Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians

Matushko, M. G.; Соколов, В. В.

doi:10.1134/s004057791704002x

Cited by 5 publications

(8 citation statements)

References 7 publications

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“…The conjectures 3.1 and 3.2 have been verified in [48] for N = 2, 3. Moreover, differential operators with polynomial coefficients that commute with P 2 and with P 3 were found.…”

Section: Observation 31 (Aturbiner) For Many Of These Hamiltonians There Exists a Change Of Variables And A Gauge Transformation That Brimentioning

confidence: 58%

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Algebraic structures related to integrable differential equations

Соколов¹

2017

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“…The conjectures 3.1 and 3.2 have been verified in [48] for N = 2, 3. Moreover, differential operators with polynomial coefficients that commute with P 2 and with P 3 were found.…”

Section: Observation 31 (Aturbiner) For Many Of These Hamiltonians There Exists a Change Of Variables And A Gauge Transformation That Brimentioning

confidence: 58%

“…In the last term we have to substitute − N i=1 y i for y N +1 . In [48] the following transformation (y 1 , . .…”

Section: Observation 31 (Aturbiner) For Many Of These Hamiltonians There Exists a Change Of Variables And A Gauge Transformation That Brimentioning

confidence: 99%

Algebraic structures related to integrable differential equations

Соколов¹

2017

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“…, where X b and Q b satisfy (16). Alternatively X can be of the form from Theorem 6.2, but X b and Q b should be related by [[X b , P ], [X b , P ]] = − 4 3 P (Q b ) ∧ P .…”

Section: First Examplesmentioning

confidence: 99%

“…They depend on the invariants g 2 , g 3 of the elliptic curve, however one can easily check that these numbers can take any values and in fact the Sokolov family determines a 3-parametric family of quadratic Poisson bivectors (the third component corresponds to the free term).The involutive family of polynomial functions canonically associated with the Poisson pair (π 1 , π 2 ) (see Definition 1.2) contains the hamiltonian of the classical elliptic Calogero-Moser system. More precisely, in [16] the hamiltonian operator of the so-called quantum elliptic Calogero-Moser system for 3 particles is brought to a polynomial form. Moreover, a system of commuting elements of the universal enveloping algebra U(gl(3)) is built such that under a specific representation these elements pass to the differential operators: the second order quantum hamiltonian and its quantum integrals of third order.…”

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confidence: 99%

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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“…A polynomial form for the elliptic case have been found in [1] by a deformation of known [2] polynomial form of the trigonometric model. Concerning polynomial forms for the elliptic Calogero-Moser see [3].…”

Section: Introductionmentioning

confidence: 99%

Algebraic quantum Hamiltonians on the plane

Соколов

2015

Theor Math Phys

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ABSTRACT. We consider second order differential operators P with polynomial coefficients that preserve the vector space V k of polynomials of degrees not greater then k. We assume that the metric associated with the symbol of P is flat and that the operator P is potential. In the case of two independent variables we obtain some classification results and find polynomial forms for the elliptic A 2 and G 2 CalogeroMoser Hamiltonians and for the elliptic Inosemtsev model.

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Polynomial forms for quantum elliptic Calogero–Moser Hamiltonians

Cited by 5 publications

References 7 publications

Algebraic structures related to integrable differential equations

Algebraic structures related to integrable differential equations

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

Algebraic quantum Hamiltonians on the plane

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