Bott–Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

Lu, Jiang; Yu, Shan

doi:10.1007/s00029-020-00595-1

Cited by 4 publications

(1 citation statement)

References 31 publications

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“…Secondly, generalized Schubert cells with the standard Poisson structures form basic building blocks for many of the Poisson manifolds associated to the Poisson Lie group (G, π st ). For example, it is shown in [20] that a number of Poisson homogeneous spaces (G/Q, π G/Q ) of the Poisson Lie group (G, π st ), including (G, π st ) itself and (G/B, π 1 ), admit so-called Bott-Samelson atlases which are built out of generalized Schubert cells, and the Poisson structure π G/Q is presented as symmetric Poisson CGL extensions in all of the coordinate charts of the Bott-Samelson atlas. We refer to [20] for more detail.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Configuration Poisson groupoids of flags

Lu¹,

Mouquin²,

Yu³

2021

Preprint

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Let G be a connected complex semi-simple Lie group and B its flag variety. For every positive integer n, we introduce a Poisson groupoid over B n , called the nth total configuration Poisson groupoid of flags of G, which contains a family of Poisson sub-groupoids whose total spaces are generalized double Bruhat cells and bases generalized Schubert cells in B n . Certain symplectic leaves of these Poisson sub-groupoids are then shown to be symplectic groupoids over generalized Schubert cells. We also give explicit descriptions of symplectic leaves in three series of Poisson varieties associated to G. Contents 1. Introduction and statements of results 1 2. A series of Poisson groupoids associated to Poisson Lie groups 9 3. The total configuration Poisson groupoids of flags and isomorphic models 13 4. Special configuration Poisson groupoids of flags and isomorphic models 17 5. Configuration symplectic groupoids of flags 22 Appendix A. Mixed product Poisson structures 25 Appendix B. The Poisson isomorphism J n 29 Appendix C. Generalized double Bruhat cells 32 Appendix D. Symplectic leaves of (O w e × T, π n ⊲⊳ 0) 40 References 55

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Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

Configuration Poisson groupoids of flags

Lu¹,

Mouquin²,

Yu³

2021

Preprint

Self Cite

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A Bruhat Atlas for the Mehta–van der Kallen Stratification of T∗GLn/B

Knutson

Sam

2022

Transformation Groups

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On the full Kostant–Toda hierarchy and its ℓ -banded reductions for the Lie algebras of type A , B

Kodama,

Xie

2024

Nonlinearity

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This paper is concerned with the solutions of the full Kostant–Toda (f-KT) hierarchy in the Hessenberg form and their reductions to the ℓ -banded Kostant–Toda ( ℓ -KT) hierarchies for 1 ⩽ ℓ ⩽ n , where the case with ℓ = 1 is the classical tridiagonal Toda hierarchy and the case with ℓ = n is the f-KT hierarchy. Based on the Hessenberg form of the f-KT hierarchy, we study the f-KT hierarchy and the corresponding ℓ -KT hierarchies on simple Lie algebras of type A , B and G as the root space reductions with proper choices of Chevalley systems. Explicit formulas of the polynomial solutions for the τ-functions are also given in terms of the Schur functions and Schur’s Q-functions.

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Bott–Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces

Cited by 4 publications

References 31 publications

Configuration Poisson groupoids of flags

Configuration Poisson groupoids of flags

A Bruhat Atlas for the Mehta–van der Kallen Stratification of T∗GLn/B

On the full Kostant–Toda hierarchy and its ℓ -banded reductions for the Lie algebras of type A , B

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