Eigenvalue coincidences and K-orbits, I

Colarusso, Mark; Evens, Sam

doi:10.1016/j.jalgebra.2014.08.051

Cited by 6 publications

(23 citation statements)

References 31 publications

(56 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Remark 5.12. For g = gl(n) an analogous description of the nilfibre is proven in Proposition 3.10 of [CE15]. The reasoning used here gives a simpler determination of the nilfibre than in [CE15].…”

Section: Qedmentioning

confidence: 77%

“…The purpose of this paper and its sequel is to remedy this situation. We establish the complete integrability of the orthogonal Gelfand-Zeitlin system on regular adjoint orbits in so(n, C) and extend a number of basic results on the strongly regular set to so(n, C), which were established in the general linear case in [KW06a,Col11,CE12,CE15]. In particular, we describe the generic leaves of the foliation given by the integrable system as well as aspects of the geometry of the nilfibre of the moment map of the system.…”

Section: Introductionmentioning

confidence: 82%

“…In [CE15], we defined the set of n-strongly regular elements for g = gl(n) to be the set of elements x ∈ g for which ω g//K (x) = 0. It follows from Theorem 4.5 and Equation (4.5) that our definition in Definition 4.7 is consistent with the previous one and…”

Section: Qedmentioning

confidence: 99%

See 2 more Smart Citations

The Complex Orthogonal Gelfand-Zeitlin System

Colarusso¹,

Evens²

2018

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we use the theory of algebraic groups to prove a number of new and fundamental results about the orthogonal Gelfand-Zeitlin system. We show that the moment map (orthogonal Kostant-Wallach map) is surjective and simplify criteria of Kostant and Wallach for an element to be strongly regular. We further prove the integrability of the orthogonal Gelfand-Zeitlin system on regular adjoint orbits and describe the generic flows of the integrable system. We also study the nilfibre of the moment map and show that in contrast to the general linear case it contains no strongly regular elements. This extends results of Kostant, Wallach, and Colarusso from the general linear case to the orthogonal case.

show abstract

Section: Qedmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 82%

See 1 more Smart Citation

The Complex Orthogonal Gelfand-Zeitlin System

Colarusso¹,

Evens²

2018

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…where {h i } r i=1 is any orthonormal basis of h with respect to , , and for α ∈ ∆ + , e α ∈ g α and e −α ∈ g −α are such that α([e α , e −α ]) = 2. Correspondingly, one has the standard multiplicative holomorphic Poisson bi-vector field π st on G given by (10) π st (g) = l g r st − r g r st = l g Λ st − r g Λ st , g ∈ G,…”

Section: Complete Hamiltonian Flows and Integrable Systems On Generalmentioning

confidence: 99%

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

2018

Lie Groups, Geometry, and Representation Theory

View full text Add to dashboard Cite

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.

show abstract

“…B. Kostant e N. Wallach em [Kostant e Wallach (2006a)] e [Kostant e Wallach (2006b)] afirmam que todas as componentes regulares da variedade de Gelfand-Tsetlin são equidimensionais com dimensão n(n−1) 2 Existe uma função conhecida como "aplicação de Kostant-Wallach" [ver Kostant e Wallach (2006a)] ϕ : M n ( ) −→ n(n+1) 2 , cuja fibra no ponto zero ϕ −1 (0) coincide com a variedade de Gelfand-Tsetlin para gl n . Recentemente, M. Colarusso e S. Evens provaram em [Colarusso e Evens (2015)] que a fibra em zero ϕ −1 2 (0) da função, chamada de aplicação parcial de Kostant-Wallach, ϕ 2 : M n ( ) −→ n −1 × n é equidimensional e nesse caso com dimensão n 2 − 2n + 1. Usando técnicas diferentes, no capítulo quatro, apresentamos uma generalização deste resultado.…”

Section: Versão Fraca Do Teorema De Ovsienkounclassified

Variedades de Gelfand-Tsetlin

Monsalve¹

View full text Add to dashboard Cite

AgradecimentosAgradeço, primeiramente a Deus, por me permitir conquistar este sonho.Muito obrigado a minha familia pelo carinho e apoio recebido durante todos esses anos, mesmo com todas minhas doideras, são eles meus motores e minha inspiração. Ao meu pai Osvaldo Benitez Tapias e a minha mãe Fabiola Monsalve Valencia, porque desde criança me mostraram o valor de estudar. Sou grato a Suki por ser minha parceira e fazer de minhas aventuras nossas aventuras. A meus irmãos Jhon Henry e Fabian pela confiança, a meu sobrinho Thomas por me ensinar a sorrir e brincar, mesmo nos momentos mais difíceis. Também, quero agradecer a Dom Jaime Palacio Escobar, muito querido e estimado pela familia Benitez Monsalve.Gostaria agradecer a meu orientador Vyacheslav Futorny pela paciência, amizade e confiança depositada em mim durante este projeto de pesquisa.

show abstract

Eigenvalue coincidences and K-orbits, I

Cited by 6 publications

References 31 publications

The Complex Orthogonal Gelfand-Zeitlin System

The Complex Orthogonal Gelfand-Zeitlin System

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

Variedades de Gelfand-Tsetlin

Contact Info

Product

Resources

About