Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras

Lu, Jia; Mouquin, Victor

doi:10.48550/arxiv.1504.06843

Cited by 3 publications

(12 citation statements)

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“…When n = 1, we have This paper, a sequel to [38], is the second of a series of papers devoted to a detailed study of the four series of Poisson manifolds in (1.1). In [38], we have identified the Poisson structures in (1.1) as mixed product Poisson structures defined by quasitriangular r-matrices.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

On the T-leaves of some Poisson structures related to products of flag varieties

Mouquin

2017

Advances in Mathematics

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Abstract. For a connected abelian Lie group T acting on a Poisson manifold (Y, π) by Poisson isomorphisms, the T-leaves of π in Y are, by definition, the orbits of the symplectic leaves of π under T, and the leaf stabilizer of a T-leaf is the subspace of the Lie algebra of T that is everywhere tangent to all the symplectic leaves in the T-leaf. In this paper, we first develop a general theory on T-leaves and leaf stabilizers for a class of Poisson structures defined by Lie bialgebra actions and quasitriangular r-matrices. We then apply the general theory to four series of holomorphic Poisson structures on products of flag varieties and related spaces of a complex semi-simple Lie group G. We describe their T -leaf decompositions, where T is a maximal torus of G, in terms of (open) extended Richardson varieties and extended double Bruhat cells associated to conjugacy classes of G, and we compute their leaf stabilizers and the dimension of the symplectic leaves in each T -leaf.

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Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

On the T-leaves of some Poisson structures related to products of flag varieties

Mouquin

2017

Advances in Mathematics

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“…and the Poisson structure π is a mixed product Poisson structure in the sense of [20], or, more precisely, π is the sum of the product Poisson structure Using the embeddings I v , we also interpret the Fomin-Zelevinsky twist map on double Bruhat cells [8,14] in terms of the inverse map of the groupoid (G/B) × B − over G/B. See Remark 5.8.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…We recall from [2,7,20] some basic facts on Poisson Lie groups and Lie bialgebras, and we refer to [20, §2] in particular on certain conventions on constants and signs.…”

Section: Poisson Lie Groups R-matrices and Mixed Product Poisson Stru...mentioning

confidence: 99%

“…as the mixed term of π X × (ρ,λ) π Y . Mixed product Poisson structures of the form in (2.9) are studied in [20].…”

Section: 2mentioning

confidence: 99%

“…Corollary 3.6. With the notation as above, and let For the rest of the paper, let G be a connected complex semisimple Lie group with Lie algebra g. We recall the so-called standard multiplicative Poisson structures on G and refer to [7,20,21] for details. Fix a pair (B, B − ) of opposite Borel subgroups of G and a non-degenerate symmetric adinvariant bilinear form , g on g, and let T = B ∩ B − .…”

Section: Actionmentioning

confidence: 99%

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Double Bruhat cells and symplectic groupoids

Mouquin

2016

Preprint

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Let G be a connected complex semisimple Lie group, equipped with a standard multiplicative Poisson structure πst determined by a pair of opposite Borel subgroups (B, B−). We prove that for each v in the Weyl group W of G, the double Bruhat celltogether with the Poisson structure πst, is naturally a Poisson groupoid over the Bruhat cell BvB/B in the flag variety G/B. Correspondingly, every symplectic leaf of πst in G v,v is a symplectic groupoid over BvB/B. For u, v ∈ W , we show that the double Bruhat cell (G u,v , πst) has a naturally defined left Poisson action by the Poisson groupoid (G u,u , πst) and a right Poisson action by the Poisson groupoid (G v,v , πst), and the two actions commute. Restricting to symplectic leaves of πst, one obtains commuting left and right Poisson actions on symplectic leaves in G u,v by symplectic leaves in G u,u and in G v,v as symplectic groupoids.

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On the T-leaves of some Poisson structures related to products of flag varieties

Mouquin

2015

Preprint

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Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras

Cited by 3 publications

References 20 publications

On the T-leaves of some Poisson structures related to products of flag varieties

On the T-leaves of some Poisson structures related to products of flag varieties

Double Bruhat cells and symplectic groupoids

On the T-leaves of some Poisson structures related to products of flag varieties

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