Mixed Product Poisson Structures Associated to Poisson Lie Groups and Lie Bialgebras

Lu, Jia; Mouquin, Victor

doi:10.1093/imrn/rnw189

Cited by 24 publications

(67 citation statements)

References 36 publications

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“…As a result, we are able to construct of a number of well known spaces including: Lu's symplectic double groupoid integrating a Poisson Lie group [32], Boalch's Fission spaces [11,12], Poisson Lie groups [17,36], and Poisson homogeneous spaces [34], among others. Our approach builds upon the results and ideas of various authors including Fock and Rosly, Boalch, and the second author [7-12, 20, 38-40].…”

Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

“…In fact, π defines the Poisson Lie group structure on G corresponding the double Lie bialgebra structure on g resulting from the Manin triple (g, e, f) [30,32]. The symplectic leaves are computed as the restriction of the l-orbits, which in this case can be seen to correspond to the orbits of the dressing action on G.…”

Section: Poisson Structuresmentioning

confidence: 99%

“…Example 1.14 (Affine Poisson structure on G [32]). Generalizing the setup found in Example 1.11, we suppose that h ⊆ g is a Lagrangian subalgebra which is also complementary (as a vector space) to e ⊆ g. As in Example 1.11, we colour the edges with the full group G, but the vertices as…”

Section: Example 113 (Poisson Homogenous Spaces) Suppose the Lie Grmentioning

confidence: 99%

“…where the bases {ζ i f } ⊆ f and {ζ i h } ⊆ h are both dual to {η i } ⊆ g. In fact, π defines the affine Poisson structure on G corresponding to the Manin triples (g, e, f) and (g, e, h) (as described by Lu [32]). The symplectic leaves are computed as the restriction of the l-orbits.…”

Section: Example 113 (Poisson Homogenous Spaces) Suppose the Lie Grmentioning

confidence: 99%

“…on G to E ⊆ G. Thus, π defines the Poisson Lie group structure on E corresponding the Manin triple (g, e, f) [30,32]. As before, the symplectic leaves are computed as the restriction of the l-orbits.…”

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confidence: 99%

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Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces

Li-Bland

Ševera

2015

Mathematical Physics Studies

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Abstract. In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to extend the results of [30,40] to allow more general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective.The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.

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Section: Introduction and Summary Of Resultsmentioning

confidence: 99%

Section: Poisson Structuresmentioning

confidence: 99%

Section: Example 113 (Poisson Homogenous Spaces) Suppose the Lie Grmentioning

confidence: 99%

Section: Example 113 (Poisson Homogenous Spaces) Suppose the Lie Grmentioning

confidence: 99%

mentioning

confidence: 99%

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Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces

Li-Bland

Ševera

2015

Mathematical Physics Studies

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Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

2018

Lie Groups, Geometry, and Representation Theory

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Let G be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of G are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on C. It follows that the Kogan-Zelevinsky integrable systems on G have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of SL(n, C). We in fact construct a class of integrable systems with complete Hamiltonian flows associated to generalized Bruhat cells which are defined using arbitrary sequences of elements in the Weyl group of G, and we obtain the results for double Bruhat cells through the so-called open Fomin-Zelevinsky embeddings of (reduced) double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide global coordinates on double Bruhat cells, called Bott-Samelson coordinates, in which all the Fomin-Zelevinsky minors become polynomials and the Poisson structure can be computed explicitly.

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Double Bruhat Cells and Symplectic Groupoids

Mouquin

2017

Transformation Groups

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Mixed Product Poisson Structures Associated to Poisson Lie Groups and Lie Bialgebras

Cited by 24 publications

References 36 publications

Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces

Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces

Generalized Bruhat Cells and Completeness of Hamiltonian Flows of Kogan-Zelevinsky Integrable Systems

Double Bruhat Cells and Symplectic Groupoids

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