# Manin Triples for Lie Bialgebroids

@article{Liu1995ManinTF,
title={Manin Triples for Lie Bialgebroids},
author={Zhang-Ju Liu and Alan D. Weinstein and Ping Xu},
journal={Journal of Differential Geometry},
year={1995},
volume={45},
pages={547-574}
}
• Published 28 August 1995
• Mathematics
• 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…
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## References

SHOWING 1-10 OF 52 REFERENCES
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
• Mathematics
• 1994
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
• Mathematics
• 1993
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
• Mathematics
• 1994
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
LIE GROUPOIDS AND LIE ALGEBROIDS IN DIFFERENTIAL GEOMETRY
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