Lie Bialgebroid Research Articles

The operators∂and∂of a generalized complex structure

We prove that the ∂ and ∂ operators introduced by Gualtieri for a generalized complex structure coincide with the d * and ∂ operators introduced by Alekseev and Xu for Evens-Lu-Weinstein modules of a Lie bialgebroid.

Dirac generating operators and Manin triples

Given a pair of (real or complex) Lie algebroid structures on a vector bundle A (over M) and its dual A*, and a line bundle ℒ such that ℒ ⊗ ℒ = (∧top A* ⊗ ∧top T*M)1/2 exists, there exist two canonically defined differential operators d̆* and ∂̆ on г(∧A ⊗ ℒ). We prove that the pair (A, A*) constitutes a Lie bialgebroid if and only if the square of D̆ = d̆* + ∂̆ is the multiplication by a function on M. As a consequence, we obtain that the pair (A, A*) is a Lie bialgebroid if and only if D̆ is a Dirac generating operator as defined by Alekseev and Xu. Our approach is to establish a list of new identities relating the Lie algebroid structures on A and A*.

Relèvement d'une algébroïde de Courant

Let q : E → P be a Courant algebroid. We show that the tangent bundle T q : T E → T P , has a lifted structure of a Courant algebroid, deduced from that of q : E → P . If E = A ⊕ A ∗ , is the double of a Lie bialgebroid ( A , A ∗ ) , then TE is the double of the tangent Lie bialgebroid ( T A , ( T A ) ∗ ) . The Dirac structures of the Courant algebroid TE are determined by integrable sub-bundles of E. To cite this article: M. Boumaiza, N. Zaalani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Q-algebroids and their cohomology

A Q-algebroid is a Lie superalgebroid equipped with a compatible homological vector field and is the infinitesimal object corresponding to a Q-groupoid. We associate to every Q-algebroid a double complex. As a special case, we define the BRST model of a Lie algebroid, which generalizes the BRST model for equivariant cohomology. We extend to this setting the Mathai-Quillen-Kalkman isomorphism of the BRST and Weil models, and we suggest a definition of a basic subcomplex which, however, requires a choice of a connection. Other examples include Roytenberg's homological double of a Lie bialgebroid, Ginzburg's model of equivariant Lie algebroid cohomology, the double of a Lie algebroid matched pair, and Q-algebroids arising from lifted actions on Courant algebroids.

Modular classes of Loday algebroids

We introduce the concept of Loday algebroids, a generalization of Courant algebroids. We define the naive cohomology and modular class of a Loday algebroid, and we show that the modular class of the double of a Lie bialgebroid vanishes. For Courant algebroids, we describe the relation between the naive and standard cohomologies and we conjecture that they are isomorphic when the Courant algebroid is transitive. To cite this article: M. Stiénon, P. Xu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Elliptic hydrodynamics and quadratic algebras of vector fields on a torus

We construct a quadratic Poisson algebra of Hamiltonian functions on a two-dimensional torus compatible with the canonical Poisson structure. This algebra is an infinite-dimensional generalization of the classical Sklyanin-Feigin-Odesskii algebras. It yields an integrable modification of the two-dimensional hydrodynamics of an ideal fluid on the torus. The Hamiltonian of the standard two-dimensional hydrodynamics is defined by the Laplace operator and thus depends on the metric. We replace the Laplace operator with a pseudodifferential elliptic operator depending on the complex structure. The new Hamiltonian becomes a member of a commutative bi-Hamiltonian hierarchy. In conclusion, we construct a Lie bialgebroid of vector fields on the torus.

Dirac structures on protobialgebroids

Protobialgebroids include several kinds of algebroid structures such as Lie algebroid, Lie bialgebroid, Lie quasi-bialgebroid, etc. In this paper, the Dirac theories are generalized from Lie bialgebroid to protobialgebroid. We give the integrable conditions for a maximally isotropic subbundle being a Dirac structure for a protobialgebroid by the notion of a characteristic pair. From the integrable conditions, we found out that the Dirac structure has closed relations with the twisting of a protobialgebroid. At last, some special cases of the Dirac structures for protobialgebroids are discussed.

Reduction of Jacobi manifolds via Dirac structures theory

We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one–one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold ( M , Λ , E ) for which 1 is an admissible function and Jacobi quotient manifolds of M. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.

Formality for Lie Algebroids

Using Dolgushev’s generalization of Fedosov’s method for deformation quantization, we give a positive answer to a question of P. Xu: can one prove a formality theorem for Lie algebroids ? As a direct application of this result, we obtain that any triangular Lie bialgebroid is quantizable.

Dirac-nijenhuis structures on lie bialgebroids

This paper is a continuation of the work done in [1]. We define Dirac-Nijenhuis structures (DN structures) on Lie bialgebroids, a generalization of the notion of Dirac-Nijenhuis structures on manifolds given in [1]. However, the problems we are going to deal with are more complex than the ones in [1]. The necessary and sufficient conditions for a structure to be a DN structure on a Lie bialgebroid as well as several examples of these structures are given. At the end of the paper we discuss the fundamental sections of DN structures.

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid ( A , A ∗ ), the differential associated to the Lie algebroid structure on A ∗ has the form d ∗ = [ Λ , ⋅ ] A + Ω , where Λ is a section of ∧ 2 A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A . Globally, for any transitive Poisson groupoid ( Γ , Π ) , the Poisson structure has the form Π = Λ ← − Λ → + Π F , where Π F is a bivector field on Γ associated to a Lie groupoid 1-cocycle.

Dirac–Nijenhuis structures

In this paper we study the concept of Dirac–Nijenhuis structures. We consider them to be a pair where D is a Dirac structure, defined with respect to a Lie bialgebroid (A, A*), and is a Nijenhuis operator which defines a deformation of the Lie algebroid structure of D in a compatible way. The transformation can be considered to deform also the double of the Lie bialgebroid, which leads to the concept of Dirac–Nijenhuis structures of type I, or to not affect it, leading to Dirac–Nijenhuis structures of type II. We prove that the concept contains Poisson–Nijenhuis structures as a particular case and provides another example for the case of Kahler manifolds.

Courant algebroid and Lie bialgebroid contractions

Contractions of Leibniz algebras and Courant algebroids by means of (1,1)-tensors are introduced and studied. An appropriate version of Nijenhuis tensors leads to natural deformations of Dirac structures and Lie bialgebroids. One recovers presymplectic-Nijenhuis structures, PoissonNijenhuis structures, and triangular Lie bialgebroids as particular examples. MSC 2000: Primary 17B99; Secondary 17B62, 53C15, 53D17.

A characterization of strict Jacobi-Nijenhuis manifolds through the theory of Lie algebroids

We obtain a characterization of strict Jacobi-Nijenhuis structures using the equivalent notions of generalized Lie bialgebroid and Jacobi bialgebroid.

Generalized Lie bialgebroids and Jacobi structures

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.

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 ${\cal D}\otimes_{\hbar} U_{\hbar}(\frakg)$, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

Integration of Lie bialgebroids

We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.

Classical lifting processes and multiplicative vector fields

We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a new description of the Lie bialgebroid structure associated with a Poisson groupoid.

Classical Lifting Processes and Multiplicative Vector Fields

We extend the calculus of multiplicative vector fields and differential forms and their intrinsic derivatives from Lie groups to Lie groupoids; this generalization turns out to include also the classical process of complete lifting from arbitrary manifolds to tangent and cotangent bundles. Using this calculus we give a new description of the Lie bialgebroid structure associated with a Poisson groupoid.

Quantum groupoids and deformation quantization

The purpose of this Note is to unify quantum groups and star-products under a general umbrella: quantum groupoids. It is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e.. the quantization problem, is posed. In particular, any regular triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a Poisson manifold, its quantization is equivalent to a star-product.