Bott-Samelson varieties and Poisson Ore extensions

Elek, Balázs; Lu, Jia

doi:10.48550/arxiv.1601.00047

Cited by 12 publications

(10 citation statements)

References 25 publications

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“…As far as we could tell, that would require that H have a rather fearsome Dynkin diagram, a sort of "broom" with handle of length n − 2 plus n + 1 bristles (nicely indexed by the covers of v(Y Spr ); see the final paragraph of the paper). It seems very hard to compute defining equations of strata inside such general Kac-Moody flag varieties; the closest work seems to be [EL19]. Much more recently we tried to embed the M-vdK poset (for small n) as an order ideal inside {ρ ∈ S m : ρ ≥ σ} for some fixed σ with X σ smooth (so that each X ρ…”

Section: Statement Of Resultsmentioning

confidence: 99%

A Bruhat atlas for the Mehta-van der Kallen stratification of $T^* GL_n/B$

Knutson¹,

Sam²

2021

Preprint

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Mehta and van der Kallen put a Frobenius splitting on the type A cotangent bundle T * GL n /B, thereby defining a stratification by compatibly split subvarieties, and they determined a few of the elements of this stratification. We embed T * GL n /B as a stratum in a larger stratified (and Frobenius split) space GL n /B × Mat n whose stratification we determine, thereby giving a full description of the one of Mehta-van der Kallen. The main technique is to endow GL n /B × Mat n with a Bruhat atlas, covering it with open sets that are stratified-isomorphic to Bruhat cells (in GL 2n /B 2n ).

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Section: Statement Of Resultsmentioning

confidence: 99%

A Bruhat atlas for the Mehta-van der Kallen stratification of $T^* GL_n/B$

Knutson¹,

Sam²

2021

Preprint

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“…In §4, we discuss the theory of global R-matrices developed by Weinstein and Xu in [19]. We show that for the Drinfeld double (D, π D ) of a pair ((G, π G ), (G * , π G * )) of dual Poisson Lie groups, the global R-matrix is, under an appropriate isomorphism, the cartesian product in Γ 4 of L and the identity bisections in Γ G and Γ G * , where L is as in (1). Section §5 is where the first main result of this paper, Theorem 5.4, appears.…”

Section: Introductionmentioning

confidence: 91%

“…Throughout this paper, by a local Lie groupoid, we will mean a 3-associative local Lie groupoid in the sense of [2,Definition 2.7]. That is, a manifold G equipped with submersions θ, τ : G → Y to a base manifold Y , an embedding ε : Y → G, a multiplication defined on an open neighbourhood G 1) of an open neighbourhood G (−1) ⊂ G of ε(Y ), and satisfying the usual axioms of a Lie groupoid wherever these make sense. A local Poisson groupoid is a pair (G ⇒ Y, π) where G ⇒ Y is a local Lie groupoid, and π a Poisson bivector field on G such that the graph of the multiplication Let (G ⇒ Y, π) be a symplectic groupoid over (Y, π Y ) with source map θ : G → Y , let (X, π X ) be a Poisson manifold with a map µ : X → Y , and let…”

Section: Poisson Actions Of Double Symplectic Groupoidsmentioning

confidence: 99%

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Mouquin¹

2019

Preprint

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Given a standard complex semisimple Poisson Lie group (G, π st ), generalised double Bruhat cells G u,v and generalised Bruhat cells O u equipped with naturally defined holomorphic Poisson structures, where u, v are finite sequences of Weyl group elements, were defined and studied by Jiang-Hua Lu and the author. We prove in this paper that G u,u is naturally a Poisson groupoid over O u , extending a result from the aforementioned authors about double Bruhat cells in (G, π st ).Our result on G u,u is obtained as an application of a construction interesting in its own right, of a local Poisson groupoid over a mixed product Poisson structure associated to the action of a pair of Lie bialgebras. This construction involves using a local Lagrangian bisection in a double symplectic groupoid closely related to the global R-matrix studied by Weinstein and Xu, to "twist" a direct product of Poisson groupoids.

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“…Varieties that appear in the theory of Poisson Lie groups and Poisson homogeneous spaces provide numerous examples of Poisson-CGL extensions, see for example [13] for coordinate rings of Schubert cells and affine charts of Bott-Samelson varieties. The coordinate rings of such varieties are semiclassical limits of quantized algebras of functions and the above axioms are Poisson incarnations of the Levendorskii-Soibelman straightening law for quantum groups [5,Proposition I.6.10].…”

mentioning

confidence: 99%

Cluster algebra structures on Poisson nilpotent algebras

Goodearl,

Yakimov

2018

Preprint

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Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert cells for symmetrizable Kac-Moody groups, affine charts of Bott-Samelson varieties, coordinate rings of double Bruhat cells (in the last case after a localization). We prove that every Poisson nilpotent algebra satisfying a mild condition on certain scalars is canonically isomorphic to a cluster algebra which coincides with the corresponding upper cluster algebra, without additional localizations by frozen variables. The constructed cluster structure is compatible with the Poisson structure in the sense of Gekhtman, Shapiro and Vainshtein. All Poisson nilpotent algebras are proved to be equivariant Poisson Unique Factorization Domains. Their seeds are constructed from sequences of Poisson-prime elements for chains of Poisson UFDs; mutation matrices are effectively determined from linear systems in terms of the underlying Poisson structure. Uniqueness, existence, mutation, and other properties are established for these sequences of Poisson-prime elements. Contents 1. Introduction 2. Poisson algebras 3. Cluster algebras and Poisson cluster algebras 4. Poisson-primes in Poisson-Ore extensions 5. Iterated Poisson-Ore extensions 6. Symmetry and maximal tori for Poisson-CGL extensions 7. One-step mutations in Poisson-CGL extensions 8. Homogeneous Poisson-prime elements for subalgebras of symmetric Poisson-CGL extensions 9. Chains of mutations in symmetric Poisson-CGL extensions 10. Division properties of mutations between Poisson-CGL extension presentations 11. Symmetric Poisson-CGL extensions and cluster algebras References

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Bott-Samelson varieties and Poisson Ore extensions

Cited by 12 publications

References 25 publications

A Bruhat atlas for the Mehta-van der Kallen stratification of $T^* GL_n/B$

A Bruhat atlas for the Mehta-van der Kallen stratification of $T^* GL_n/B$

Local Poisson groupoids over mixed product Poisson structures and generalised double Bruhat cells

Cluster algebra structures on Poisson nilpotent algebras

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