Linearization Problems for Lie Algebroids and Lie Groupoids

In this Letter, we discuss a series of linearization problems – for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and almost invariant submanifolds for compact group actions. The Letter ends with a discussion of possible extensions of the convexity theorems for momentum maps of hamiltonian actions of compact groups.     

Ricci flow,Courant algebroids,and renormalization of Poisson–Lie T-duality

We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson–Lie T-duality is compatible with the 1-loop renormalization group flow.     

Compatible Lie algebroids and the periodic Toda lattice

We give the set of maps from to the structure of a Poisson manifold endowed with a pair of compatible Lie algebroids. A suitable reduction process, of the Marsden–Ratiu type, yields a smaller manifold with the same geometrical properties as the original manifold. Moreover, is a bi-Hamiltonian manifold and the flows naturally defined on it are the periodic Toda flows.     

Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries

Chern–Simons (CS) gauge theories in three dimensions and the Poisson sigma model (PSM) in two dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well defined for non-parallelizable Poisson manifolds and propose a covariant definition of the off-shell gauge symmetries as left action of some finite-dimensional Lie algebroid.

Our approach allows us to avoid complications arising in the infinite-dimensional super-geometry of the BV- and AKSZ-formalism. This preprint is a starting point in a series of papers meant to introduce Yang–Mills type gauge theories of Lie algebroids, which include the standard YM theory, gerbes, and the PSM.     

Pre-symplectic algebroids and their applications

In this paper, we introduce the notion of a pre-symplectic algebroid and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. Then we study exact pre-symplectic algebroids and show that they are classified by the third cohomology group of a left-symmetric algebroid. Finally, we study para-complex pre-symplectic algebroids. Associated with a para-complex pre-symplectic algebroid, there is a pseudo-Riemannian Lie algebroid. The multiplication in a para-complex pre-symplectic algebroid characterizes the restriction to the Lagrangian subalgebroids of the Levi–Civita connection in the corresponding pseudo-Riemannian Lie algebroid.     

Nambu–Dirac Structures for Lie Algebroids

The theory of Nambu–Poisson structures on manifolds is extended to the context of Lie algebroids in a natural way based on the derived bracket associated with the Lie algebroid differential. A new way of combining Nambu–Poisson structures and triangular Lie bialgebroids is described in this work. Also, we introduce the concept of a higher order Dirac structure on a Lie algebroid. This allows to describe both Nambu–Poisson structures and Dirac structures on manifolds in the same setting.     

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.     

On the Integration of Poisson Manifolds,Lie Algebroids,and Coisotropic Submanifolds

In recent years, methods for the integration of Poisson manifolds and of Lie algebroids have been proposed, the latter being usually presented as a generalization of the former. In this Letter it is shown that the latter method is actually related to (and may be derived from) a particular case of the former if one regards dual of Lie algebroids as special Poisson manifolds. The core of the proof is the fact, discussed in the second part of this Letter, that coisotropic submanifolds of a (twisted) Poisson manifold are in one-to-one correspondence with possibly singular Lagrangian subgroupoids of source-simply-connected (twisted) symplectic groupoids.     

Foliation-coupling Dirac structures

We extend the notion of “coupling with a foliation” from Poisson to Dirac structures and get the corresponding generalization of the Vorobjev characterization of coupling Poisson structures [Yu.M. Vorobjev, Coupling tensors and Poisson geometry near a single symplectic leaf, Lie algebroids and related topics in differential geometry, Banach Center Publ., Polish Acad. Sci. (Warsaw) 54 (2001) 249–274; I. Vaisman, Coupling Poisson and Jacobi structures, Int. J. Geom. Meth. Mod. Phys. 1 (5) (2004) 607–637]. We show that any Dirac structure is coupling with the fibers of a tubular neighborhood of an embedded presymplectic leaf, give new proofs of the results of Dufour and Wade [J.-P. Dufour, A. Wade, On the local structure of Dirac manifolds. arXiv:math.SG/0405257] on the transversal Poisson structure, and compute the Vorobjev structure of the total space of a normal bundle of the leaf. Finally, we use the coupling condition along a submanifold, instead of a foliation, in order to discuss submanifolds of a Dirac manifold which have differentiable, induced Dirac structures. In particular, we get an invariant that reminds the second fundamental form of a submanifold of a Riemannian manifold.     

The Muhly–Renault–Williams Theorem for Lie Groupoids and its Classical Counterpart

A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.     

Unimodularity criteria for Poisson structures on foliated manifolds

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a (singular) symplectic leaf. Our results generalize some known unimodularity criteria for regular Poisson manifolds related to the notion of the Reeb class. In particular, we show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded decomposition of modular vector fields of a coupling Poisson structure on a foliated manifold. Moreover, we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb class.     

Quantum Groupoids

We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid , which gives an interpretation of the quantum dynamical Yang–Baxter equation in terms of Hopf algebroids. Received: 6 April 2000 / Accepted: 15 August 2000     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of Poisson structures with background'.     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of ‘Poisson structures with background’.     

Poisson Involutions, Spin Calogero-Moser Systems Associated with Symmetric Lie Subalgebras and the Symmetric Space Spin Ruijsenaars-Schneider Models

We develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical Lie algebroids and symmetric coboundary dynamical Poisson groupoids. The method is based on the successive use of Dirac reduction and Poisson reduction. Then we show that certain spin Calogero-Moser systems associated with symmetric Lie subalgebras can be studied in this fashion. We also consider some spin-generalized Ruisjenaars-Schneider equations which correspond to the N-soliton solutions of affine Toda field theory. In this case, we show how the equations are obtained from the Dirac reduction of some Hamiltonian system on a symmetric coboundary dynamical Poisson groupoid.     

Courant Algebroids and Strongly and Strongly Homotopy Lie Algebras

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the bundle TM T^*M with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infinitesimal objects for Poisson groupoids. We show that Courant algebroids can be considered as strongly homotopy Lie algebras.     

<Emphasis Type="Italic">Q</Emphasis>-Manifolds and Mackenzie Theory

Double Lie algebroids were discovered by Kirill Mackenzie from the study of double Lie groupoids and were defined in terms of rather complicated conditions making use of duality theory for Lie algebroids and double vector bundles. In this paper we establish a simple alternative characterization of double Lie algebroids in a supermanifold language. Namely, we show that a double Lie algebroid in Mackenzie’s sense is equivalent to a double vector bundle endowed with a pair of commuting homological vector fields of appropriate weights. Our approach helps to simplify and elucidate Mackenzie’s original definition; we show how it fits into a bigger picture of equivalent structures on ‘neighbor’ double vector bundles. It also opens ways for extending the theory to multiple Lie algebroids, which we introduce here.     

Geometry of Maurer-Cartan Elements on Complex Manifolds

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite dimensional; we describe how homology and cohomology are related through the Evens-Lu-Weinstein duality module; and we describe a duality on Koszul-Brylinski homology, which generalizes the Serre duality of Dolbeault cohomology.     

Reduction of Lagrangian mechanics on Lie algebroids

We prove that if a surjective submersion which is a homomorphism of Lie algebroids is given, then there exists another homomorphism between the corresponding prolonged Lie algebroids and a relation between the dynamics on these Lie algebroid prolongations is established. We also propose a geometric reduction method for dynamics on Lie algebroids defined by a Lagrangian and the method is applied to regular Lagrangian systems with nonholonomic constraints.