Algebraic structures related to integrable differential equations

Соколов, В. В.

doi:10.21711/217504322017/em311

Cited by 4 publications

(3 citation statements)

References 60 publications

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“…By (6) we have π(q) = π 1 (m − Xq), so one can put r 0 := −3(m − Xq) in order to satisfy (23). Note that π 1 (q)r 0 = −π 1 (r 0 )q = π(r 1 )q = 3π(q)q = 0 by (23). It remains to check (22).…”

Section: First Examplesmentioning

confidence: 99%

“…The resulting linear-quadratic Poisson pencil is related to the so-called elliptic rotator bihamiltonian integrable system ([13]). The coefficients of the tensors π 2 are written by means of elliptic constants.Recently, V. Sokolov introduced a family of quadratic Poisson bivectors π 2 on the space gl(3, C) * which have the same property: π 2 (I) = 0 and the linearization of π 2 at I coincides with the standard Lie-Poisson bivector π 1 on gl(3, C) ∼ = gl * (3, C) [24], [23]. This family is also related to an elliptic curve but on the contrary to the above case the coefficients of π 2 are written explicitly (with no use of elliptic constants).…”

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confidence: 99%

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On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

Panasyuk¹,

Shevchishin²

2019

Preprint

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The paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that there exists a 10-parametric family of quadratic Poisson structures on gl(3) * compatible with the canonical linear Poisson structure and containing the 3-parametric family of quadratic bivectors recently introduced by Vladimir Sokolov. The involutive family of polynomial functions related to the corresponding Poisson pencils contains the hamiltonian of the polynomial form of the elliptic Calogero-Moser system.1 We will use the term "polyvector" instead of "polyvector field" for short, with the exception of vector fields.By linear, quadratic etc. polyvectors on a vector space we mean polyvectors with homogeneous linear, quadratic etc. coefficients with respect to some linear system of coordinates.We will say that a Poisson pencil is linear-quadratic if its generators are linear and quadratic bivectors on some vector space respectively. The cases of "linear-constant" and "linear-linear" Poisson pencils are already very important, see for instance [1], [20] and references therein. The linear-quadratic Poisson pencils are the main objects of this paper.There are basically two classes of linear-quadratic Poisson pencils known in the literature. The first one can be described as follows ( [4], [9], [10]). Let r ∈ g ∧ g be a skew-symmetric solution to the classical Yang-Baxter equation on a Lie algebra g, i.e. an "algebraic Poisson bivector" on g. Assume that a linear representation of g in a vector space V is given and ξ 1 , . . . , ξ n are the fundamental vector fields of this representation corresponding to a basis e 1 , . . . , e n of g. Then, if r = r ij e i ∧ e j , the bivector π 2 := r ij ξ i ∧ ξ j is a quadratic Poisson bivector on V . If, moreover, another bivector π 1 on V is given, invariant under the action of g, the bivectors π 1 and π 2 are compatible. In particular, if V = g * with the canonical Lie-Poisson bivector π 1 , and the representation is the coadjoint one, we get a linear-quadratic Poisson pair (π 1 , π 2 ), where π 2 = r ij π 1 (x i ) ∧ π 1 (x j ) (here x i are the corresponding coordinates on g * ).The second class concerns a specific situation when a quadratic Poisson bivector π 2 is given on a vector space V and there exists a point a ∈ V , a = 0, such that π 2 (a) = 0 (not for every quadratic Poisson bivector such points exist). Then the linearization of π 2 at a is a linear Poisson bivector which is automatically compatible with π 2 .One can find numerous examples of such a situation in the literature. One class of examples is related to Poisson-Lie groups. Given such a group (G, π), its Poisson tensor π necessarily vanishes at the neutral element e ∈ G and there is a correctly defined linearization π 1 of π defined on the linear space T e G (it coincides with the r-matrix bra...

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Section: First Examplesmentioning

confidence: 99%

mentioning

confidence: 99%

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

Panasyuk¹,

Shevchishin²

2019

Preprint

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“…In the associative context, the linearly compatible structures were studied in [28] for matrix algebras and especially for linear deformations. Further a linearly compatible algebra gives rise to a hierarchy of integrable systems of ODEs via the Lenard-Magri scheme [25,31].…”

Section: Introductionmentioning

confidence: 99%

Core Content Selection and Textbook Design for High School Mathematics

Zhang¹,

Zhou²,

Wang³

et al. 2021

School Mathematics Textbooks in China

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This paper is devoted to studying the following fractional L 2 -critical nonlinear Schrödinger equation≥ 0 is an external potential. We obtain normalized L 2 -norm solutions of the above equation by solving the associated constraint minimization problem (1.4). It shows that there is a threshold a * > 0 such that (1.4) has minimizers for 0 < a < a * , and minimizers do not exist for any a > a * . For the case of a = a * , it gives a fact that the existence and non-existence of minimizers depend strongly on the value of V (0). Especially for V (0) = 0, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as a ր a * . Applying implicit function theorem, the uniqueness of minimizers is also proved for a > 0 small enough.

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Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality

Zhang,

Gao,

Guo

2024

Appl Categor Struct

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Algebraic structures related to integrable differential equations

Abstract: Commutative subalgebras in U (gl N +1 ) and quantum Calogero-Moser Hamiltonians .

Cited by 4 publications

References 60 publications

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

On linear-quadratic Poisson pencils on trivial central extensions of semisimple Lie algebras

Core Content Selection and Textbook Design for High School Mathematics

Compatible Structures of Nonsymmetric Operads, Manin Products and Koszul Duality

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