Alina Dobrogowska

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Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal

2011-04-07

Anatol Odzijewicz , Alina Dobrogowska

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Factorization method applied to the second order difference equations

2017-06-10

Alina Dobrogowska , Grzegorz Jakimowicz

Tangent lifts of bi-Hamiltonian structures

2017-08-01

Alina Dobrogowska , Grzegorz Jakimowicz

We construct two Poisson structures πTM and π̃TM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold … We construct two Poisson structures πTM and π̃TM on the tangent bundle TM to a Poisson manifold (M,π) using Lie algebroid (T*M, qM, M). Next, we construct the Poisson manifold (TM,πTM,λ) from a bi-Hamiltonian manifold (M,π1,π2). This structure can be considered as a deformation of the Poisson structure related to an algebroid structure. We show that the bi-Hamiltonian structure from M can be transferred to the manifold TM. Moreover we present how to construct the Casimir functions for structures πTM, πTM,λ, π̃Tg*, and π̃Tg*,λ from the Casimir functions for π1 and π2 and discuss some particular examples.

Integrable Systems Related to Deformed so(5)

2014-06-03

Alina Dobrogowska

We investigate a family of integrable Hamiltonian systems on Lie-Poisson spaces L + (5) dual to Lie algebras so λ,α (5) being two-parameter deformations of so(5).We integrate corresponding Hamiltonian equations … We investigate a family of integrable Hamiltonian systems on Lie-Poisson spaces L + (5) dual to Lie algebras so λ,α (5) being two-parameter deformations of so(5).We integrate corresponding Hamiltonian equations on L + (5) and T * R 5 by quadratures as well as discuss their possible physical interpretation.

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Factorization Method for (q, h)-Hahn Orthogonal Polynomials

2015-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization Method and General Second Order Linear Difference Equation

2018-01-01

Alina Dobrogowska , Mahouton Norbert Hounkonnou

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Lie bundle on the space of deformed skew-symmetric matrices

2014-11-01

Alina Dobrogowska , Tomasz Goliński

We study a Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {A}_{a_1,\ldots ,a_{n-1}}$\end{document}Aa1,...,an−1 of deformed skew-symmetric n × n matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations … We study a Lie algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {A}_{a_1,\ldots ,a_{n-1}}$\end{document}Aa1,...,an−1 of deformed skew-symmetric n × n matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real numbers a1, …, an − 1. Using isomorphism \documentclass[12pt]{minimal}\begin{document}$(\mathcal {A}_{a_1,\ldots ,a_{n-1}})^*\cong L_+$\end{document}(Aa1,...,an−1)*≅L+ we introduce a Lie–Poisson structure on the space of upper-triangular matrices L+. In this way, we generate hierarchies of Hamiltonian systems with bihamiltonian structure.

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Integrable Systems of Neumann Type

2013-08-28

Alina Dobrogowska , Tudor S. Raţiu

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The factorization of the (q,h)-difference operators

2007-11-16

Alina Dobrogowska , Klara Janglajew

We prove that a linear (q, h)-difference operator with leading coefficient equal to one admits a factorization into first order linear (q, h)-difference operators. Some applications and limiting cases are … We prove that a linear (q, h)-difference operator with leading coefficient equal to one admits a factorization into first order linear (q, h)-difference operators. Some applications and limiting cases are also presented.

Change of Variables in Factorization Method for Second-order Functional Equations

2004-11-01

Alina Dobrogowska , Tomasz Goliński , Anatol Odzijewicz

R-matrix, Lax pair, and multiparameter decompositions of Lie algebras

2015-11-01

Alina Dobrogowska

We construct R operators on the Lie algebra 𝔤𝔩(n, ℝ) or more generally Hilbert–Schmidt operators L2 in Hilbert space. These operators are related to a multiparameter deformation given by a … We construct R operators on the Lie algebra 𝔤𝔩(n, ℝ) or more generally Hilbert–Schmidt operators L2 in Hilbert space. These operators are related to a multiparameter deformation given by a sequence of parameters α = {a1, a2, …}. We determine for which choices of parameters R operators are R-matrices. We also construct the Lax pair for the corresponding Hamilton equations.

Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling

2012-11-01

Alina Dobrogowska , Anatol Odzijewicz

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Generalization of the concept of classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3442" altimg="si7.svg"><mml:mi>r</mml:mi></mml:math>-matrix to Lie algebroids

2021-03-26

Alina Dobrogowska , Grzegorz Jakimowicz

We present some new constructions of Lie algebroids starting from vector fields on manifold M. The tangent bundle TM possess a natural structure of Lie algebroid, but we use these … We present some new constructions of Lie algebroids starting from vector fields on manifold M. The tangent bundle TM possess a natural structure of Lie algebroid, but we use these fields to construct a collection of interesting new algebroid structures. Next, we show that these constructions can be used in a more general situation, starting from an arbitrary Lie algebroid over M. In the final step, we show that after limiting ourselves to Lie algebras these formulas as a special case contain brackets well known in theory of classical r-matrices. We can think of our constructions as extending the concept of classical r-matrices to Lie algebroids. Several examples illustrate the importance of these constructions.

A new look at Lie algebras

2023-08-09

Alina Dobrogowska , Grzegorz Jakimowicz

We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. … We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair (F,v) consisting of a linear mapping F∈End(V) with an eigenvector v. This pair allows to build a Lie bracket on a dual space to a linear space V. The Lie algebra obtained in this way is solvable. In particular, when F is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie–Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.

Eigenvalue problem versus Casimir functions for Lie algebras

2024-03-25

Alina Dobrogowska , Marzena Szajewska

Factorization method applied to the second order q-difference operators

2013-12-15

Alina Dobrogowska , Grzegorz Jakimowicz

On the (q; h)-discretization of ladder operators

2014-01-01

Stefan Hilger , Galina Filipuk , R. A. Kycia +1 more

In this paper we shall study the problem of the (q, h)-discretization (generalization) of ladder operators. We shall present a few illustrative and important examples including the Weber, Bessel and … In this paper we shall study the problem of the (q, h)-discretization (generalization) of ladder operators. We shall present a few illustrative and important examples including the Weber, Bessel and Laguerre ladders and their (q, h)-analogues.

Darboux transformations and second order difference equations

2018-01-01

Alina Dobrogowska , David J. Fernández C.

In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schrödinger equation. The technique is based on the use … In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schrödinger equation. The technique is based on the use of first order operators intertwining two difference operators of second order. This method, which has been applied successfully for differential cases, leads also to interesting non trivial results in the discrete case. The technique allows us to construct the solutions for a wide class of difference Schrödinger equations. The exact solutions for some special potentials are also found explicitly.

Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Marzena Szajewska +1 more

The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras. The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras.

Second order q-difference equations solvable by factorization method

2003-01-01

Alina Dobrogowska , Anatol Odzijewicz

By solving an infinite nonlinear system of $q$-difference equations one constructs a chain of $q$-difference operators. The eigenproblems for the chain are solved and some applications, including the one related … By solving an infinite nonlinear system of $q$-difference equations one constructs a chain of $q$-difference operators. The eigenproblems for the chain are solved and some applications, including the one related to $q$-Hahn orthogonal polynomials, are discussed. It is shown that in the limit q->1 the present method corresponds to the one developed by Infeld and Hull.

Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

2021-08-09

Alina Dobrogowska , Karolina Wojciechowicz

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of … We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

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Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor

2018-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

In this paper we prove that both complete and vertical lifts of a Poisson vector field from a Poisson manifold $(M, \pi)$ to its tangent bundle $(TM, \pi_{TM})$ are also … In this paper we prove that both complete and vertical lifts of a Poisson vector field from a Poisson manifold $(M, \pi)$ to its tangent bundle $(TM, \pi_{TM})$ are also Poisson. We use this fact to describe the infinitesimal deformations of Poisson tensor $\pi_{TM}$. We study some of their properties and present a extensive set of examples in a low dimensional case.

On Some Structures of Lie Algebroids on the Cotangent Bundles

2023-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Marzena Szajewska

Deformation of algebroid bracket of differential forms and Poisson manifold

2018-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,π)$ starting from an algebroid bracket of differential forms. We use these brackets … We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,π)$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is equipped with a bi-Hamiltonian structure $(M,π_1, π_2)$ we show how to construct another family of Poisson structures. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.

Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor

2018-07-05

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

New classes of second order difference equations solvable by factorization method

2019-06-27

Alina Dobrogowska

Bi-Hamiltonian Structures on the Tangent Bundle to a Poisson Manifold

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

In the case when M is equipped with a bi-Hamiltonian structure (M, π 1 , π 2 ) we show how to construct family of Poisson structures on the tangent … In the case when M is equipped with a bi-Hamiltonian structure (M, π 1 , π 2 ) we show how to construct family of Poisson structures on the tangent bundle T M to a Poisson manifold.Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.

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On some deformations of the Poisson structure associated with the algebroid bracket of differential forms

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

A new look at Lie algebras

2023-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and … We present a new look at description of real finite-dimensional Lie algebras. The basic element turns out to be a pair $(F,v)$ consisting of a linear mapping $F\in End(V)$ and its eigenvector $v$. This pair allows to build a Lie bracket on a dual space to a linear space $V$. This algebra is solvable. In particular, when $F$ is nilpotent, the Lie algebra is also nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. %Which allows, having a collection of pairs $(F_i,v_i)$, $i=1, \dots, n$, to construct any Lie algebra. Using relations between the Lie algebra, the Lie--Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has a geometric sense. Several examples illustrate the importance of these constructions.

Cyclic Lie-Rinehart algebras

2024-02-16

Daniel Beltiţă , Alina Dobrogowska , Grzegorz Jakimowicz

We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to … We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is a cyclic submodule of the derivation module, and therefore we call them cyclic Lie-Rinehart algebras. In a very special case of our construction, these brackets turn out to be related to certain differential operators that occur in mathematical physics.

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Lie Algebras, Eigenvalue Problems and Left-Symmetric Algebras

2024-01-01

Alina Dobrogowska , Karolina Wojciechowicz

Integrable Hamiltonian systems generated by antisymmetric matrices

2013-11-29

Alina Dobrogowska

We construct a family of integrable systems generated by the Casimir functions of Lie algebra of skew-symmetric matrices, where the Lie bracket is deformed by a symmetric matrix. We construct a family of integrable systems generated by the Casimir functions of Lie algebra of skew-symmetric matrices, where the Lie bracket is deformed by a symmetric matrix.

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Symplectic dual pair related to soa1,...,an-1(n)

2014-11-26

Alina Dobrogowska , Grzegorz Jakimowicz

We investigate a symplectic dual pair related to a Lie algebra soa1,...,an-1 (n) of deformed skew-symmetric n × n matrices. The deformations are parametrized by a sequence of real numbers … We investigate a symplectic dual pair related to a Lie algebra soa1,...,an-1 (n) of deformed skew-symmetric n × n matrices. The deformations are parametrized by a sequence of real numbers a1,...,an-1.

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Examples of Hamiltonian Systems on the Space of Deformed Skew-symmetric Matrices

2015-01-01

Alina Dobrogowska , Tomasz Goliński

Symplectic Dual Pair Related to Deformed SO(5)

2014-01-01

Alina Dobrogowska , Anatol Odzijewicz

Erratum to: Integrable Systems of Neumann Type

2016-06-22

Alina Dobrogowska , Tudor S. Raţiu

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Factorization method and second order q– and (q, h)–difference equations

2017-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

We investigate a version of factorization method on a sequence of Hilbert spaces ℋk related to q– and (q, h)–ladder. We present second order q– and (q, h)–difference operators Hk … We investigate a version of factorization method on a sequence of Hilbert spaces ℋk related to q– and (q, h)–ladder. We present second order q– and (q, h)–difference operators Hk which can be factorized using first order operators Ak, A*k. As an examples we present (q, h)–Hermite II polynomials and q–deformed Schrödinger equation, where q is a root of unity.

Factorization method and general second order linear difference equation

2017-09-21

Alina Dobrogowska , Mahouton Norbert Hounkonnou

This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be … This paper addresses an investigation on a factorization method for difference equations. It is proved that some classes of second order linear difference operators, acting in Hilbert spaces, can be factorized using a pair of mutually adjoint first order difference operators. These classes encompass equations of hypergeometic type describing classical orthogonal polynomials of a discrete variable.

Discrete Quantum Harmonic Oscillator

2019-11-03

Alina Dobrogowska , David J. Fernández C.

In this paper, we propose a discrete model for the quantum harmonic oscillator. The eigenfunctions and eigenvalues for the corresponding Schrödinger equation are obtained through the factorization method. It is … In this paper, we propose a discrete model for the quantum harmonic oscillator. The eigenfunctions and eigenvalues for the corresponding Schrödinger equation are obtained through the factorization method. It is shown that this problem is also connected with the equation for Meixner polynomials.

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Periodic One-Point Rank One Commuting Difference Operators

2020-01-01

Alina Dobrogowska , Andrey E. Mironov

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Periodic one-point rank one commuting difference operators

2019-01-01

Alina Dobrogowska , Andrey E. Mironov

In this paper we study one-point rank one commutative rings of difference operators. We find conditions on spectral data which specify such operators with periodic coefficients. In this paper we study one-point rank one commutative rings of difference operators. We find conditions on spectral data which specify such operators with periodic coefficients.

Factorization method and general second order linear difference equation

2017-01-01

Alina Dobrogowska , Mahouton Norbert Hounkonnou

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Geometric Methods in Physics XXXIX

2023-01-01

Piotr Kielanowski , Alina Dobrogowska , Gerald A. Goldin +1 more

Lie Algebroid Structures on Cotangent Bundles Determined by Vector Fields

2025-01-01

Alina Dobrogowska

Lie algebras and eigenvalue problems

2025-03-01

Alina Dobrogowska

Abstract It was recently demonstrated that such classical concepts as eigenproblems and Lie algebras are closely related. A family of appropriate eigenproblems generates any finite-dimensional Lie algebra and, conversely, a … Abstract It was recently demonstrated that such classical concepts as eigenproblems and Lie algebras are closely related. A family of appropriate eigenproblems generates any finite-dimensional Lie algebra and, conversely, a Lie algebra generates a set of pairs ( F, v ), where F is a linear mapping and v its eigenvector. This point of view can be used to describe the invariants of Lie algebras. The application of this procedure is shown in detail on the example of five-dimensional nilpotent Lie algebras.

Construction of Low-Dimensional Lie Algebras versus Eigenproblem

2025-05-02

Alina Dobrogowska , Marzena Szajewska

Construction of Low-Dimensional Lie Algebras versus Eigenproblem

2025-05-02

Alina Dobrogowska , Marzena Szajewska

Lie algebras and eigenvalue problems

2025-03-01

Alina Dobrogowska

Lie Algebroid Structures on Cotangent Bundles Determined by Vector Fields

2025-01-01

Alina Dobrogowska

Eigenvalue problem versus Casimir functions for Lie algebras

2024-03-25

Alina Dobrogowska , Marzena Szajewska

Cyclic Lie-Rinehart algebras

2024-02-16

Daniel Beltiţă , Alina Dobrogowska , Grzegorz Jakimowicz

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Lie Algebras, Eigenvalue Problems and Left-Symmetric Algebras

2024-01-01

Alina Dobrogowska , Karolina Wojciechowicz

A new look at Lie algebras

2023-08-09

Alina Dobrogowska , Grzegorz Jakimowicz

A new look at Lie algebras

2023-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

Geometric Methods in Physics XXXIX

2023-01-01

Piotr Kielanowski , Alina Dobrogowska , Gerald A. Goldin +1 more

On Some Structures of Lie Algebroids on the Cotangent Bundles

2023-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Marzena Szajewska

Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

2021-08-09

Alina Dobrogowska , Karolina Wojciechowicz

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Generalization of the concept of classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3442" altimg="si7.svg"><mml:mi>r</mml:mi></mml:math>-matrix to Lie algebroids

2021-03-26

Alina Dobrogowska , Grzegorz Jakimowicz

Periodic One-Point Rank One Commuting Difference Operators

2020-01-01

Alina Dobrogowska , Andrey E. Mironov

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Discrete Quantum Harmonic Oscillator

2019-11-03

Alina Dobrogowska , David J. Fernández C.

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New classes of second order difference equations solvable by factorization method

2019-06-27

Alina Dobrogowska

Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Marzena Szajewska +1 more

Bi-Hamiltonian Structures on the Tangent Bundle to a Poisson Manifold

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

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On some deformations of the Poisson structure associated with the algebroid bracket of differential forms

2019-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

Periodic one-point rank one commuting difference operators

2019-01-01

Alina Dobrogowska , Andrey E. Mironov

Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor

2018-07-05

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

Deformation of algebroid bracket of differential forms and Poisson manifold

2018-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

Darboux transformations and second order difference equations

2018-01-01

Alina Dobrogowska , David J. Fernández C.

Factorization Method and General Second Order Linear Difference Equation

2018-01-01

Alina Dobrogowska , Mahouton Norbert Hounkonnou

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Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor

2018-01-01

Alina Dobrogowska , Grzegorz Jakimowicz , Karolina Wojciechowicz

Factorization method and general second order linear difference equation

2017-09-21

Alina Dobrogowska , Mahouton Norbert Hounkonnou

Tangent lifts of bi-Hamiltonian structures

2017-08-01

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization method applied to the second order difference equations

2017-06-10

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization method and second order q– and (q, h)–difference equations

2017-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization method and general second order linear difference equation

2017-01-01

Alina Dobrogowska , Mahouton Norbert Hounkonnou

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Erratum to: Integrable Systems of Neumann Type

2016-06-22

Alina Dobrogowska , Tudor S. Raţiu

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R-matrix, Lax pair, and multiparameter decompositions of Lie algebras

2015-11-01

Alina Dobrogowska

Factorization Method for (q, h)-Hahn Orthogonal Polynomials

2015-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

Examples of Hamiltonian Systems on the Space of Deformed Skew-symmetric Matrices

2015-01-01

Alina Dobrogowska , Tomasz Goliński

Symplectic dual pair related to soa1,...,an-1(n)

2014-11-26

Alina Dobrogowska , Grzegorz Jakimowicz

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Lie bundle on the space of deformed skew-symmetric matrices

2014-11-01

Alina Dobrogowska , Tomasz Goliński

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Integrable Systems Related to Deformed so(5)

2014-06-03

Alina Dobrogowska

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On the (q; h)-discretization of ladder operators

2014-01-01

Stefan Hilger , Galina Filipuk , R. A. Kycia +1 more

Symplectic Dual Pair Related to Deformed SO(5)

2014-01-01

Alina Dobrogowska , Anatol Odzijewicz

Factorization method applied to the second order q-difference operators

2013-12-15

Alina Dobrogowska , Grzegorz Jakimowicz

Integrable Hamiltonian systems generated by antisymmetric matrices

2013-11-29

Alina Dobrogowska

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Integrable Systems of Neumann Type

2013-08-28

Alina Dobrogowska , Tudor S. Raţiu

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Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling

2012-11-01

Alina Dobrogowska , Anatol Odzijewicz

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Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal

2011-04-07

Anatol Odzijewicz , Alina Dobrogowska

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The factorization of the (q,h)-difference operators

2007-11-16

Alina Dobrogowska , Klara Janglajew

Change of Variables in Factorization Method for Second-order Functional Equations

2004-11-01

Alina Dobrogowska , Tomasz Goliński , Anatol Odzijewicz

Second order q-difference equations solvable by factorization method

2003-01-01

Alina Dobrogowska , Anatol Odzijewicz

Coauthor	Papers Together
Grzegorz Jakimowicz	17
Karolina Wojciechowicz	7
Anatol Odzijewicz	5
Tomasz Goliński	4
Marzena Szajewska	4
Mahouton Norbert Hounkonnou	3
Andrey E. Mironov	2
David J. Fernández C.	2
Tudor S. Raţiu	2
Karolina Wojciechowicz	1
Klara Janglajew	1
Piotr Kielanowski	1
R. A. Kycia	1
Gerald A. Goldin	1
Daniel Beltiţă	1
Stefan Hilger	1
Galina Filipuk	1

Integrable Hamiltonian systems related to the Hilbert–Schmidt ideal

2011-04-07

Anatol Odzijewicz , Alina Dobrogowska

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Tangent lifts of Poisson and related structures

1995-12-07

Janusz Grabowski , Paweł Urbański

The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector … The derivation dT on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with application to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

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A simple model of the integrable Hamiltonian equation

1978-05-01

Franco Magri

A method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Bäcklund transformation or the use of the Lax equation is suggested. This analysis is based … A method of analysis of the infinite-dimensional Hamiltonian equations which avoids the introduction of the Bäcklund transformation or the use of the Lax equation is suggested. This analysis is based on the possibility of connecting in several ways the conservation laws of special Hamiltonian equations with their symmetries by using symplectic operators. It leads to a simple and sufficiently general model of integrable Hamiltonian equation, of which the Korteweg–de Vries equation, the modified Korteweg–de Vries equation, the nonlinear Schrödinger equation and the so-called Harry Dym equation turn out to be particular examples.

Invariants of real low dimension Lie algebras

1976-06-01

J. Patera , R. T. Sharp , P. Winternitz +1 more

All invariant functions of the group generators (generalized Casimir operators) are found for all real algebras of dimension up to five and for all real nilpotent algebras of dimension six. All invariant functions of the group generators (generalized Casimir operators) are found for all real algebras of dimension up to five and for all real nilpotent algebras of dimension six.

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Tangent Lie algebroids

1994-07-07

Theodore James Courant

This paper shows that a Lie algebroid structure on a smooth vector bundle A to ( pi over)Q gives rise to a Lie algebroid structure on the bundle TA to … This paper shows that a Lie algebroid structure on a smooth vector bundle A to ( pi over)Q gives rise to a Lie algebroid structure on the bundle TA to (T pi over)TQ, called the tangent Lie algebroid. The analysis uses global arguments. A Lie algebroid A is equivalent to a certain Poisson structure on A*, and the tangent bundle of any Poisson manifold has a tangent Poisson structure. The tangent Poisson structure on TA* is then dualized to produce the tangent Lie algebroid structure on TA. Local calculations are used, and formulae for local brackets are given.

Tangent lifts of bi-Hamiltonian structures

2017-08-01

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization method for second order functional equations

2004-09-14

Tomasz Goliński , Anatol Odzijewicz

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Symplectic groupoids and Poisson manifolds

1987-01-01

Alan Weinstein

A symplectic groupoid is a manifold T with a partially defined multiplication (satisfying certain axioms) and a compatible symplectic structure.The identity elements in T turn out to form a Poisson … A symplectic groupoid is a manifold T with a partially defined multiplication (satisfying certain axioms) and a compatible symplectic structure.The identity elements in T turn out to form a Poisson manifold To? and the correspondence between symplectic groupoids and Poisson manifolds is a natural extension of the one between Lie groups and Lie algebras.As with Lie groups, under certain (simple) connectivity assumptions, every homomorphism of symplectic groupoids is determined by its underlying Poisson mapping, and every Poisson mapping may be integrated to a canonical relation between symplectic groupoids.On the other hand, not every Poisson manifold arises from a symplectic groupoid, at least if we restrict our attention to ordinary manifolds (even non-Hausdorff ones), so "Lie's third fundamental theorem" does not apply in this context.Using the notion of symplectic groupoid, we can answer many of the questions raised by Karasev and Maslov [9,10] about "universal enveloping algebras" for quasiclassical approximations to nonlinear commutation relations.(I wish to acknowledge here that [9] already contains implicitly some of the ideas concerning Poisson structures and their symplectic realizations which were presented in [18].)In fact, the reading of Karasev and Maslov's papers was one of the main stimuli for the work described here.Following their reasoning, it seems that a suitably developed "quantization theory" for symplectic groupoids should provide a tool for studying nonlinear commutation relations which is analogous to the use of topology and analysis on global Lie groups in the study of linear commutation relations.Such a theory would also clarify the relation, mostly an analogy at present, between symplectic groupoids, star products [2], and the operator algebras of noncommutative differential geometry [3].More immediately, the notion of symplectic groupoid unifies many constructions in symplectic and Poisson geometry; in particular, it provides a framework for studying the collection of all symplectic realizations of a given Poisson manifold.A detailed exposition of these results will appear in [4].Many of the details were worked out during a visit to the Université Claude-Bernard Lyon I.I would like to thank Pierre Dazord for his hospitality in Lyon, as well as for many stimulating discussions.The idea of introducing groupoids into symplectic geometry arose in the course of conversations with Marc Rieffel about operator algebras and the subsequent reading of J. Renault's thesis m

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The finite difference algorithm for higher order supersymmetry

2000-05-01

Bogdan Mielnik , L. M. Nieto , Óscar Rosas-Ortiz

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Classical Orthogonal Polynomials of a Discrete Variable

1991-01-01

Arnold F. Nikiforov , Vasilii B. Uvarov , Sergeĭ K. Suslov

Quasidiagonality of $$C^*$$ C ∗ -Algebras of Solvable Lie Groups

2018-02-01

Ingrid Beltiţă , Daniel Beltiţă

Completely integrable bi-Hamiltonian systems

1994-01-01

Rui Loja Fernandes

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Linear bundles of Lie algebras and their applications

2000-11-01

A.B. Yanovski

We consider the algebraic properties of families of Lie brackets, also called linear bundles of Lie algebras, and especially ones that can be defined over the classical algebras. Some applications … We consider the algebraic properties of families of Lie brackets, also called linear bundles of Lie algebras, and especially ones that can be defined over the classical algebras. Some applications to the theory of the integrable systems are given as such bundles naturally define compatible Poisson–Lie (Kirillov) tensors.

Integrable Systems Related to Deformed so(5)

2014-06-03

Alina Dobrogowska

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Generalization of the concept of classical <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e3442" altimg="si7.svg"><mml:mi>r</mml:mi></mml:math>-matrix to Lie algebroids

2021-03-26

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization: little or great algorithm?

2004-10-15

Bogdan Mielnik , Óscar Rosas-Ortiz

The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial … The progress of the factorization method since the 1935 work of Dirac is briefly reviewed. Though linked with older mathematical theories the factorization seems an autonomous 'driving force', offering substantial support to the present day Darboux and Backlund approaches.

Differential calculus on a Lie algebroid and Poisson manifolds

2008-01-01

Charles-Michel Marle

A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very … A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds.

New hydrogen-like potentials

1984-07-01

David J. Fernández C.

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Integrable relativistic systems given by Hamiltonians with momentum-spin-orbit coupling

2012-11-01

Alina Dobrogowska , Anatol Odzijewicz

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Factorization method applied to the second order difference equations

2017-06-10

Alina Dobrogowska , Grzegorz Jakimowicz

On the euler equation: Bi-Hamiltonian structure and integrals in involution

1996-06-01

Carlo Morosi , Livio Pizzocchero

Factorization of (q, h)-difference operators – an algebraic approach

2014-04-14

Stefan Hilger , Galina Filipuk

In this paper we introduce -deformation of Weyl algebra and study the ladders in this algebra, which give the factorization of certain q- and h-difference operators of second order. In this paper we introduce -deformation of Weyl algebra and study the ladders in this algebra, which give the factorization of certain q- and h-difference operators of second order.

Vertical and horizontal lifts of multivector fields and applications

2017-01-01

A. Mba , P. M. Kouotchop Wamba , Romain Pefoukeu Nimpa

Commutation Methods for Jacobi Operators

1996-06-01

Fritz Gesztesy , Gerald Teschl

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Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra

2004-12-09

R. Álvarez-Nodarse , Natig M. Atakishiyev , Roberto S. Costas-Santos

We argue that one can factorize the difference equation of hypergeometric type on the nonuniform lattices in general case. It is shown that in the most cases of q-linear spectrum … We argue that one can factorize the difference equation of hypergeometric type on the nonuniform lattices in general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues this directly leads to the dynamical symmetry algebra $su_q(1,1)$, whose generators are explicitly constructed in terms of the difference operators, obtained in the process of factorization. Thus all models with the $q$-linear spectrum (some of them, but not all, previously considered in a number of publications) can be treated in a unified form.

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Integrable Systems of Neumann Type

2013-08-28

Alina Dobrogowska , Tudor S. Raţiu

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Factorization Method for (q, h)-Hahn Orthogonal Polynomials

2015-01-01

Alina Dobrogowska , Grzegorz Jakimowicz

Lie bundle on the space of deformed skew-symmetric matrices

2014-11-01

Alina Dobrogowska , Tomasz Goliński

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Classification of Solvable Lie Algebras

2005-01-01

Willem A. de Graaf

In this paper we describe a simple method for obtaining a classification of small-dimensional solvable Lie algebras. Using this method, we obtain the classification of three- and fourdimensional solvable Lie … In this paper we describe a simple method for obtaining a classification of small-dimensional solvable Lie algebras. Using this method, we obtain the classification of three- and fourdimensional solvable Lie algebras (over fields of any characteristic). Precise conditions for isomorphism are given.

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The factorization method for the general second-orderq-difference equation and the Laguerre Hahn polynomials on the generalq-lattice

2003-01-08

Gaspard Bangerezako , Mahouton Norbert Hounkonnou

For the general linear second-order q-difference equation, we show the interconnection between the factorization method and the Laguerre–Hahn polynomials on the general q-lattice. Applications are then given in the cases … For the general linear second-order q-difference equation, we show the interconnection between the factorization method and the Laguerre–Hahn polynomials on the general q-lattice. Applications are then given in the cases of the hypergeometric and Askey–Wilson second-order q-difference equations.

On the differential operators of first order in tensor calculus

1953-01-01

J. A. Schouten

Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey

2008-01-16

Yvette Kosmann–Schwarzbach

After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, … After a brief summary of the main properties of Poisson manifolds and Lie algebroids in general, we survey recent work on the modular classes of Poisson and twisted Poisson manifolds, of Lie algebroids with a Poisson or twisted Poisson structure, and of Poisson-Nijenhuis manifolds.A review of the spinor approach to the modular class concludes the paper.

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Hierarchy of integrable Hamiltonians describing the nonlinearn-wave interaction

2012-01-04

Anatol Odzijewicz , Tomasz Goliński

In the paper we construct an hierarchy of integrable Hamiltonian systems which describe the variation of n-wave envelopes in nonlinear dielectric medium. The exact solutions for some special Hamiltonians are … In the paper we construct an hierarchy of integrable Hamiltonian systems which describe the variation of n-wave envelopes in nonlinear dielectric medium. The exact solutions for some special Hamiltonians are given in terms of elliptic functions of the first kind.

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Compatible Brackets in Hamiltonian Mechanics

1993-01-01

H. P. McKean

ANALOGUES OF THE OBJECTS OF LIE GROUP THEORY FOR NONLINEAR POISSON BRACKETS

1987-06-30

M. V. Karasev

For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation. A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, … For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation. A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, and analogues are obtained for the three classical theorems of Lie. The problem of constructing global pseudogroups is examined.

A new look at Lie algebras

2023-08-09

Alina Dobrogowska , Grzegorz Jakimowicz

Factorization of the hypergeometric-type difference equation on the uniform lattice

2007-01-01

Natig M. Atakishiyev , R. Álvarez-Nodarse , Roberto S. Costas-Santos

Jacobi-type identities for bilinear differential concomitants of certain tensor fields. II

1955-01-01

Albert Nijenhuis

On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-devries type equations

1978-10-01

Mark Adler

Lie Theory and Special Functions

1969-07-01

Willard Miller , H. S. Valk

The Category of Ladders

2010-04-13

Stefan Hilger

The quasi-bi-Hamiltonian formulation of the Lagrange top

2002-02-11

Carlo Morosi , Giorgio Tondo

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the possible reductions of the Poisson tensors, the vector field and its Hamiltonian functions … Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the possible reductions of the Poisson tensors, the vector field and its Hamiltonian functions on a four-dimensional space. We show that the vector field of the Lagrange top possesses, on the reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set of separation variables for the corresponding Hamilton-Jacobi equation.

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The local structure of Poisson manifolds

1983-01-01

Alan Weinstein

Varietes de Poisson et applications. Decomposition. Structures de Poisson lineaires. Approximation lineaire. Systemes hamiltoniens. Le probleme de linearisation. Groupes de fonction, realisations et applications impulsion. Paires duales et groupes de … Varietes de Poisson et applications. Decomposition. Structures de Poisson lineaires. Approximation lineaire. Systemes hamiltoniens. Le probleme de linearisation. Groupes de fonction, realisations et applications impulsion. Paires duales et groupes de jauge. Existence des realisations. Unicite des realisations. Les problemes des 3 corps restreints et autres exemples

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Poisson Structures and Their Normal Forms

2005-01-01

Jean-Paul Dufour , Nguyen Tien Zung , H. E. Bass +2 more

On the Contraction of Groups and Their Representations

1953-06-01

Erdal İnönü , E. P. Wigner

Classical mechanics is a limiting case of relativistic mechanics. Hence the group of the former, the Galilei group, must be in some sense a limiting case of the relativistic mechanics’ … Classical mechanics is a limiting case of relativistic mechanics. Hence the group of the former, the Galilei group, must be in some sense a limiting case of the relativistic mechanics’ group, the representations of the former must be limiting cases of the latter’s representations. There are other examples for similar relations between groups. Thus, the inhomogeneous Lorentz group must be, in the same sense, a limiting case of the de Sitter groups. The purpose of the present note is to investigate, in some generality, in which sense groups can be limiting cases of other groups (Section I), and how their representations can be obtained from the representations of the groups of which they appear as limits (Section II). Section III deals briefly with the transition from inhomogeneous Lorentz group to Galilei group. It shows in which way the representation up to a factor of the Galilei group, embodied in the Schrodinger equation, appears as a limit of a representation of the inhomogeneous Lorentz group and also gives the reason why no physical interpretation is possible for the real representations of that group.

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A Class of Integrable Flows on the Space of Symmetric Matrices

2009-07-07

Anthony M. Bloch , Vasile Brînzănescu , Arieh Iserles +2 more

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Integrable Systems of Classical Mechanics and Lie Algebras

1990-01-01

A. M. Perelomov

1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The Moment Map.- 1.6 Hamiltonian … 1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The Moment Map.- 1.6 Hamiltonian Systems with Symmetry.- 1.7 Reduction of Hamiltonian Systems with Symmetry.- 1.8 Integrable Hamiltonian Systems.- 1.9 The Projection Method.- 1.10 The Isospectral Deformation Method.- 1.11 Hamiltonian Systems on Coadjoint Orbits of Lie Groups.- 1.12 Constructions of Hamiltonian Systems with Large Families of Integrals of Motion.- 1.13 Completeness of Involutive Systems.- 1.14 Hamiltonian Systems and Algebraic Curves.- 2. Simplest Systems.- 2.1 Systems with One Degree of Freedom.- 2.2 Systems with Two Degrees of Freedom.- 2.3 Separation of Variables.- 2.4 Systems with Quadratic Integrals of Motion.- 2.5 Motion in a Central Field.- 2.6 Systems with Closed Trajectories.- 2.7 The Harmonic Oscillator.- 2.8 The Kepler Problem.- 2.9 Motion in Coupled Newtonian and Homogeneous Fields.- 2.10 Motion in the Field of Two Newtonian Centers.- 3. Many-Body Systems.- 3.1 Lax Representation for Many-Body Systems.- 3.2 Completely Integrable Many-Body Systems.- 3.3 Explicit Integration of the Equations of Motion for Systems of Type I and V via the Projection Method.- 3.4 Relationship Between the Solutions of the Equations of Motion for Systems of Type I and V.- 3.5 Explicit Integration of the Equations of Motion for Systems of Type II and III.- 3.6 Integration of the Equations of Motion for Systems with Two Types of Particles.- 3.7 Many-Body Systems as Reduced Systems.- 3.8 Generalizations of Many-Body Systems of Type I-III to the Case of the Root Systems of Simple Lie Algebras.- 3.9 Complete Integrability of the Systems of Section 3.8.- 3.10 Anisotropic Harmonic Oscillator in the Field of a Quartic Central Potential (the Garnier System).- 3.11 A Family of Integrable Quartic Potentials Related to Symmetric Spaces.- 4. The Toda Lattice.- 4.1 The Ordinary Toda Lattice. Lax Representation. Complete Integrability.- 4.2 The Toda Lattice as a Dynamical System on a Coadjoint Orbit of the Group of Triangular Matrices.- 4.3 Explicit Integration of the Equations of Motion for the Ordinary Nonperiodic Toda Lattice.- 4.4 The Toda Lattice as a Reduced System.- 4.5 Generalized Nonperiodic Toda Lattices Related to Simple Lie Algebras.- 4.6 Toda-like Systems on Coadjoint Orbits of Borel Subgroups.- 4.7 Canonical Coordinates for Systems of Toda Type.- 4.8 Integrability of Toda-like Systems on Generic Orbits.- 5. Miscellanea.- 5.1 Equilibrium Configurations and Small Oscillations of Some Integrable Hamiltonian Systems.- 5.2 Motion of the Poles of Solutions of Nonlinear Evolution Equations and Related Many-Body Problems.- 5.3 Motion of the Zeros of Solutions of Linear Evolution Equations and Related Many-Body Problems.- 5.4 Concluding Remarks.- Appendix A.- Examples of Symplectic Non-Kahlerian Manifolds.- Appendix B.- Solution of the Functional Equation (3.1.9).- Appendix C.- Semisimple Lie Algebras and Root Systems.- Appendix D.- Symmetric Spaces.- References.

On the geometry of Lie algebras and Poisson tensors

1994-11-21

José F. Cariñena , Alberto Ibort , G. Marmo +1 more

A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced … A geometric programme to analyse the structure of Lie algebras is presented with special emphasis on the geometry of linear Poisson tensors. The notion of decomposable Poisson tensors is introduced and an algorithm to construct all solvable Lie algebras is presented. Poisson-Liouville structures are also introduced to discuss a new class of Lie algebras which include, as a subclass, semi-simple Lie algebras. A decomposition theorem for Poisson tensors is proved for a class of Poisson manifolds including linear ones. Simple Lie algebras are also discussed from this viewpoint and lower-dimensional real Lie algebras are analysed.

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A historical review of the classifications of Lie algebras

2013-01-01

Luis Boza Prieto , Eugenio Manuel Fedriani Martel , Juan Núñez Valdés +1 more

The problem of Lie algebras’ classification, in their different varieties, has been dealt with by theory researchers since the early 20th century. This problem has an intrinsically infinite nature since … The problem of Lie algebras’ classification, in their different varieties, has been dealt with by theory researchers since the early 20th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimension. Despite the hundreds of attempts published, there are currently fields and dimensions in which only partial classifications of some families of algebras of low dimensions have been obtained. This article intends to bring some order to the achievements of this prolific line of research so far, in order to facilitate future research.

Über Orthogonalpolynome, die q‐Differenzengleichungen genügen

1949-01-01

Wolfgang Hahn