Jordan–Kronecker invariants for semidirect sums defined by standard representation of orthogonal or symplectic Lie algebras

Ворушилов, Константин Сергеевич

doi:10.1134/s1995080217060142

Cited by 3 publications

(6 citation statements)

References 5 publications

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“…Her results, as well as some other examples of computing Jordan-Kronecker invariants, can be found in the arXiv version of [38]. For some semidirect sums, Jordan-Kronecker invariants have been recently computed by K. Vorushilov [224]. To further develop these ideas, it would be interesting to study Poisson pencils generated by one quadratic and one linear brackets.…”

Section: Argument Shift Methods and Jordan-kronecker Invariants Of Liementioning

confidence: 99%

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Open problems, questions and challenges in finite- dimensional integrable systems

Bolsinov

Matveev

Miranda

et al. 2018

Phil. Trans. R. Soc. A.

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The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

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Section: Argument Shift Methods and Jordan-kronecker Invariants Of Liementioning

confidence: 99%

“…Her results, as well as some other examples of computing Jordan-Kronecker invariants, can be found in the arXiv version of [141]. For some semidirect sums, Jordan-Kronecker invariants have been recently computed by Vorushilov [153]. Question 5.7.…”

Section: Problem 53 Find Necessary and Sufficient Conditions For The Bi-lagrangian Grassmannianmentioning

confidence: 99%

Open problems, questions and challenges in finite- dimensional integrable systems

Bolsinov

Matveev

Miranda

et al. 2018

Phil. Trans. R. Soc. A.

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“…• In [29], [30], [31] and [32] the JK invarians are the sizes of blocks. This is not ideal, there is no need to lose the valuable information -which Jordan blocks have the same eigenvalue.…”

Section: Jordan-kronecker Invariants Of Lie Algebrasmentioning

confidence: 99%

“…Apart from [5] and references therein, the Jordan-Kronecker invariants for various Lie algebras were calculated in [29], [30], [31], [32], [10], [11]. Yet the following question, asked in [1], [2] and [4] remains open.…”

Section: Introductionmentioning

confidence: 99%

Jordan–Kronecker Invariants for Lie Algebras of Small Dimensions

Groznova

2023

J Math Sci

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In this paper, Jordan-Kronecker invariants are calculated for all nilpotent 6-and 7-dimensional Lie algebras. We consider the Poisson bracket family, depending on the lambda parameter on a Lie coalgebra, i.e., on the linear space dual to a Lie algebra. For some space g proposed in the paper, two skew-symmetric matrices are defined for all points x on this linear space. To understand the behaviour of the matrix pencil (A − λB)(x), we consider Jordan-Kronecker invariants for this pencil and how they change with x (the latter is done for 6-dimensional Lie algebras).

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“…All matrices Λ k are skew-symmetric and hence belong to so(n). The condition ( 27) is satisfied due to (10). The "terminal" conditions Λ 0 A = 0 and Λ α X = 0 are satisfied since…”

Section: Proof Of Theorem 321mentioning

confidence: 99%

Jordan–kronecker Invariants of Lie Algebra Representations and Degrees of Invariant Polynomials

Bolsinov

Izosimov

Козлов

2021

Transformation Groups

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For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

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Jordan–Kronecker invariants for semidirect sums defined by standard representation of orthogonal or symplectic Lie algebras

Cited by 3 publications

References 5 publications

Open problems, questions and challenges in finite- dimensional integrable systems

Open problems, questions and challenges in finite- dimensional integrable systems

Jordan–Kronecker Invariants for Lie Algebras of Small Dimensions

Jordan–kronecker Invariants of Lie Algebra Representations and Degrees of Invariant Polynomials

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