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On Dynamical Poisson Groupoids I
 
Luen-Chau Li Pennsylvania State University, University Park, PA
Serge Parmentier Université Lyon, Villeurbanne, France
On Dynamical Poisson Groupoids I
eBook ISBN:  978-1-4704-0425-3
Product Code:  MEMO/174/824.E
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $37.20
On Dynamical Poisson Groupoids I
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On Dynamical Poisson Groupoids I
Luen-Chau Li Pennsylvania State University, University Park, PA
Serge Parmentier Université Lyon, Villeurbanne, France
eBook ISBN:  978-1-4704-0425-3
Product Code:  MEMO/174/824.E
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $37.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1742005; 72 pp
    MSC: Primary 53; Secondary 58;

    We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of (CDYBE) on simple Lie algebras as classified by the same authors. Our approach is based on the study of a class of Poisson structures on trivial Lie groupoids within the category of biequivariant Poisson manifolds. In the former case, it is shown that the dual Poisson groupoid of such a dynamical Poisson groupoid is isomorphic to a Poisson groupoid (with trivial Lie groupoid structure) within this category. In the latter case, we find that the dual Poisson groupoid is also of dynamical type modulo Poisson groupoid isomorphisms. For the coboundary dynamical Poisson groupoids associated with constant \(r\)-matrices, we give an explicit construction of the corresponding symplectic double groupoids. In this case, the symplectic leaves of the dynamical Poisson groupoid are shown to be the orbits of a Poisson Lie group action.

  • Table of Contents
     
     
    • Chapters
    • 1. Introduction
    • 2. A class of biequivariant Poisson groupoids
    • 3. Duality
    • 4. An explicit case study of duality
    • 5. Coboundary dynamical Poisson groupoids – the constant $r$-matrix case
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1742005; 72 pp
MSC: Primary 53; Secondary 58;

We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of (CDYBE) on simple Lie algebras as classified by the same authors. Our approach is based on the study of a class of Poisson structures on trivial Lie groupoids within the category of biequivariant Poisson manifolds. In the former case, it is shown that the dual Poisson groupoid of such a dynamical Poisson groupoid is isomorphic to a Poisson groupoid (with trivial Lie groupoid structure) within this category. In the latter case, we find that the dual Poisson groupoid is also of dynamical type modulo Poisson groupoid isomorphisms. For the coboundary dynamical Poisson groupoids associated with constant \(r\)-matrices, we give an explicit construction of the corresponding symplectic double groupoids. In this case, the symplectic leaves of the dynamical Poisson groupoid are shown to be the orbits of a Poisson Lie group action.

  • Chapters
  • 1. Introduction
  • 2. A class of biequivariant Poisson groupoids
  • 3. Duality
  • 4. An explicit case study of duality
  • 5. Coboundary dynamical Poisson groupoids – the constant $r$-matrix case
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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