Characteristic classes for singular foliations

Iakovos Androulidakis

Abstract

In earlier work we introduced the notion of an atlas of bi-submersions

as a substitute for groupoids. The holonomy groupoid of a singular folia-

tion was given as a quotient of such an atlas, as well as the construction of

the relevant C�-algebra. The analytic index map of families of pseudod-

i�erential operators along the leaves takes values in the K-theory of this

C�-algebra, however for its computation one really needs the cohomology

of the classifying space of this groupoid.

This poster explains how bi-submersions allow for the formulation of

the �Cech-de Rham bi-complex associated with a singular foliation and

the de�nition of the cohomology of the classifying space of a singular

foliation, even though the holonomy groupoid itself is topologically a very

ill-behaved object.

Extending an idea of Connes and Skandalis, we use bi-submersions to

de�ne principal bundles for a singular foliation which allow for the inter-

pretation of elements of the above cohomology as natural transformations.

This is joint work with Paulo Carrillo Rouse.

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Paulo Antunes Universidade de Coimbra

Poisson quasi-Nijenhuis structure with background on a Lie algebroid

The aim of the poster is to introduce a definition of Poisson quasi-Nijenhuis structure with background on a Lie algebroid. We also relate it to quasi-Lie bialgebroids, compatible second-order tensors and generalized complex geometry.

Paula Balseiro CSIC, Spain

Generalized nonholonomic systems on Lie

In this poster we will present mechanical systems subject to generalized nonholonomic constraints on a Lie algebroid. We will show how the geo-metry underlying these systems gives rise to another algebroid structure on the constraint manifold, a Leibniz algebroid. Finally using the linear almost Poisson structure, induced by this Leibniz algebroid, on the dual bundle it will be possible to formulate a Hamilton-Jacobi theory.

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QUASI-LIE BIALGEBRAS AND LIE LOOPS

MOMO BANGOURA

Abstract. In this work, we define the quasi-Poisson Lie quasi-groups,dual objects to the quasi-Poisson Lie groups and we establish the corre-spondence between the local quasi-Poisson Lie quasi-groups and quasi-Lie bialgebras (up to isomorphism).

Departement de Mathematiques, Universite de Conakry, BP 1147, Republique

de Guinee

E-mail address: angoura@gn.refer.org

2000 Mathematics Subject Classification. 17B70 17A30.Keywords. Lie algebra, Akivis algebra, quasi-Lie bialgebra, Lie loop, Lie group, Lie quasi-

bialgebra, quasi-double, quasi-group, quasi-Poisson .

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Florian Becher University of Freiburg

Completely positive time evolution of the quantized algebre of observalbes of the open

subsystem gives a completely positive time evolution of the quantized algebra of observables of the open subsystem

Consider a combined Hamiltonian system consisting of two originally separate Hamiltonian systems (subsystem & bath) coupled by interaction terms in the Hamiltonian of the combined system. The subsystem together with its projection of the time evolution of the combined system may then be viewed as an open system.

We show in the setting of deformation quantization that a certain choice of the deformation of a completely positive time evolution of the classical algebra of observables of the open subsystem gives a completely positive time evolution of the quantized algebra of observables of the open subsystem.

Connections on k-symplectic manifolds

Adara M. Blaga

Abstract.

A canonical connection on a k-symplectic manifold is defined andsufficient conditions such that it should be preserved by perform-ing Marsden-Weinstein reduction are given. In particular, the re-lation between the induced canonical connections on the reducedstandard k-symplectic manifolds with respect to the action of aLie group G is established. Similarly, defining Poisson structureson these manifolds, the relation between the corresponding Pois-son structures on the reduced manifolds is stated.

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HOCHSCHILD COHOMOLOGY OF KLEIN SURFACESFrédéric BUTIN

Université Lyon 1, Francebutin@math.univ-lyon1.fr

AbstractGiven a mechanical system (M, F(M)), where M is a Poisson manifold and F(M) the algebra of regularfunctions on M , it is important to be able to quantize it, in order to obtain more precise results thanthrough classical mechanics. An available method is the deformation quantization, which consists inconstructing a star-product on the algebra of formal power series F(M)[[~]]. A first step toward study ofstar-products is the calculation of Hochschild cohomology of F(M).The aim of this article is to determine this Hochschild cohomology in the case of singular curves of theplane — so we rediscover, by a different way, a result proved by Fronsdal and make it more precise —and in the case of Klein surfaces. The use of a complex suggested by Kontsevich and the help of Gröbnerbases allow us to solve the problem.

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On Poisson quasi-Nijenhuis Lie algebroids

Raquel Caseiro, Antonio De Nicola and Joana M. Nunes da CostaCMUC, Department of Mathematics, University of Coimbra, Portugal

The notion of Poisson quasi-Nijenhuis manifold was recently introducedby Stienon and Xu [9]. It is a manifold M together with a Poisson bivectorfield π, a (1, 1)-tensor N compatible with π and a closed 3-form φ suchthat iNφ is also closed and the Nijenhuis torsion of N , which is nonzero,is expressed by means of φ and π. When φ = 0 one obtains a Poisson-Nijenhuis manifold, a concept introduced by Magri and Morosi [6] to studyintegrable systems and which was extended to the Lie algebroid framework byKosmann-Schwarzbach [3] and Grabowski and Urbanski [2] who introducedthe notion of a Poisson-Nijenhuis Lie algebroid. In this paper we propose adefinition of Poisson quasi-Nijenhuis Lie algebroid, which is a straightforwardgeneralization of a Poisson quasi-Nijenhuis manifold.

Quasi-Lie bialgebroids were introduced by Roytenberg [7] who showedthat they are the natural framework to study twisted Poisson structures [8].On the other hand, quasi-Lie bialgebroids are intimately related to Courantalgebroids [4]. Extending a result of [9], we show that a Poisson quasi-Nijenhuis Lie algebroid has an associated quasi-Lie bialgebroid, so that ithas also an associated Courant algebroid.

In an unpublished manuscript, Alekseev and Xu [1], gave the definition ofa Courant algebroid morphism between E1 and E2 and, in the case where E1

and E2 are doubles of Lie bialgebroids (A,A∗) and (B,B∗), i.e E1 = A⊕A∗

and E2 = B ⊕ B∗, they established a relationship with a Lie bialgebroidmorphism A → B [5]. Since doubles of quasi-Lie bialgebroids are Courantalgebroids, it seems natural to obtain a relationship between Courant alge-broid morphisms and quasi-Lie bialgebroid morphisms. This is the case whenconsidering Courant algebroids associated with a Poisson quasi-Nijenhuis Liealgebroid of a certain type and with a twisted Poisson Lie algebroid, respec-tively. In a first step towards our result, we give the definition of a morphismof quasi-Lie bialgebroids which is, up to our knowledge, a new concept thatincludes morphism of Lie bialgebroids as a particular case.

References

[1] A. Alekseev and P. Xu, Derived brackets and Courant algebroids. Un-published manuscript.

[2] J. Grabowski and P. Urbanski, Lie algebroids and Poisson-Nijenhuisstructures. Rep. Math. Phys. 40 (1997) 195–208.

— E–mail: raquel@mat.uc.pt, antondenicola@gmail.com, jmcosta@mat.uc.pt

[3] Y. Kosmann-Schwarzbach, The Lie bialgebroid of a Poisson-Nijenhuismanifold. Lett. Math. Phys. 38 (1996) 421–428.

[4] Z.-J. Liu, A. Weinstein and P. Xu, Manin triples for Lie bialgebroids. J.Diff. Geom. 45 (1997), no. 3, 547–574.

[5] K. Mackenzie, General theory of Lie groupoids and Lie algebroids. Lon-don Math. Soc. Lecture notes series 213, Cambridge University Press,Cambridge 2005.

[6] F. Magri and C. Morosi, A geometrical characterization of integrableHamiltonian systems through the theory of Poisson-Nijenhuis manifolds.Quaderno S 19, 1984, University of Milan.

[7] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds.Lett. Math. Phys. 61 (2002), 123–137.

[8] P. Severa and A. Weinstein, Poisson geometry with a 3-form background.Noncommutative geometry and string theory. Prog. Theor. Phys., Suppl.144 (2001) 145–154.

[9] M. Stienon and P. Xu, Poisson Quasi-Nijenhuis Manifolds. Comm. Math.Phys. 50 (2007), 709–725.

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ON CLASSIFICATION AND CONSTRUCTION OFALGEBRAIC FROBENIUS MANIFOLDS

YASSIR IBRAHIM DINAR

Abstract. We develop the theory of generalized bi-Hamiltonian re-duction. Applying this theory to a suitable loop algebra we recover ageneralized Drinfeld-Sokolov reduction. This gives a way to constructnew examples of algebraic Frobenius manifolds.

University of Khartoum, Khartoum, SudanE-mail address: dinar@ictp.it

2000 Mathematics Subject Classification. Primary 37K10; Secondary 35D45.Key words and phrases. Integrable systems, bi-Hamiltonian manifolds, Dirac reduction,

Frobenius manifolds.

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Lie Symmetries Analysis of Shallow WaterEquations

E. H. El Kinani and A. Ouhadan

Univesite Moulay Ismail, Faculte des Sciences et TechniquesGroupe algebre et applications, Departement de mathematiques

B.P 509 ErrachidiaMaroc

Abstract

In this work we consider an auxiliary system of the shallow waterequations in which the first equation is written in a conserved form.Using the Lie theory, we obtain a five dimentional symmetry algebra.Similarity reductions are performed for each generator and we constructexplicit non trivial exact solutions.

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Sean Fitzpatrick University of Toronto

Quantization of Contact Manifolds

Affine Euler-Poincare and Lie-Poisson ReductionsApplication to Complex Fluids

Francois Gay-Balmaz

As opposed to the case of (simple) fluids, the dynamics of complex fluids dependsalso on variables called order parameters that describe the macroscopic variations of theinternal structure of the fluid parcels. These macroscopic variations may form observablepatterns, as seen in liquid crystals. Other examples of perfect complex fluids includesuperfluids, Yang-Mills magnetofluids, microfluids and spin-glasses.

From a serie of papers by Holm, Marsden and Ratiu, it is known that the Euler-Poincare and Lie-Poisson frameworks provide a unified approach for fluids and leads tonew models.

We extend this framework to the case of complex fluids by:(1) adding an affine term to the cotangent lifted action;(2) replacing the group of fluid motions by a semidirect product of this group and theLie group of order parameters.

We also explore the Hamiltonian counterpart of the problem and obtain, using theprocess of reduction by stages, that the symplectic reduced spaces are affine coadjointorbits. In particular, we obtain the noncanonical Poisson structure of complex fluids byreduction of the canonical Poisson structure on a cotangent bundle.

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Classes of semi-symmetric andpseudo-symmetric Riemann spaces

Iulia Elena Hirica

University of Bucharest, Faculty of Mathematics and InformaticsDepartment of Geometry, 14 Academiei Str., Bucharest 1, Romania,

e-mail: ihirica@fmi.unibuc.ro

Abstract

Let (M, g) be a Riemannian manifold. It is called pseudo-symmetricif at every point of M the tensor R·R and the Tachibana tensor Q(g,R)are linearly dependent. Any semi-symmetric manifold (R · R = 0) ispseudo-symmetric. This general notion aroses during the study oftotally umbilical submanifolds of semi-symmetric spaces, as well asduring the consideration of geodesic mappings. Also any spacetimeverifies conditions of pseudo-symmetric type.

We continue the study in this direction, considering subgeodesicmappings, which are a natural generalization of geodesic mappingson Riemannian manifolds. We study ξ-subgeodesically related spaces,extending some known results concerning pseudo-symmetric spacesadmitting geodesic mappings. Conharmonic semi-symmetric spacesgeodesically related are also characterized, considering several con-harmonic and concircular transformations of metrics.

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Dirac and Nonholonomic Reduction

Madeleine Jotz ∗ and Tudor Ratiu

AbstractSeveral aspects of Dirac reduction are compared and formulated from the same geometric point of

view. A link with nonholonomic reduction is found. The theory of optimal momentum maps andreduction is extended from the category of Poisson manifolds to that of closed Dirac manifolds. Anoptimal reduction method for a class of nonholonomic systems is formulated. Several examples arestudied in detail.

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Rieffel’s Deformation Quantization for States

Daniel Kaschek∗, Nikolai Neumaier

‡, Stefan Waldmann§,

Fakultat fur Mathematik und Physik

Albert-Ludwigs-Universitat Freiburg

Physikalisches Institut

Hermann Herder Straße 3

D 79104 Freiburg

Germany

23.06.2008

Abstract

Rieffel’s deformation quantization provides a method to construct star products in a conver-

gent C*-algebraic setting. The noncommutative product is put into effect by oscillatory integrals

over bounded functions coming from an isometric action of a vector space on the algebra to be

deformed. The noncommutativity of the product is determined by the symplectic structure of

the vector space as can be seen in the asymptotic expansion of the integral formula.

In the framework of C*-algebras the normalized, positive, linear functionals are the physical

states. In general, a deformation of the original algebra into a new C*-algebra does not conserve

the positive elements, consequently the “classical” states are no longer states. We have shown

that for every deformation corresponding to a symplectic structure one can construct an operator

S that deforms the states ω of the classical C*-algebra into states ω ◦S of the deformed algebra.

However, the choice of S is not unique but it depends on the choice of a metric that is compatible

with the symplectic structure.

∗Daniel.Kaschek@physik.uni-freiburg.de‡Nikolai.Neumaier@physik.uni-freiburg.de§Stefan.Waldmann@physik.uni-freiburg.de

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Noah Kieserman

U. Wisconsin, Madison

The Liouville Phenomenon in the Deformat

Olga Kravchenko Institut Camille Jordan, Universite Lyon

Courant Algebroid Operad

We describe a Courant algebroid structure as an algebra over a certain coloured operad. It is inspired by the work of Pepijn van der Laan who defined a colored operad corresponding to a Lie algebroid in his thesis. He studied the notion of Koszul duality for coloured operads based on the work of Markl. Lie algebroid operad is Koszul self-dual.While Courant algebroid operad is not quadratic it is still possible to define certain operadic constructions for it.

Lie groupoid�s point of view of linear direct connections andcharacteristic classes

International Conference POISSON 2008Poisson Geometry in Mathematics and PhysicsEcole Polytechnique Fédérale de LausanneLausanne, Switzerland, July 7-11, 2008

Jan KubarskiInstitute of Mathematics

Technical University of ×ódz, Poland

N. Teleman shows how the Chern character of the tangent bundle of a smoothmanifold may be extracted from the geodesic distance function by means ofcyclic homology. N.Teleman in one of his papers said: "The arguments discussedhere may be extended to the language of groupoids". My talk is the step in thisdirection. In the paper J.Kubarski, N.Teleman, "Linear direct connections",Banach Center Publications Vol. 76, Warszawa 2007, the geometry of directconnections � is studied. The "in�nitesimal part" r� is constructed and it isshowed thatr� is a usual linear connection. We determined the curvature tensorof r� and as applications we presented a direct proof N.Teleman�s theorem onthe Chern character of the tangent bundle.This talk concerns Lie groupoid�s point of view of linear direct connections.Let � be an arbitrary transitive Lie groupoid with the anchor � and the

target �. We denote by uy the unit of � at y: By a linear direct connection in� we mean a mapping � : (M �M)jU ! �; where U � M �M is an openneighborhood of the diagonal � = f(x; x) ; x 2Mg ; such that � � � (x; y) =y; � � � (x; y) = x;and � (x; x) = ux:I show how we can extract the Chern-Weil homomorphism of a Lie groupoid

via any local direct connection in this Lie groupoid (on the level of di¤erentialforms).

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VB-ALGEBROIDS AND HIGHER LIE ALGEBROIDREPRESENTATIONS

ALFONSO GRACIA-SAZ AND RAJAN AMIT MEHTA

Abstract. A VB-algebroid is essentially defined as a Lie algebroid object

in the category of vector bundles. It turns out that a VB-algebroid may bethought of as a “higher” generalization of a Lie algebroid representation. In

this setting, we are able to construct characteristic classes, which in special

cases reproduce characteristic classes constructed by Crainic and Fernandes.We give a complete classification of regular VB-algebroids, and in the process

we obtain another characteristic class that does not appear in the ordinary

representation theory of Lie algebroids.

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Symmetries of Poisson pencils and integrable systems

Andriy Panasyuk

There are several bihamiltonian descriptions of the n-dimensional free rigid body system whichgive a complete set of the first integrals of this system. But if the inertia matrix of the body hasmultiplicities in its spectrum these integrals (e.g. the Manakov ones) form an incomplete set, whichhowever can be completed by the noetherian integrals.

This picture will be generalized to a general situation of Poisson pencils with symmetries andcriteria of completeness of the corresponding sets of integrals will be presented.

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Loops on the Hamiltonian group of a Cartesian product

Andres Pedroza

Let (M,ω) be a closed symplectic manifold and Ham(M,ω) the group of Hamil-tonian diffeomorphisms of (M,ω). Then the Seidel homomorphism is a map fromthe fundamental group of Ham(M,ω) to the quantum homology ring QH∗(M ; Λ).Using this homomorphism we give a sufficient condition for when a nontrivial loopψ in Ham(M,ω) determines a nontrivial loop ψ× idN in Ham(M×N,ω⊕η), where(N, η) is a closed symplectic manifold such that π2(N) = 0.

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Lie algebroid geometry for hamiltonian dynamics

Liviu Popescu

It is well known that the cotangent bundle of a differentiable manifold M playsa very important role in symplectic geometry and its applications, since this carriesa canonical symplectic structure induced by the Liouville form. The Hamiltonianformalism seems to be, in many ways, mathematically more straightforward thatthe Lagrangian formalism, because on the tangent bundle one cannot define anaturally symplectic structure, out of the Lagrangian function. On the contrary,the tangent bundle has a naturally defined integrable almost tangent structureand the notion of semispray (second order differential equation vector field) whichinduce a (nonlinear) connection on it [2] [6].

The notion of Lie algebroid [11] (E, π,M) is a generalization of the conceptsof Lie algebra and integrable distribution. The notion of prolongation of a Liealgebroid [7] over the vector bundle projections of vector bundle and its dual gen-eralize the concepts of tangent and cotangent bundle. In [17] A. Weinstein givesa generalized theory of Lagrangian on Lie algebroids and obtains the equationsof motion, using the Poisson structure on the dual and Legendre transformationinduced by a regular Lagrangian. The same equations were later obtained by E.Martinez [12] using a modified version of symplectic formalism, in which the bun-dles tangent to E and E∗ are replaced by the prolongations T E and T E∗. In thelast years diverse aspects of these subjects are studied by many authors. In [10]the Hamiltonian formalism on the prolongation of Lie algebroid over the vectorbundle projection of the dual bundle, and equivalece with Lagrangian formalismare presented. In [3] [4] [5] [13] [15] the geometric properties of connections onT E are investigated and in [8] the study of some geometric structures of T E∗ isonly starting.

The purpose of this paper is to study the geometry of the prolongation ofLie algebroid and the relations with Hamiltonian dynamics. We introduce thenotion of dynamical covariant derivative [9] [1] on T E induced by a semisprayand shows that the metric compatibility of the semispray and associated nonlinearconnection gives the one the so called Helmholtz conditions of the inverse problem

of Lagrangian Mechanics [16].We study the properties of connections on T E∗

and introduce the notions of adapted almost tangent structure, almost complexstructure and characterize the integrability conditions in terms of torsion andcurvature of connection. We show that every regular section (in particular, anyregular Hamiltonian on E∗) determines a canonical Ehresmann connections onT E∗. We introduce some generalizations of the Hamilton sections, as a mechanicalstructures and semi-Hamiltonian sections and characterize their properties.

In the last part, using the diffeomorpfism from T E∗ to T E [10], induced by aregular Hamiltonian, we transfer many geometrical results between these spaces.Thus, a semispray on T E is transformed into a semi-Hamiltonian section on T E∗

if and only if the nonlinear connection determined by semispray is metric.

References

[1] I. Bucataru, Metric nonlinear connection, Diff. Geom. and its Appl. 25 (2007)335-343.

[2] M. Crampin, Tangent bundle geometry for Lagrangian dynamics, J. Phys. A:Math. Gen. 16 (1983) 3755-3772.

[3] F. Cantrijn, B. Langerock, Generalized connections over a bundle map, Diff.Geom. and its Appl. 18 (2003) 295-317.

[4] J. Cortez, E. Martinez, Mechanical control systems on Lie algebroids, IMAJ. Math. Control Inform. 21 (2004) 457-492.

[5] R. L. Fernandes, Lie Algebroids, Holonomy and Characteristic Classes, Ad-vances in Mathematics, 170 (2002) 119-179.

[6] J. Grifone, Structure presque tangente et connections I , Ann. Inst. Fourier22 (1) (1972) 287-334.

[7] P. J. Higgins, K. Mackenzie, Algebraic constructions in the category of Liealgebroids, Journal of Algebra 129 (1990) 194-230.

[8] D. Hrimiuc, L. Popescu, Nonlinear connections on dual Lie algebroids, BalkanJournal of Geometry and Its Applications, 11 (1) (2006) 73-80.

[9] O. Krupkova, Variational metric structure, Publ. Math. Debrecen 62, 3-4(2003), 461-498.

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[10] M. de Leon, J. C. Marrero, E. Martinez, Lagrangian submanifolds and dy-namics on Lie algebroids J. Phys. A: Math. Gen. 38 (2005) 241-308.

[11] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Lon-don Mathematical Society Lecture Note Series, Cambridge, no.124, 1987.

[12] E. Martinez, Lagrangian mechanics on Lie algebroids, Acta Appl. Math., 67,(2001) 295-320.

[13] T. Mestdag, Generalized connections on affine Lie algebroids, Rep. Math.Phys. 51 (2003) 297-305.

[14] V. Oproiu, Regular vector fields and connections on cotangent bundle, AnnalsSt. Univ. Al. I. Cuza, Iasi 37, 1 (1991) 87-104.

[15] L. Popescu, Geometrical structures on Lie algebroids, Publ. Math. Debrecen72, 1-2 (2008), 95-109.

[16] W. Sarlet, The Helmholtz conditions revisited. A new approach to the inverseproblem of Lagrangian dynamics, J. Phys. A 15 (1982) 1503-1517.

[17] A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Comm. 7(1996), 206-231.

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Florian Schätz

University of Zurich

E-Courant Algebroids

Zhuo Chen, Zhangju Liu and Yunhe ShengDepartment of Mathematics, Peking University, Beijing, China

Abstract

A kind of generalized Courant algebroids is introduced associated to a vector bundle E, called anE-Courant algebroid, with the differential operator bundle DE as the target of its anchor. Such anew structure not only unifies several known structures such as Courant algebroids, Courant-Jacobialgebroids and omni-Lie algebroids but also provides some interesting topics for the future studies.

1

Libor Snobl Czech Technical University in Prague

Description of D-branes invariant under the Poisson-Lie T–plurality

transformations

Abstract

We write the conditions for open strings with charged endpoints in the language of gluing matrices. We identify constraints imposed on the gluing matrices that are essential in this setup and investigate the question of their invariance under the Poisson–Lie T–plurality transformations. We show that the chosen set of constraints is equivalent to the statement that the lifts of D–branes into the Drinfel’d double are right cosets with respect to a maximally isotropic subgroup and therefore it is invariant under the Poisson–Lie T–plurality transformations.

1

Tazvan Micu Tudoran

The West University of Timisoara, Romani

On 3D Hamiltonian systems.

Deformation Quantization

of

Surjective Submersions and Principal Fibre Bundles

Stefan Weiß

Faculty of Mathematics and Physics

Albert-Ludwigs-University FreiburgInstitute of Physics

Hermann Herder Straße 3D 79104 Freiburg

F.R. Germany

Principal fibre bundles and, more general, surjective submersions over smooth manifoldsare omnipresent in differential geometry and its applications in mathematical physics, especi-ally in classical gauge theories. If one considers such a structure over some Poisson-manifold(M, π) with a given star product defining a deformation quantization of the algebra of smoothfunctions on M one can ask for the notion of a deformation quantization of the whole geo-metric structure. In our work we present a new definition for this using the fact, that thefunctions on the total space build a right module with respect to the functions on the basespace M . Beside the proof of existence and uniqueness of the new defined structures wecan show that a deformation quantization of a principal fibre bundle naturally leads to adeformation quantization of every associated vector bundle.

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