Manin Triples for Lie Bialgebroids

  title={Manin Triples for Lie Bialgebroids},
  author={Zhang-Ju Liu and Alan D. Weinstein and Ping Xu},
  journal={Journal of Differential Geometry},
In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly… 
Pseudo-Dirac Structures
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
Courant Algebroids in Parabolic Geometry
Let $p$ be a Lie subalgebra of a semisimple Lie algebra $g$ and $(G,P)$ be the corresponding pair of connected Lie groups. A Cartan geometry of type $(G,P)$ associates to a smooth manifold $M$ a
Courant Algebroids and Poisson Geometry
Given a manifold M with an action of a quadratic Lie algebra , such that all stabilizer algebras are coisotropic in , we show that the product becomes a Courant algebroid over M. If the bilinear form
Conformal Courant Algebroids and Orientifold T-duality
We introduce conformal Courant algebroids, a mild generalization of Courant algebroids in which only a conformal structure rather than a bilinear form is assumed. We introduce exact conformal Courant
Notions of double for Lie algebroids
We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated
Poisson-generalized geometry and $R$-flux
We study a new kind of Courant algebroid on Poisson manifolds, which is a variant of the generalized tangent bundle in the sense that the roles of tangent and the cotangent bundle are exchanged. Its
Shifted Symplectic Lie Algebroids
Shifted symplectic Lie and $L_{\infty }$ algebroids model formal neighborhoods of manifolds in shifted symplectic stacks and serve as target spaces for twisted variants of the classical topological
M. Kontsevich's graph complexes and universal structures on graded symplectic manifolds I
In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds.


Lie Algebroids Associated to Poisson Actions
This work is motivated by a result of Drinfeld on Poisson homogeneous spaces. For each Poisson manifold $P$ with a Poisson action by a Poisson Lie group $G$, we describe a Lie algebroid structure on
Poisson cohomology and quantization.
Let R be a commutative ring, and let A be a Poisson algebra over R. We construct an (R,A)-Lie algebra structure, in the sense of Rinehart, on the A-module of K\"ahler differentials of A depending
Complementary 2-forms of Poisson structures
Let (M, P) be a Poisson manifold. A 2-form w of M such that the Koszul bracket {03C9, 03C9}P = 0 is called a complementary form of P. Every complementary form yields a new Lie algebroid structure of
Strongly homotopy Lie algebras
The present paper can be thought of as a continuation of the paper "Introduction to sh Lie algebras for physicists" by T. Lada and J. Stasheff (International Journal of Theoretical Physics Vol. 32,
Quantum Groups
This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions
Resolution of Diagonals and Moduli Spaces
This paper is a continuation of [BG]. In that paper, for any smooth complex curve X and n > 1, we constructed a canonical completion of the configuration space of all ordered n-tuples of distinct
Lie bialgebroids and Poisson groupoids
Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a
Exact Gerstenhaber algebras and Lie bialgebroids
We show that to any Poisson manifold and, more generally, to any triangular Lie bialgebroid in the sense of Mackenzie and Xu, there correspond two differential Gerstenhaber algebras in duality, one
4. G. D. Mostow and P. Deligne, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Etudes Sci. Publ. Math. 46 (1983). 5. E. Picard, Sur les fonctions hyperfuchsiaes