Home page of the Research Program PRIN 2006-2008 
Noncommutative geometry of quantum groups
and their homogeneous spaces 

Trieste Unit

Human Resources:

University Personnel

Post-docs

PhD students

Scientific Background

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 Levi-Civita 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 zeta-function, 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 2-spheres [bm93, hm99], certain instantonic fibre bundles over q-deformed 4-spheres have been constructed [dlm01, bct02, dl02, hs02, cd-v02, ab02, lm03, hl04, bcdt04], fi06]. A recent important result is the construction in [lpr06] of a `true' noncommutative principal bundle on a quantum 4-sphere, with structure group SUq(2). Technically, an A(SUq(2)) Hopf-Galois extension of the algebra of functions on a quantum 4-sphere (the base manifold) into the algebra of functions on a quantum 7-sphere (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 2-sphere), which satisfies a weak form of the reality and `order one condition'. On SUq(2) itself we have constructed in [dlssv05] a 3-summable 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 3-sphere. 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.

References

[bct02] F. Bonechi, N. Ciccoli, M. Tarlini. Noncommutative Instantons on the 4-Sphere from Quantum Groups. Commun. Math. Phys. 226 (2002) 419-432.

[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) 71-181.

[bm93] T. Brzezinski, S. Majid. Quantum Group Gauge Theory on Quantum Spaces. Commun. Math. Phys. 157 (1993) 591--638. erratum 167 (1955) 235.

[cpa03] P.S. Chakraborty, A. Pal. Equivariant spectral triples on the quantum SU(2) group. K-Theory 28 (2003) 107-126.

[co94] A. Connes. Noncommutative geometry. Academic Press 1994

[co96] A. Connes. Gravity coupled with matter and foundation of non-commutative geometry. Commun. Math. Phys. 182 (1996) 155-176

[co04] A. Connes. Cyclic Cohomology, Quantum group Symmetries and the Local Index Formula for SUq(2). J. Inst. Math. Jussieu 3 (2004), 17-68.

[cl01] A. Connes, G. Landi. Noncommutative manifolds, the instanton algebra and isospectral deformations. Commun. Math. Phys. 221 (2001) 141-159

[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) 174-243.

[cmo98] A. Connes, H. Moscovici. Hopf algebras, Cyclic cohomology and the Tranverse Index Theorem. Commun. Math. Phys. 198 (1998) 199-246

[cr87] A. Connes, M. Rieffel. Yang-Mills for Non-commutative Two-Tori. Contemp. Math. 62 (1987) 237-266

[dl02] L. Dąbrowski, G. Landi. Instanton algebras and quantum 4-spheres. Differ. Geom. Appl. 16 (2002) 277-284

[dlm01] L. Dąbrowski, G. Landi, T. Masuda. Instantons on the Quantum 4-Spheres S^4_q. Commun. Math. Phys. 221 (2001) 161-168

[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) 729-759.

[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. K-Theory, 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) 819-822.

[ds03] L. Dąbrowski, A. Sitarz. Dirac Operator on the Standard Podles Quantum Sphere. Banach Center Publications 61 (2003) 49-58.

[frt90] L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan. Quantization Of Lie Groups And Lie Algebras. Lengingrad Math. J. 1 (1990) 193-225.

[fi06] G. Fiore. q-Quaternions and q-deformed su(2) instantons. hep-th/0603138.

[gvf01] J.M. Gracia-Bondia, J.C. Varilly, H. Figueroa. Elements of Noncommutative Geometry. Birkhauser 2001.

[hm99] P. Hajac, S. Majid. Projective module description of the q-monopole. Commun. Math. Phys. 206 (1999) 247-264.

[hl04] E. Hawkins, G. Landi. Fredholm Modules for Quantum Euclidean Spheres. J. Geom. Phys. 49 (2004) 272-293.

[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), 49-59.

[ku03] J. Kustermans, G. J. Murphy and L. Tuset. Differential calculi over quantum groups and twisted cyclic cocycles. J. Geom. Phys. 44 (2003) 570-594.

[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) 151-163

[lpr06] G. Landi, C. Pagani, C. Reina. A Hopf Bundle over a Quantum Four-Sphere from the Symplectic Group. Commun. Math. Phys. 263 (2006) 65-88.

[ls05] G. Landi, W. van Suijlekom. Principal fibrations from noncommutative spheres. Commun. Math. Phys. 260 (2005) 203-225.

[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. K-Theory 4 (1990) 157-180.

[ns98] N. Nekrasov, A. Schwarz. Instantons on noncommutative R4 and (2,0) superconformal six dimensional theory. Commun.Math.Phys. 198 (1998) 689-703

[nt05] S. Neshveyev and L. Tuset. A local index formula for the quantum sphere. Commun. Math. Phys. 254 (2005) 323-341.

[po87] P. Podles. Quantum Spheres. Lett. Math. Phys. 14 (1987) 521-531.

[sw03a] Schmuedgen, K. and E. Wagner. Examples of twisted cyclic cocycles from covariant differential calculi. Lett. Math. Phys. 64 (2003) 245-254.

[sw03b] Schmuedgen, K. and E. Wagner. Hilbert space representations of cross product algebras. J. Funct. Anal. 200 (2003) 451-493.

[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), 219-235.

[ri81] M.A. Rieffel. C*-algebras Associated with Irrational Rotations. Pacific J. Math. 93 (1981) 415-429.

[wo87] S.L. Woronowicz. Twisted SU(2) group. An example of a noncommutative differential calculus. Publ. RIMS, Kyoto University, 23 (1987) 117--181.

Research Program Description

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 post-docs, 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 Yang-Mills 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 4-sphere 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 semi-simple) 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 Cartan-Killing 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 q-analogue of the Hopf fibration on the 2-sphere). 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 7-sphere 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 two-parameter 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.