On Lie groups with left-invariant symplectic or Kählerian structures

Left-invariant symplectic structure on a group G; properties of the corresponding Lie algebra g. A unimodular symplectic Lie algebra has to be solvable (see [1]). Symplectic subgroups and left-invariant Poisson structures on a group. Affine Poisson structures: an affine Poisson structure associated to g and admitting g ^* as a unique leaf corresponds to a unimodular symplectic Lie algebra and the associate group is right-affine. If G is unimodular and endowed with a left-invariant metric g, harmonic theory for the left-invariant forms. Kählerian group is metabelian and Riemannianly flat. Decomposition of a simply connected Kählerian group. A symplectic group admitting a left-invariant metric with a nonnegative Ricci curvature is unimodular and admits a left-invariant flat Kählerian structure.     

Existence and invariance of twisted products on a symplectic manifold

It is now well-known [1] that the twisted product on the functions defined on a symplectic manifold, play a fundamental role in an invariant approach of quantum mechanics. We prove here a general existence theorem of such twisted products. If a Lie group G acts by symplectomorphisms on a symplectic manifold and if there is a G-invariant symplectic connection, the manifold admits G-invariant Vey twisted products. In particular, if a homogeneous space G/H admits an invariant linear connection, T ^*(G/H) admits a G-invariant Vey twisted product. For the connected Lie group G, the group T ^*G admits a symplectic structure, a symplectic connection and a Vey twisted product which are bi-invariant under G.     

A family of poisson structures on hermitian symmetric spaces

We investigate the compatibility of symplectic Kirillov-Kostant-Souriau structure and Poisson-Lie structure on coadjoint orbits of semisimple Lie group. We prove that they are compatible for an orbit compact Lie group iff the orbit is hermitian symmetric space. We prove also the compatibility statement for non-compact hermitian symmetric space. As an example we describe a structure of symplectic leaves onCP ⁿ for this family. These leaves may be considered as a perturbation of Schubert cells. Possible applications to infinite-dimensional examples are discussed.     

Poisson Lie groups and pentagonal transformations

Symplectic pentagonal transformations are intimately related to global versions of Poisson Lie groups (Manin groups, S ^*-groups, or symplectic pseudogroups). Symplectic pentagonal transformations of cotangent bundles, preserving the natural polarization, are shown to be in one to one correspondence with pentagonal transformations of the base manifold with a cocycle (if the base is connected and simply connected). By the results of Baaj and Skandalis, this allows to quantize (at the C ^*-algebra level!) those Poisson Lie groups, whose associated symplectic pentagonal transformation admits an invariant polarization. The (2n)²-parameter family of Poisson deformations of the (2n+1)-dimensional Heisenberg group described by Szymczak and Zakrzewski is shown to fall into this case.Supported by Alexander von Humboldt Foundation. On leave from Department of Mathematical methods in Physics, Warsaw University, Poland.     

An explicit *-product on the cotangent bundle of a Lie group

We give explicit formulas for a ^*-product on the cotangent bundle T ^* G of a Lie group G; these formulas involve on the one hand the multiplicative structure of the universal enveloping algebra U(G) of the Lie algebra G of G and on the other hand bidifferential operators analogous to the ones used by Moyal to define a ^*-product on IR²ⁿ.Chargé de recherches au FNRS, on leave of absence from Université libre de Bruxelles.     

Classical solutions of the quantum Yang-Baxter equation

The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a Poisson Lie group, we work directly with a Poisson Lie groupG and its associated symplectic groupoid. The classical analog of the quantumR-matrix is a lagrangian submanifold in the cartesian square of the symplectic groupoid. For any symplectic leafS inG, induces a symplectic automorphism ofS×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the flip map exchanging components and suitably implanted in each cartesian powerS ⁿ , generates a symplectic action of the braid groupB _n onS ⁿ . Application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants, but work on this last step is still in progress.Research partially supported by NSF Grant DMS-90-01089Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of University of Pennsylvania     

A Unified Approach to Poisson Reduction

Given any Poisson action G×PP of a Poisson–Lie group G we construct an object =T ^*G*T^* P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.     

Flat connections on a punctured sphere and geodesic polygons in a Lie group

Let G be a compact connected Lie group and X denote the complement of n distinct points of the sphere S². The space of isomorphism classes of flat G connections on X with fixed conjugacy class of holonomy around each of n punctures has a natural symplectic structure. This space is related to the space of geodesic n-gons in G. The space of geodesic polygons in G has a natural 2-form. It is shown that this 2-form coincides with symplectic form on the space of isomorphism classes of flat G-connections on X satisfying holonomy condition at the punctures.     

Operator-valued connections,Lie connections,and Gauge field theory

Noether's first theorem tells us that the global symmetry groupG _r of an action integral is a Lie group of point transformations that acts on the Cartesian product of the space-time manifold with the space of states and their derivatives. Gauge theory constructs are thus required for symmetry groups that act indiscriminately on the independent and dependent variables where the group structure can not necessarily be realized as a subgroup of the general linear group. Noting that the Lie algebra of a general symmetry groupG _r can be realized as a Lie algebrag _r of Lie derivatives on an appropriately structured manifold,G _r -covariant derivatives are introduced through study of connection 1-forms that take their values in the Lie algebrag _r of Lie derivatives (operator-valued connections). This leads to a general theory of operator-valued curvature 2-forms and to the important special class of Lie connections. The latter are naturally associated with the minimal replacement and minimal coupling constructs of gauge theory when the symmetry groupG _r is allowed to act locally. Lie connections give rise to the gauge fields that compensate for the local action ofG _r in a natural way. All governing field equations and their integrability conditions are derived for an arbitrary finite dimensional Lie group of symmetries. The case whereG _r contains the ten-parameter Poincaré group on a flat space-timeM ₄ is considered. The Lorentz structure ofM ₄ is shown to give a pseudo-Riemannian structure of signature 2 under the minimal replacement associated with the Lie connection of the local action of the Poincaré group. Field equations for the matter fields and the gauge fields are given for any system of matter fields whose action integral is invariant under the global action of the Poincaré group.     

Normal forms generalizing action-angle coordinates for Hamiltonian actions of Lie groups

Let (M, Ω) be a symplectic manifold on which a Lie group G acts by a Hamiltonian action. Under some restrictive assumptions, we show that there exists a symplectic diffeomorphism ψ of a G-invariant open neighbourhood U of a given G-orbit in M, onto an open subset ψ(U) of a vector bundle F ^*, with base space G. Explicit expressions are given for the symplectic 2-form, for the momentum map and for a Hamiltonian vector field whose Hamiltonian function is G-invariant, on the model symplectic manifold ψ(U).     

Non-compact symmetric spaces and the Toda molecule equations

It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, G ^N, by its maximal compact subgroup. This is explained in more detail and it is shown that the fundamental Poisson bracket relation involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forG ^N endows the symmetric space with a hidden group theoretic structure.Supported by CNP _q (Brasil)     

q → ∞ limit of the quasitriangular WZW model

Abstract

We study the q → ∞ limit of the q-deformation of the WZW model on a compact simple and simply connected target Lie group. We show that the commutation relations of the q → ∞ current algebra are underlied by certain affine Poisson structure on the group of holomorphic maps from the disc into the complexification of the target group. The Lie algebroid corresponding to this affine Poisson structure can be integrated to a global symplectic groupoid which turns out to be nothing but the phase space of the q → ∞ limit of the q-WZW model. We also show that this symplectic grupoid admits a chiral decomposition compatible with its (anomalous) Poisson-Lie symmetries. Finally, we dualize the chiral theory in a remarkable way and we evaluate the exchange relations for the q → ∞ chiral WZW fields in both the original and the dual pictures.     

Boundary G/G Theory and Topological Poisson–Lie Sigma Model

We study a boundary version of the gauged WZW model with a Poisson–Lie group G as the target. The Poisson–Lie structure of G is used to define the Wess–Zumino term of the action on surfaces with boundary. We clarify the relation of the model to the topological Poisson sigma model with the dual Poisson–Lie group G ^* as the target and show that the phase space of the theory on a strip is essentially the Heisenberg double of G introduced by Semenov–Tian–Shansky.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Method of Quantum Characters in Equivariant Quantization

 Let G be a reductive Lie group, g its Lie algebra, and M a G-manifold. Suppose 𝔸 _h (M) is a 𝕌 _h (g)-equivariant quantization of the function algebra 𝔸(M) on M. We develop a method of building 𝕌 _h (g)-equivariant quantization on G-orbits in M as quotients of 𝔸 _h (M). We are concerned with those quantizations that may be simultaneously represented as subalgebras in 𝕌^* _h (g) and quotients of 𝔸 _h (M). It turns out that they are in one-to-one correspondence with characters of the algebra 𝔸 _h (M). We specialize our approach to the situation g=gl(n,ℂ), M=End(ℂ ⁿ ), and 𝔸 _h (M) the so-called reflection equation algebra associated with the representation of 𝕌 _h (g) on ℂ ⁿ . For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the 𝕌(g)-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits. Received: 28 April 2002 / Accepted: 3 October 2002 Published online: 24 January 2003 RID="*" ID="*" This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01. Communicated by L. Takhtajan     

Drinfeld-Sokolov gravity

A lagrangian euclidean model of Drinfeld-Sokolov (DS) reduction leading to generalW-algebras on a Riemann surface of any genus is presented. The background geometry is given by the DS principal bundleK associated to a complex Lie groupG and anSL(2,) subgroupS. The basic fields are a hermitian fiber metricH ofK and a (0, 1) Koszul gauge fieldA ^* ofK valued in a certain negative graded subalgebrar ofg related tos. The action governing theH andA ^* dynamics is the effective action of a DS field theory in the geometric background specified byH andA ^*. Quantization ofH andA ^* implements on one hand the DS reduction and on the other defines a novel model of 2d gravity, DS gravity. The gauge fixing of the DS gauge symmetry yields an integration on a moduli space of DS gauge equivalence classes ofA ^* configurations, the DS moduli space. The model has a residual gauge symmetry associated to the DS gauge transformations leaving a given fieldA ^* invariant. This is the DS counterpart of conformal symmetry. Conformal invariance and certain non-perturbative features of the model are discussed in detail.     

A note on coherent states

Given a connected Lie groupG with an Abelian invariant Lie subgroup and a continuous unitary representation ofG on the Hilbert space ?, we investigate a relationship between the first cohomology groupH ¹(G, ?) and classes of sectors, determined by coherent states with a projectivelyG-covariant Weyl system. This result is applied to calculateH ¹(G, ?), if the groupG has in addition a compact subgroup with certain properties.