Poisson actions and Lie bialgebroid morphisms

He, Long-guang; Liu, Zhangju; Zhong, Dingxing

doi:10.1090/conm/315/05483

Cited by 5 publications

(3 citation statements)

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“…Note that, by the two equivalent descriptions of morphisms of Lie algebroids in Definition 4.5 and in Theorem 4.6, we recover Theorem 3.1 in [10].…”

Section: ⇒ (V N) Is a Morphism Of Lie Algebroids If And Only Ifmentioning

confidence: 65%

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On (co-)morphisms of Lie pseudoalgebras and groupoids

Chen

Liu

2007

Journal of Algebra

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“…Note that, by the two equivalent descriptions of morphisms of Lie algebroids in Definition 4.5 and in Theorem 4.6, we recover Theorem 3.1 in [10].…”

Section: ⇒ (V N) Is a Morphism Of Lie Algebroids If And Only Ifmentioning

confidence: 65%

“…(1) Ψ is clearly well defined. The above lemma says that (1) of Definition 3.1 is exactly the condition that G (Ψ,ψ) ⊂ E ⊕ ψ F. And under this condition, it is easily seen that (2) of Definition 3.1 is exactly the condition for G (Ψ,ψ) to be closed under the bracket given by relation (10) in Proposition 2.9.…”

Section: The Main Theoremmentioning

confidence: 93%

On (co-)morphisms of Lie pseudoalgebras and groupoids

Chen

Liu

2007

Journal of Algebra

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“…Remark. In [6], the authors characterize Poisson actions of Poisson groupoid G on Poisson manifold X by Lie bialgebroid morphisms from T * X to A * , where A is the Lie algebroid of the groupoid G. Thus, it is interesting to study P-N Lie bialgebroid morphism and characterize P-N actions in terms of P-N Lie bialgebroid morphisms.…”

Section: Coisotropic-invariant Subgroupoidsmentioning

confidence: 99%

Poisson-Nijenhuis groupoids

Das

2017

Preprint

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We define multiplicative Poisson-Nijenhuis structures on a Lie groupoid which extends the notion of symplectic-Nijenhuis groupoid introduced by Stiénon and Xu. We also introduce a special class of Lie bialgebroid structure on a Lie algebroid A, called P-N Lie bialgebroid, which defines a hierarchy of compatible Lie bialgebroid structures on A. We show that under some topological assumption on the groupoid, there is a one-to-one correspondence between multiplicative Poisson-Nijenhuis structures on a Lie groupoid and P-N Lie bialgebroid structures on the corresponding Lie algebroid. Contents 1. Introduction 1 2. Preliminaries 3 3. (1,1)-tensors 7 4. Compatible Lie algebroids 9 5. Poisson-Nijenhuis groupoids 13 6. P-N Lie bialgebroids 15 7. Infinitesimal form of Poisson-Nijenhuis groupoids 17 8. Coisotropic-invariant subgroupoids 20 9. Poisson-Nijenhuis actions 21 References 22

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Generalized Classical Dynamical Yang-Baxter Equations and Moduli Spaces of Flat Connections on Surfaces

2015

Commun. Math. Phys.

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In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures, Poisson Gspaces and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix, induced by the gauge fixing procedure, to give a new finite dimensional description of the Atiyah-Bott symplectic structure on the moduli space of flat connections on a surface. Using this, we find a Poisson groupoid symmetry of the moduli space.

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Poisson actions and Lie bialgebroid morphisms

Cited by 5 publications

References 0 publications

On (co-)morphisms of Lie pseudoalgebras and groupoids

On (co-)morphisms of Lie pseudoalgebras and groupoids

Poisson-Nijenhuis groupoids

Generalized Classical Dynamical Yang-Baxter Equations and Moduli Spaces of Flat Connections on Surfaces

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