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Alina Dobrogowska

Publications and source records attributed to Alina Dobrogowska.

12 recordsLinked to original sources
arXiv2024

Cyclic Lie-Rinehart algebras

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.

math.DG↗
arXiv2023

A new look at Lie algebras

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.

math-ph↗
arXiv2018

Darboux transformations and second order difference equations

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.

math.DS↗
arXiv2018

Deformation of algebroid bracket of differential forms and Poisson manifold

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.

math-ph↗
arXiv2017

Factorization method and general second order linear difference equation

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.

math-ph↗
arXiv2014

Lie bundle on the space of deformed skew-symmetric matrices

We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real numbers $a_1,\ldots,a_{n-1}$. Using isomorphism $\mathcal A_{a_1,\ldots,a_{n-1}}^* \cong L_+$ we introduce a Lie-Poisson structure on the space of upper-triangular matrices $L_+$. In this way we generate hierarchies of Hamilton systems with bihamiltonian structure.

math-ph↗
arXiv2014

Integrable Systems Related to Deformed $\mathfrak{so}(5)$

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

math-ph↗
arXiv2010

Integrable Hamiltonian systems related to the Hilbert--Schmidt ideal

By application of the coinduction method as well as Magri method to the ideal of real Hilbert-Schmidt operators we construct the hierarchies of integrable Hamiltonian systems on the Banach Lie-Poisson spaces which consist of these type of operators. We also discuss their algebraic and analytic properties as well as solve them in dimensions N=2,3,4.

math-ph↗
arXiv2003

Second order q-difference equations solvable by factorization method

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.

math-ph↗