quantum Groups and Noncommutative Spaces
Matilde Marcolli | Deepak Parashar (Eds.)
Quantum Groups
and Noncommutative
Spaces
Perspectives on Quantum Geometry
A Publication of the Max-Planck-Institute
for Mathematics, Bonn
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Prof. Dr. Matilde Marcolli Dr. Deepak Parashar
Mathematics Department University of Cambridge
California Institute of Technology Cambridge Cancer Trials Centre
1200 E.California Blvd. Department of Oncology
Pasadena, CA 91125 Addenbrooke's Hospital (Box 279)
USA Hills Road
Cambridge CB2 0QQ
[email protected]
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Bergische Universität Wuppertal University Forvie Site
Fachbereich Mathematik Robinson Way
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42119 Wuppertal UK
Germany
[email protected]
[email protected]
Mathematics Subject Classification
17B37 Quantum groups (quantized enveloping algebras) and related deformations, 58B34Noncom-
mutative geometry (à la Connes) , 58B32Geometry of quantum groups, 20G42Quantum groups
(quantized function algebras) and their representations, 16T05Hopf algebras and their applications,
19D55K-theory and homology; cyclic homology and cohomology, 81T75Noncommutative geometry
methods
1st Edition 2011
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Contents
Preface vii
Hopf-cyclic homology with contramodule coefficients
Tomasz Brzezinski 1
Moduli spaces of Dirac operators for finite spectral triples
Branimir C´ac´ic´ 9
Tensor representations of the general linear supergroup
Rita Fioresi 69
Quantum duality priciple for quantum Grassmanians
Rita Fioresi and Fabio Gavarini 80
Some remarks on the action of quantum isometry groups
Debashish Goswami 96
Generic Hopf Galois extensions
Christian Kassel 104
Quantizing the moduli space of parabolic Higgs bundle
Avijit Mukherjee 121
Locally compact quantum groups. Radford’s S4 formula
Alfons Van Daele 130
Categorical Aspects of Hopf Algebras
Robert Wisbauer 146
Laplacians and gauged Laplacians on a quantum Hopf bundle
Alessandro Zampini 164
Preface
The present volume is based on an activity organized at the Max Planck Insti-
tute for Mathematics in Bonn, during the days August 6–8, 2007, dedicated to the
topic of Quantum Groups and Noncommutative Geometry. The main purpose of
theworkshopwastofocusontheinteractionbetweenthemanydifferentapproaches
to the topic of Quantum Groups, ranging from the more algebraic techniques, re-
volving around algebraic geometry, representation theory and the theory of Hopf
algebras, and the more analytic techniques, based on operator algebras and non-
commutative differential geometry. We also focused on some recent developments
in the field of Noncommutative Geometry, especially regarding spectral triples and
their applications to models of elementary particle physics, where quantum groups
are expected to play an important role.
The contributions to this volume are written, as much as possible, in a peda-
gogical and expository way, which is intended to serve as an introduction to this
area of research for graduate students, as well as for researchers in other areas
interested in learning about these topics.
The first contribution to the volume, by Brzezinski, deals with the important
topic of Hopf-cyclic homology, which is the right cohomology theory in the context
ofHopfalgebras,playingarole,withrespecttocyclichomologyofalgebras,similar
to the cohomology of Lie algebras in the context of de Rham cohomology. The
contribution in this volume focuses on the observation that anti-Yetter-Drinfeld
contramodules can serve as coefficients for cyclic homology.
The second contribution, by C´a´ci´c, focuses on recent developments in particle
physics models based on noncommutative geometry. In particular, the paper de-
scribes a general framework for the classification of Dirac operators on the finite
geometries involved in specifying the field content of the particle physics models.
These Dirac operators have interesting moduli spaces, which are analyzed exten-
sively in this paper.
The paper by Fioresi deals with supergeometry aspects. More precisely, it
describes how one can treat the general linear supergroup from the point of view
of group schemes and Hopf algebras.
Fioresi and Gavarini contributed a paper on a generalization of the quantum
duality principle to quantizations of projective quantum homogeneous spaces. The
procedure is illustrated completely explicitly in the important case of the quantum
Grassmannians.
viii PREFACE
The paper by Goswami considers the problem of finding an analogue in Non-
commutative Geometry of the isometry group in Riemannian geometry. The non-
commutativeanalogofRiemannianmanifoldsisprovidedbyspectraltriples,hence
the replacement is provided by a compact quantum group, which acts on the spec-
tral triple.
Kassel’s paper deals with the geometry of Hopf Galois extensions. Hopf Galois
extensions can be constructed from Hopf algebras, whose product is twisted with
a cocycle. The algebra obtained in this way is a flat deformation over a central
subalgebra. This paper presents a construction of elements in this commutative
subalgebra. It also shows that an integrality condition is satisfied by all finite-
dimensional Hopf algebras generated by grouplike and skew-primitive elements.
Explicit computations are given for the case of the Hopf algebra of a cyclic group.
Mukherjee’s paper gives a survey or recent results on the quantization of the
modulispaceofstableparabolicHiggsbundlesofranktwooveraRiemannsurface
of genus at least two. This is obtained via the deformation quantization of the
Poisson structure associated to a natural holomorphic symplectic structure. The
choice of a projective structure on the Riemann surface induces a canonical star
product over a Zariski open dense subset of the moduli space.
Van Daele’s paper discusses the Radford formula expressing the forth power
of the antipode in terms of modular operators. It is first shown how the formula
simplifies in the case of compact and discrete quantum groups. Then the setting of
locally compact quantum groups is recalled and it is shown that the square of the
antipode is an analytical generator of the scaling group of automorphisms.
A paper dealing with the idea of Hopf monads over arbitrary categories was
contributed by Wisbauer, as a generalization to arbitrary categories of the notion
of Hopf algebras in module categories.
The last paper in the volume, by Zampini, deals with the important topic of
covariant differential calculus on quantum groups. The example of the quantum
HopffibrationonthestandardPodle´ssphereisanalysedinfulldetails. Itisshown
thenhowoneobtainsfromthedifferentialcalculusgaugedLaplaciansonassociated
line bundles and a Hodge star operator on the total space and base space of the
Hopf bundle. The paper includes an explicit review of the ordinary differential
calculusonSU(2)basedontheclassisalgeometryoftheHopffibration,sothatthe
comparison with the quantum groups case becomes more transparent.
We are grateful to the numerous referees for their expertise in ensuring a high
standard of the contributions, and to all speakers and participants for a very lively
interaction during the workshop. Finally, we wish to thank the MPIM, Bonn, for
financial support for the activity and for hosting the workshop, and Vieweg Verlag
for publishing this volume.
Matilde Marcolli and Deepak Parashar
Hopf-cyclic homology with contramodule coefficients
Tomasz Brzezin´ski
Abstract. AnewclassofcoefficientsfortheHopf-cyclichomologyofmodule
algebras and coalgebras is introduced. These coefficients, termed stable anti-
Yetter-Drinfeldcontramodules,arebothmodulesandcontramodulesofaHopf
algebrathatsatisfycertaincompatibilityconditions.
1. Introduction
It has been demonstrated in [8], [9] that the Hopf-cyclic homology developed
by Connes and Moscovici [5] admits a class of non-trivial coefficients. These co-
efficients, termed anti-Yetter-Drinfeld modules are modules and comodules of a
Hopfalgebrasatisfyingacompatibilityconditionreminiscentofthatforcrossmod-
ules. The aim of this note is to show that the Hopf-cyclic (co)homology of module
coalgebras and module algebras also admits coeffcients that are modules and con-
tramodules of a Hopf algebra with a compatibility condition.
All(associativeandunital)algebras,(coassociativeandcounital)coalgebrasin
this note are over a field k. The coproduct in a coalgebra C is denoted by Δ ,
C
and counit by ε . A Hopf algebra H is assumed to have a bijective antipode S.
C
We use the standard Sweedler notation for coproduct Δ (c)=c ⊗c , Δ2(c)=
C (1) (2) C
c ⊗c ⊗c , etc., and for the left coaction N(cid:3) of a C-comodule N, N(cid:3)(x) =
(1) (2) (3)
x ⊗x (in all cases summation is implicit). Hom(V,W) denotes the space of
(−1) (0)
k-linear maps between vector spaces V and W.
2. Contramodules
The notion of a contramodule for a coalgebra was introduced in [6], and dis-
cussed in parallel with that of a comodule. A right contramodule of a coalgebra C
is a vector space M together with a k-linear map α : Hom(C,M) → M rendering
the following diagrams commutative
Hom(C,α) (cid:2)(cid:2)
Hom(C,Hom(C,M)) Hom(C,M)
Θ α
(cid:3)(cid:3) (cid:3)(cid:3)
Hom(C⊗C,M) Hom(ΔC,M) (cid:2)(cid:2)Hom(C,M) α (cid:2)(cid:2)M,
2000 Mathematics Subject Classification. 19D55.
M. Marcolli, D. Parashar (Eds.), Quantum Groups and Noncommutative Spaces,
DOI: 10.1007/978-3-8348-9831-9_1, © Vieweg+Teubner Verlag | Springer Fachmedien
Wiesbaden GmbH 2011
2 TOMASZBRZEZIN´SKI
Hom(εC,M) (cid:2)(cid:2)
Hom(k,M) Hom(C,M)
(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:4)(cid:4) (cid:5)(cid:5)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)(cid:3)α(cid:3)(cid:3)(cid:3)(cid:3)
M,
where Θ is the standard isomorphism given by Θ(f)(c⊗c(cid:3)) = Θ(f)(c)(c(cid:3)). Left
contramodules are defined by similar diagrams, in which Θ is replaced by the iso-
morphism Θ(cid:3)(f)(c⊗c(cid:3)) = f(c(cid:3))(c) (or equivalenty, as right contramodules for the
co-opposite coalgebra Cop). Writing blanks for the arguments, and denoting by
matching dots the respective functions α and their arguments, the associativity
and unitality conditions for a right C-contramodule can be explicitly written as,
for all f ∈Hom(C⊗C,M), m∈M,
(cid:2) (cid:2) (cid:2) (cid:3)(cid:3)(cid:3) (cid:2) (cid:2) (cid:3)(cid:3)
α˙ α¨ f −˙⊗−¨ =α f (−) ⊗(−) , α(ε (−)m)=m.
(1) (2) C
With the same conventions the conditions for left contramodules are
(cid:2) (cid:2) (cid:2) (cid:3)(cid:3)(cid:3) (cid:2) (cid:2) (cid:3)(cid:3)
α˙ α¨ f −¨⊗−˙ =α f (−) ⊗(−) , α(ε (−)m)=m.
(1) (2) C
If N is a left C-comodule with coaction N(cid:3):N →C⊗N, then its dual vector space
M =N∗ :=Hom(N,k) is a right C-contramodule with the structure map
α:Hom(C,M)(cid:5)Hom(C⊗N,k)→Hom(N,k)=M, α=Hom(N(cid:3),k).
Explicitly, α sends a functional f on C⊗N to the functional α(f) on N,
α(f)(x)=f(x ⊗x ), x∈N.
(−1) (0)
ThedualvectorspaceofarightC-comoduleN withacoaction(cid:3)N :N →N⊗C isa
leftC-contramodulewiththestructuremapα=Hom((cid:3)N,k). Thereaderinterested
in more detailed accounts of the contramodule theory is referred to [1], [15].
3. Anti-Yetter-Drinfeld contramodules
Given a Hopf algebra H with a bijective antipode S, anti-Yetter-Drinfeld con-
tramodules are defined as H-modules and H-contramodules with a compatibility
condition. Similarly to the case of anti-Yetter-Drinfeld modules [7] they come in
four different flavours.
(1) A left-left anti-Yetter-Drinfeld contramodule is a left H-module (with the
action denoted by a dot) and a left H-contramodule with the structure map
α, such that, for all h∈H and f ∈Hom(H,M),
(cid:2) (cid:2) (cid:3)(cid:3)
h·α(f)=α h ·f S−1(h )(−)h .
(2) (1) (3)
M is said to be stable, provided that, for all m ∈ M, α(r ) = m, where
m
r :H →M, h(cid:6)→h·m.
m
(2) A left-right anti-Yetter-Drinfeld contramodule is a left H-module and a right
H-contramodule, such that, for all h∈H and f ∈Hom(H,M),
(cid:2) (cid:2) (cid:3)(cid:3)
h·α(f)=α h ·f S(h )(−)h .
(2) (3) (1)
M is said to be stable, provided that, for all m∈M, α(r )=m.
m
(3) A right-left anti-Yetter-Drinfeld contramodule is a right H-module and a left
H-contramodule, such that, for all h∈H and f ∈Hom(H,M),
(cid:2) (cid:2) (cid:3) (cid:3)
α(f)·h=α f h (−)S(h ) ·h .
(3) (1) (2)
HOPF-CYCLIC HOMOLOGY WITH CONTRAMODULE COEFFICIENTS 3
M is said to be stable, provided that, for all m ∈ M, α((cid:5) ) = m, where
m
(cid:5) :H →M, h(cid:6)→m·h.
m
(4) A right-right anti-Yetter-Drinfeld contramodule is a right H-module and a
right H-contramodule, such that, for all h∈H and f ∈Hom(H,M),
(cid:2) (cid:2) (cid:3) (cid:3)
α(f)·h=α f h (−)S−1(h ) ·h .
(1) (3) (2)
M is said to be stable, provided that, for all m∈M, α((cid:5) )=m.
m
In a less direct, but more formal way, the compatibility condition for left-left
anti-Yetter-Drinfeld contramodules can be stated as follows. For all h ∈ H and
f ∈Hom(H,M), define k-linear maps (cid:5)(cid:5) :H →M, by
f,h
(cid:2) (cid:3)
(cid:5)(cid:5) :h(cid:3) (cid:6)→h ·f S−1(h )h(cid:3)h .
f,h (2) (1) (3)
Then the main condition in (1) is
h·α(f)=α((cid:5)(cid:5) ), ∀h∈H, f ∈Hom(H,M).
f,h
Compatibility conditions between action and the structure maps α in (2)–(4) can
be written in analogous ways.
IfN isananti-Yetter-Drinfeldmodule,thenitsdualM =N∗ isananti-Yetter-
Drinfeld contramodule (with the sides interchanged). Stable anti-Yetter-Drinfeld
modules correspond to stable contramodules. For example, consider a right-left
Yetter-Drinfeld module N. The compatibility between the right action and left
coaction N(cid:3) thus is, for all x∈N and h∈H,
N(cid:3)(x·h)=S(h )x h ⊗x h .
(3) (−1) (1) (0) (2)
The dual vector space M =N∗ is a left H-module by h⊗m(cid:6)→h·m,
(h·m)(x)=m(x·h),
for all h∈H, m∈M =Hom(N,k) and x∈N, and a right H-contramodule with
the structure map α(f) = f ◦N(cid:3), f ∈ Hom(H⊗N,k) (cid:5) Hom(H,M). The space
Hom(H⊗N,k) is a left H-module by (h·f)(h(cid:3)⊗x)=f(h(cid:3)⊗x·h). Hence
(cid:2) (cid:3)
(h·α(f))(x)=α(f)(x·h)=f N(cid:3)(x·h) ,
and
(cid:2) (cid:2) (cid:3)(cid:3) (cid:2) (cid:3)
α h ·f S(h )(−)h (x) = h ·f S(h )x h ⊗x
(2) (3) (1) ((cid:2)2) (3) (−1) (1) (0)(cid:3)
= f S(h )x h ⊗x ·h .
(3) (−1) (1) (0) (2)
Therefore, the compatibility condition in item (2) is satisfied. The k-linear map
r :H →M isidentifiedwithr :H⊗N →k, r (h⊗x)=m(x·h). Inviewofthis
m m m
identification, the stability condition comes out as, for all m∈M and x∈N,
m(x)=α(r )(x)=r (x ⊗x )=m(x ·x ),
m m (−1) (0) (0) (−1)
andissatisfiedprovidedN isastableright-leftanti-Yetter-Drinfeldmodule. Similar
calculations establish connections between other versions of anti-Yetter-Drinfeld
modules and contramodules.
4 TOMASZBRZEZIN´SKI
4. Hopf-cyclic homology of module coalgebras
Let C be a left H-module coalgebra. This means that C is a coalgebra and a
left H-comodule such that, for all c∈C and h∈H,
Δ (h·c)=h ·c ⊗h ·c , ε (h·c)=ε (h)ε (c).
C (1) (1) (2) (2) C H C
ThemultipletensorproductofC,C⊗n+1,isaleftH-modulebythediagonalaction,
that is
h·(c0⊗c1⊗...⊗cn):=h ·c0⊗h ·c1⊗...⊗h ·cn.
(1) (2) (n+1)
Let M be a stable left-right anti-Yetter-Drinfeld contramodule. For all positive
integers n, set CH(C,M):=Hom (C⊗n+1,M) (left H-module maps), and, for all
n H
0≤i,j ≤n, define d :CH(C,M)→CH (C,M), s :CH(C,M)→CH (C,M),
i n n−1 j n n+1
t :CH(C,M)→CH(C,M), by
n n n
d (f)(c0,...,cn−1) = f(c0,...,Δ (ci),...,cn−1), 0≤i<n,
i (cid:2) (cid:2) C (cid:3)(cid:3)
d (f)(c0,...,cn−1) = α f c0 ,c1,...,cn−1,(−)·c0 ,
n (2) (1)
s (f)(c0,...,cn+1) = ε (cj+1)f(c0,...,cj,cj+2,...,cn+1),
j C(cid:2) (cid:2) (cid:3)(cid:3)
t (f)(c0,...,cn) = α f c1,...,cn,(−)·c0 .
n
It is clear that all the maps s , d , i < n, are well-defined, i.e. they send left H-
j i
linear maps to left H-linear maps. That d and t are well-defined follows by the
n n
anti-Yetter-Drinfeldcondition. Toillustratehowtheanti-Yetter-Drinfeldcondition
enters here we check that the t are well defined. For all h∈H,
n
t (f)(h·(c0,...,cn)) = t (f)(h ·c0,...,h ·cn)
n (cid:2)n (cid:2) (1) (n+1) (cid:3)(cid:3)
= α f h ·c1,...,h ·cn,(−)h ·c0
(cid:2) (cid:2) (2) (n+1) (1) (cid:3)(cid:3)
= α f h ·c1,...,h ·cn,h S(h )(−)h ·c0
(cid:2) (2)(cid:2) (n+1) (n+2) ((cid:3)n(cid:3)+3) (1)
= α h ·f c1,...,cn,S(h )(−)h ·c0
(cid:2)(2)(cid:2) ((cid:3)3(cid:3)) (1)
= h·α f c1,...,cn,(−)·c0 =h·t (f)(c0,...,cn),
n
where the third equation follows by the properties of the antipode and counit, the
fourth one is a consequence of the H-linearity of f, while the anti-Yetter-Drinfeld
condition is used to derive the penultimate equality.
Theorem 1. Given a left H-module coalgebra C and a left-right stable anti-
Yetter-Drinfeld contramodule M, CH(C,M) with the d , s , t defined above is a
∗ i j n
cyclic module.
Proof. One needs to check whether the maps d , s , t satisfy the relations
i j n
of a cyclic module; see e.g. [12, p. 203]. Most of the calculations are standard, we
only display examples of those which make use of the contramodule axioms. For
example,
(cid:2) (cid:2) (cid:3)(cid:3)
(tn−1◦dn−1)(f)(c0,...,cn−1) = α(cid:2)dn(cid:2)−1(f) c1,...,cn−(cid:2)1,(−)·c(cid:3)0(cid:3)(cid:3)
= α f c1,...,cn−1,Δ (−)·c0
(cid:2) (cid:2) C (cid:3)(cid:3)
= α f c1,...,cn−1,(−) ·c0 ,(−) ·c0
(cid:4) (cid:4) (cid:4) (1) (1) (2) (2(cid:5))(cid:5)(cid:5)
= α˙ α¨ f c1,...,cn−1,(−˙)·c0 ,(−¨)·c0
(1) (2)
(cid:2) (cid:2) (cid:3)(cid:3)
= α t (f) c0 ,c1,...,cn−1,(−)·c0
n (2) (1)
= (d ◦t )(f)(c0,...,cn−1),
n n
