Home
page of the Research Program PRIN 20062008
Noncommutative
geometry of quantum groups
and their homogeneous spaces
Trieste
Unit

University
Personnel
Name 
Institute 
Ludwik Dąbrowski 
SISSA 
Cesare Reina 
SISSA 
Postdocs
Name 
Institute 
Alessandro Zampini 
University of Bonn 
Simon Brain 
Oxford University 
PhD students
Name 
Institute 
Andrea Brini 
SISSA 
Lucio Cirio 
SISSA 
Francesco D'Andrea 
SISSA 
Abdelmoubine Amar Henni 
SISSA 
Annibale Magni 
SISSA 
Luca Philippe Mertens 
SISSA 
Noncommutative geometry, as developed by A. Connes [co94]
since the 80's, generalizes the usual geometric methods to `bad'
quotient spaces on which the usual classes of function algebras are
too small, but lot of interesting information about their structure
can be extracted from certain noncommutative algebras. The prototype
is the well known noncommutative torus [ri04], that can be associated
to the quotient space of a circle by the discrete group of irrational
rotations. More complicated quotients of that kind, quite abundant in
mathematics, arise from foliations, representations of discrete
groups, tilings, dynamical systems, arithmetic geometry and others
(see [cma06] for a recent review). They include also a number of
interesting examples arising from physics.
Of course,
noncommutative geometry studies  at various levels  also genuinely
noncommutative algebras, starting with measure theory, topology,
differential calculus and riemannian spin structure. The latter layer
is best encoded [co94, co96] in terms of a spectral triple (A, H, D)
consisting of a (suitable subalgebra of a) C*algebra A, represented
by bounded operators on a Hilbert space H and a suitable operator D
(generalizing the Dirac operator). Classically, the canonical
spectral triple (A, H, D) associated to any spin manifold M with a
Riemannian metric g consists of the algebra of smooth complex
functions on M; the Hilbert space of the square integrable sections
of the spinor bundle over M and the Dirac operator of the LeviCivita
connection of g. With an additional requirement of regularity of a
spectral triple (A,H,D), stating that both A and [D, A] are in the
smooth domain of the (unbounded) derivation given by the commutator
with D, a general pseudodifferential calculus can be developed
[cmo95]. Under another requirement of finite summability (controlled
by the asymptotic behaviour of eigenvalues of D), one associates to
any pseudodifferential operators the generalized zetafunction, the
residues of which are tracial functionals. These provide tools for
the local index theorem of Connes and Moscovici, which is a powerful
algorithm for performing complicated computations in a local way by
neglecting plethora of irrelevant details. Certain additional
requirements have been in fact proposed in [co96] in order not to
loose any differential geometric information at least in the
commutative case, but they are still a subject of investigations. In
this respect the construction of a number of interesting examples is
strongly needed, in order to complete the scheme.
An
important arena for implementing and testing these ideas is quantum
groups and their homogeneous spaces (regarded as deformed function
algebras), see e.g. [wo87, frt90, ma95, ks98]. Their study from the
point of view of noncommutative differential geometry has started
only recently and the members of the SISSA local group have actively
contributed to this work. In particular, extending the case of
2spheres [bm93, hm99], certain instantonic fibre bundles over
qdeformed 4spheres have been constructed [dlm01, bct02, dl02, hs02,
cdv02, ab02, lm03, hl04, bcdt04], fi06]. A recent important result
is the construction in [lpr06] of a `true' noncommutative principal
bundle on a quantum 4sphere, with structure group SUq(2).
Technically, an A(SUq(2)) HopfGalois extension of the algebra of
functions on a quantum 4sphere (the base manifold) into the algebra
of functions on a quantum 7sphere (the total space of the fibration)
has been constructed. These examples require an analysis analogous to
that performed in [ns98, ls05]. Concerning the Riemannian structure,
the members of the group contributed to resolve an apparent
difficulty, related to the cohomological dimension drop [mnw90], by
showing that quantum groups fit up to large extend the framework of
noncommutative geometry, contrary to what was previously believed. In
particular we have constructed the first example of a real even
equivariant spectral triple on the standard noncommutative Podles
quantum sphere [po87], which is a homogeneous space of SUq(2). The
operator D satisfies the `order one condition' and has an
exponentially growing spectrum. On another homogeneous space, the
equatorial Podles quantum sphere, we have found [dlps05] a 2 summable
equivariant even spectral triple with the Dirac operator having a
linear spectrum (as on the usual round 2sphere), which satisfies a
weak form of the reality and `order one condition'. On SUq(2) itself
we have constructed in [dlssv05] a 3summable spectral triple. The
geometry in this case is also an isospectral deformation of the
classical case, in the sense that the spectrum of the Dirac operator
is the same as the usual one on the round ordinary 3sphere.
Moreover, it is equivariant with respect to both left and right
action of the Hopf algebra Uq(su(2)). Regarding the analytic aspects,
in [dlssv05a] the local index formula [cmo95] has been explictly
controlled on quantum SUq(2), with the structures of the cotangent
space and the geodesic flow being essentially the same as in [co04].
In working out the spectral geometry of quantum groups, one
can identify roughly three main steps:  Selection of appropriate
spectral triples  Explicit form of the local index formula  The
inner perturbations, scalar curvature and spectral action. We can
summarize the current status of the first step as satisfactory at
least regarding the simplest example of SUq(2) and its homogeneous
spaces. It is clear that the extension to other quantum groups is
required and for that reason one needs to investigate which kind of
spectra of Dirac operator, either with linear or exponential growth
(with the related form of the first order condition) are more
suitable. Concerning the second step, it works quite well in the
examples studied so far, though the difficulty of analytical
questions are expected to grow with more complicated examples.
Regarding the third step, it forms at the moment the hardest part,
still awaiting future investigations; for that the resolution of the
problem mentioned at the end of the first step should be crucial.
[bct02] F. Bonechi, N. Ciccoli, M. Tarlini. Noncommutative
Instantons on the 4Sphere from Quantum Groups. Commun. Math.
Phys. 226 (2002) 419432.
[bcdt04] F. Bonechi, N. Ciccoli, L.
Dąbrowski, M. Tarlini. Bijectivity of the canonical map
for the noncommutative instanton bundle. J. Geom. Phys. 51
1(2004) 71181.
[bm93] T. Brzezinski, S. Majid. Quantum
Group Gauge Theory on Quantum Spaces. Commun. Math. Phys. 157
(1993) 591638. erratum 167 (1955) 235.
[cpa03] P.S.
Chakraborty, A. Pal. Equivariant spectral triples on the quantum
SU(2) group. KTheory 28 (2003) 107126.
[co94] A.
Connes. Noncommutative geometry. Academic Press 1994
[co96]
A. Connes. Gravity coupled with matter and foundation of
noncommutative geometry. Commun. Math. Phys. 182 (1996) 155176
[co04] A. Connes. Cyclic Cohomology, Quantum group
Symmetries and the Local Index Formula for SUq(2). J. Inst. Math.
Jussieu 3 (2004), 1768.
[cl01] A. Connes, G. Landi.
Noncommutative manifolds, the instanton algebra and isospectral
deformations. Commun. Math. Phys. 221 (2001) 141159
[cma06]
A. Connes, M. Marcolli. A walk in the noncommutative garden.
math.QA/0601054
[cmo95] A. Connes, H. Moscovici. The local
index formula in noncommutative geometry. Geom. Funct. Anal. 5
(1995) 174243.
[cmo98] A. Connes, H. Moscovici. Hopf
algebras, Cyclic cohomology and the Tranverse Index Theorem.
Commun. Math. Phys. 198 (1998) 199246
[cr87] A. Connes, M.
Rieffel. YangMills for Noncommutative TwoTori. Contemp.
Math. 62 (1987) 237266
[dl02] L. Dąbrowski, G.
Landi. Instanton algebras and quantum 4spheres. Differ. Geom.
Appl. 16 (2002) 277284
[dlm01] L. Dąbrowski, G.
Landi, T. Masuda. Instantons on the Quantum 4Spheres S^4_q.
Commun. Math. Phys. 221 (2001) 161168
[dlssv05] L.
Dąbrowski, G. Landi, A. Sitarz, W. van Suijlekom, J.C.
Varilly. The Dirac operator on SU_q(2). Commun. Math. Phys.
259 (2005) 729759.
[dlssv05a] L. Dąbrowski, G.
Landi, A. Sitarz, W. van Suijlekom, J.C. Varilly. The local index
formula for SU_q(2). Archive math.QA/0501287. KTheory, in
stampa.
[dlps05] L. Dąbrowski, G. Landi, M. Paschke,
A. Sitarz. The Spectral Geometry of the Equatorial Podles Sphere.
C. R. Acad. Sci. Paris, Ser. I 340 (2005) 819822.
[ds03] L.
Dąbrowski, A. Sitarz. Dirac Operator on the Standard
Podles Quantum Sphere. Banach Center Publications 61 (2003)
4958.
[frt90] L.D. Faddeev, N.Y. Reshetikhin, L.A.
Takhtajan. Quantization Of Lie Groups And Lie Algebras.
Lengingrad Math. J. 1 (1990) 193225.
[fi06] G. Fiore.
qQuaternions and qdeformed su(2) instantons. hepth/0603138.
[gvf01] J.M. GraciaBondia, J.C. Varilly, H. Figueroa.
Elements of Noncommutative Geometry. Birkhauser 2001.
[hm99]
P. Hajac, S. Majid. Projective module description of the
qmonopole. Commun. Math. Phys. 206 (1999) 247264.
[hl04]
E. Hawkins, G. Landi. Fredholm Modules for Quantum Euclidean
Spheres. J. Geom. Phys. 49 (2004) 272293.
[ks98] A. U.
Klimyk, K. Schmuedgen. Quantum Groups and their Representations.
Springer, 1998.
[kr04] U. Krahmer. Dirac Operators on
Quantum Flag Manifolds. Lett. Math. Phys. 67 (2004), 4959.
[ku03] J. Kustermans, G. J. Murphy and L. Tuset. Differential
calculi over quantum groups and twisted cyclic cocycles. J. Geom.
Phys. 44 (2003) 570594.
[la97] G. Landi. An Introduction
to Noncommutative Spaces and their Geometry. Springer 1997.
[lm03] G. Landi, J. Madore. Twisted Configurations over
Quantum Euclidean Spheres. J. Geom. Phys. 45 (2003) 151163
[lpr06] G. Landi, C. Pagani, C. Reina. A Hopf Bundle over
a Quantum FourSphere from the Symplectic Group. Commun. Math.
Phys. 263 (2006) 6588.
[ls05] G. Landi, W. van Suijlekom.
Principal fibrations from noncommutative spheres. Commun.
Math. Phys. 260 (2005) 203225.
[ma95] S. Majid. Foundations
of Quantum Group Theory. Cambridge Univ. Press 1995
[mnw90]
T. Masuda, Y. Nakagami, J. Watanabe. Noncommutative differential
geometry on the quantum SU(2) I: An algebraic viewpoint. KTheory
4 (1990) 157180.
[ns98] N. Nekrasov, A. Schwarz. Instantons
on noncommutative R4 and (2,0) superconformal six dimensional theory.
Commun.Math.Phys. 198 (1998) 689703
[nt05] S. Neshveyev and
L. Tuset. A local index formula for the quantum sphere.
Commun. Math. Phys. 254 (2005) 323341.
[po87] P. Podles.
Quantum Spheres. Lett. Math. Phys. 14 (1987) 521531.
[sw03a] Schmuedgen, K. and E. Wagner. Examples of twisted
cyclic cocycles from covariant differential calculi. Lett. Math.
Phys. 64 (2003) 245254.
[sw03b] Schmuedgen, K. and E.
Wagner. Hilbert space representations of cross product algebras.
J. Funct. Anal. 200 (2003) 451493.
[sw03c] Schmuedgen, K. E.
Wagner. Hilbert space representations of cross product algebras
II, math.QA/0308229; to appear in Algebras and Representation
Theory.
[sw03d] Schmuedgen, K. and E. Wagner. Representations
of cross product algebras of Podles' quantum spheres,
math.QA/0305309.
[sw04] K. Schmuedgen, E. Wagner. Dirac
operator and a twisted cyclic cocycle on the standard Podles quantum
sphere. J. reine angew. Math. 574 (2004), 219235.
[ri81]
M.A. Rieffel. C*algebras Associated with Irrational Rotations.
Pacific J. Math. 93 (1981) 415429.
[wo87] S.L. Woronowicz.
Twisted SU(2) group. An example of a noncommutative differential
calculus. Publ. RIMS, Kyoto University, 23 (1987) 117181.
The project will be performed in a close collaboration
between the researchers of the two units. For the detailed programme,
times of realization, the role of each unit according to the foreseen
objectives and the relative modes of integration and collaboration,
see the form A, point 2.3.
The SISSA local unit of the
project is formed by two professors who have been active for several
years in the field of noncommutative geometry, quantum groups and
applications in mathematical physics and related areas, who will
supervise  as research directors  the scientific activity of the
two postdocs, among which one to be supported on the project funds.
A large part of the activity will consist of the scientific training
and orientation of the seven PhD students involved in the project.
The unit is hosted by the mathematical physics group of SISSA, which
is a graduate school with a regular program of doctoral fellowships
and an active centre of research at an international level. The
collaborations with the University of Trieste and the nearby Abdus
Salam International Center for Theoretical Physics create a
stimulating scientific environment with several joint activities such
as seminars, workshops and schools. In particular, in the area of
noncommutative geometry an international symposium was organized in
March 2001 and another one is planned for October 2004. The
facilities (library, computer network) is of a high level and is
adequate for a realization of the proposed research programme.
A
substantial part of the work will be devoted to the construction of
gauge theories on noncommutative spaces mentioned in point 2.4. In
particular, during the first year, generalizing the work of [ls05,
ls06], we plan to construct YangMills gauge fields (connections) on
the projective modules of instantonic type and study them as
solutions of suitable field equations, e.g. the selfduality
conditions on the instantonic projective modules constructed on
noncommutative spheres more complicated than the toric ones, for
example on the symplectic 4sphere of [lpr06] and other homogeneous
spaces (e.g. noncommutative projective spaces). During the second
year the systematic analysis of the space of connections, the action
of the group of gauge transformations and the corresponding orbit
space (moduli space) is planned for all the values of the Chern
number (topological charge). Here, the essential ingredients (like
the differential calculus and the Hodge structure) are not yet
available and have to be constructed ab initio. A related research
topic is focused on the construction of Hall Hamiltonians in the non
commutative set up (preliminary work is in [la06]. Classically these
are the Laplace operators acting on the space of sections of vector
bundles associated to principal bundles G > G/H over homogeneous
spaces (with G semisimple) and can be explicitely constructed in
terms of the Casimir operators of G and H. In particular, this
construction implicitly involves the canonical connection given by
the orthogonal splitting (with respect to the CartanKilling metric
on G) of the Lie algebra Lie(G) in terms of Lie(H) and of its
orthogonal complement. The first part of the research will be devoted
to this construction for SUq(2) as a Hopf Galois extension of the
algebra of the standard Podles sphere (i.e. the qanalogue of the
Hopf fibration on the 2sphere). Next we will tackle the problem of
non abelian Hall effects by looking at the obvious higher dimensional
case, namely Spq(2) as an extension of the symplectic quantum
7sphere found in [lpr06]. A spin off of these examples would be to
clarify in the quantum case the role of the canonical connection,
which we aim to construct as a splitting of the universal enveloping
algebra of Uq(Lie(G)).
The second line of the work concerns
the Riemannian spin geometry of quantum groups and their homogeneous
spaces. The first task we plan to complete is the construction of
spectral triples on the general twoparameter family of Podles
spheres (extending the two special cases of the standard and
equatorial spheres). We expect to obtain the Dirac operator with
linearly growing spectrum and the real structure and first order
condition satisfied up to infinitesimal operators of arbitrarily low
order. We plan to analyze the smoothness (regularity) properties, the
pseudodifferential calculus, generalized zeta functions and local
index formula. The second task to be performed during the first year
of the project regards the recently constructed [dlssv05] spectral
triple on SUq(2). We want to compare the difference between the
logarithmic derivative of the Dedekind eta function and its rational
approximation (which gives the cochain whose coboundary is the
difference between the original Chern character and the local one)
with the one in [co04]. During the second year of the project, we
plan to extend our constructions of spectral triples to more general
quantum groups. We shall study the twisted regularity condition.
Moreover, we shall investigate the important orientation requirement,
and the possibility to weaken its current formulation. Other tasks
interesting for the geometric interpretation of spectral triples we
plan to study is the geodesic flow and the curvature in the examples
related to quantum groups. Our final research target regards the
investigation of the internal perturbations and the spectral action
over the already known as well as the new geometries to be
constructed according to our project.
We plan to organize in
Trieste (in collaboration with the other unit) an international
workshop on the subjects of the project with the participation of the
major international experts.
The evaluation of the results
obtained during this project will be achieved via seminars to be
given at international workshops and conferences and via the
publication on international mathematical journals.