
Ghorbanali HaghighatdoostAzarbaijan Shahid Madani University · Department of Mathematics
Ghorbanali Haghighatdoost
Ph.D
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56
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January 2013 - March 2020
Publications
Publications (56)
In this work we study the invariant optimal control problem on Lie groupoids.We show that any invariant optimal control problem on a Lie groupoid reduces to its co-adjoint Lie algebroid. In the final section of the paper, we present an illustrative example.
As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated to Lie...
Based on our previous works, and in order to relate them with the theory of quantum graphs and the quantum computing principles, we once again try to introduce some newly developed technical structures just by relying on our toy example, i.e. the coordinate ring of n × n quantum matrix algebra M q (n), and the associated directed locally finite gra...
This paper is concerned with the structures introduced recently by the authors of the current paper concerning the multiplier Hopf *-graph algebras and also the Cuntz-Krieger algebras and their relations with the C *-graph algebras, and once again by using the C *-graph algebra constructions associated to our toy example, to initiate our first exam...
In this paper, we consider a generalized coupled Hirota-Satsuma KdV (CHSK) system of equations. We apply the moving frames method to find a finite generating set of differential invariants for the Lie symmetry group of CHSK equations. Once the generating set of differential invariants is located, we obtain recurrence relations and syzygies among the ge...
This is a noticeably short biography and introductory paper on multiplier Hopf algebras. It delves into questions regarding the significance of this abstract construction and the motivation behind its creation. It also concerns quantum linear groups, especially the coordinate ring of Mq(n) and the observation that 𝕂[Mq(n)] is a quadratic algebra, a...
This paper focuses on the introduction of right-invariant Poisson-Nijenhuis structures on Lie groupoids and their infinitesimal counterparts, also known as structures. A Poisson-Nijenhuis structure refers to a combination of a Poisson structure and a Nijenhuis structure. The paper also presents a mutual correspondence between Poisson-Nijenhuis stru...
In this paper, we study the algebraic structure of differential invariants of the fifth-order KdV types of equations. Using the moving frames method, we locate a finite generating set of differential invariants, recurrence relations, and syzygies among the generating differential invariants for the fifth-order KdV types of equations.
By Poissonization of Jacobi structures on real three-dimensional Lie groups G and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on G ⊗ R.
By Poissonization of Jacobi structures on real three-dimensional Lie groups $\mathbf{G}$ and using the realizations of their Lie algebras, we obtain integrable bi-Hamiltonian systems on $\mathbf{G}\otimes \mathbb{R}$.
In this paper, we consider the coupled Hirota-Satsuma KdV (CHSK) equations. We apply the Moving frames method to find a finite generating set of differential invariants for the Lie symmetry group of CHSK equations. Once the generating set of differential invariants is located, we obtain recurrence relations and syzygies among the generating differe...
This work is intended as an attempt to extend the notion of bialgebra for Lie algebras to Leibniz algebras and also, the correspondence between the Leibniz bialgebras and its dual is investigated. Moreover, the coboundary Leibniz bialgebras, the classical r-matrices, and Yang–Baxter equations related to the Leibniz algebras are defined, and some ex...
In this work we study the invariant optimal control problem on Lie groupoids. We show that any invariant optimal control problem on a Lie groupoid reduces to its co-adjoint Lie algebroid. At the final section of the paper, we present an illustrative example.
It is well-known that some of minimal (or maximal) hypersurfaces are stable. However, there is growing recognition on unstable hypersurfaces by introducing the concept of index of stability for minimal ones. For instance, the index of stability for minimal hypersurefces in Euclidean n-sphere has been defined by J. Simons and followed by many people...
As we said in our previous work [4], the main idea of our research is to introduce the a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called co-adjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated t...
Our purpose in this paper is to introduce a class of Lie groupoids, which we will call co-adjoint Lie groupoid, by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid. Also, we try to construct and define Hamiltonian systems on the co-adjoint Lie groupoids. By considering some examples, we show that our construction c...
In this paper, we consider Harry-Dym equation, which is a very important equation in the theory of solitons. Using the moving frame method, we locate differential invariants for the Harry-Dym equation. Every differential invariant can be expressed as a function of the generators and their invariant derivatives.
The theory of topological classification of integrable Hamiltonian systems with two degrees of freedom due to Fomenko and his school. On the basis of this theory we give a topological Liouville classification of the integrable Hamiltonian systems with two degrees of freedom. Essentially, to an integrable system with two degrees of freedom whi...
С использованием присоединенных представлений алгебр Ли проведена классификация всех структур Якоби на вещественных дву- и трехмерных группах Ли. Кроме того, изучены системы Якоби-Ли на таких вещественных группах Ли с небольшой размерностью. Полученные результаты иллюстрируются примерами гамильтоновых систем Якоби-Ли на некоторых вещественных дву-...
The Poissonization of a Jacobi structure allows us to construct a non-degenerate The Poissonization of a Jacobi structure allows us to construct a non-degenerate Poisson structure
which defines a symplectic structure. Moreover, Using Darboux’s theorem and the realizations of three-dimensional real Lie algebras, we discuss the integrability of three...
We provide an alternative method for providing of compatible Poisson structures on low-dimensional Lie groups by means of the structure constants of Lie algebras and the Liouville vector field. In this way we calculate some compatible Poisson structures on low-dimensional Lie groups. Then using Magri-Morosi's theorem and obtain new integrable bi-Ha...
In this work, we discuss bi-Hamiltonian structures on a family of integrable systems on 4-dimensional real Lie groups. By constructing the corresponding control matrix for this family of bi-Hamiltonian structures, we obtain an explicit process for finding the variables of separation and the separated relations in detail.
Our purpose in this paper is to introduce by means of co-adjoint representation of a Lie groupoid on its
isotropy Lie algebroid a class of Lie groupoids. In other words, we show that the orbits of the co-adjoint
representation on the isotropy Lie algebroid of a Lie groupoid are Lie groupoid. We will call this type of
Lie groupoid, co-adjoint Lie gr...
Using the adjoint representations of Lie algebras, we classify all Jacobi structures on real two- and three-dimensional Lie groups. Also, we study Jacobi-Lie systems on these real low-dimensional Lie groups. Our results are illustrated through examples of Jacobi-Lie Hamiltonian systems on some real two- and three-dimensional Lie groups.
We study Jacobi-Lie Hamiltonian systems admitting Vessiot-Guldberg Lie algebras of Hamiltonian vector fields related to Jacobi structures on real low-dimensional Jacobi-Lie groups. Also, we find some examples of Jacobi-Lie Hamiltonian systems on real two- and three- dimensional Jacobi-Lie groups. Finally, we present Lie symmetries of Jacobi-Lie Ham...
We study Jacobi-Lie Hamiltonian systems admitting Vessiot-Guldberg Lie algebras of Hamiltonian vector fields related to Jacobi structures on real low-dimensional Jacobi-Lie groups. Also, we find some examples of Jacobi-Lie Hamiltonian systems on real two-and three-dimensional Jacobi-Lie groups. Finally, we present Lie symmetries of Jacobi-Lie Hamil...
We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and...
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system on Lie algebroids are given. Here we use the general properties of Lie algebroids to express and prove two geometric version of the Hamilton-Jacobi theorem for Hamiltonian system on Lie algebroids. Then this results are generalized and two types of time-dependen...
In this paper, we study (n + 1)-dimensional real submanifolds M with (n − 1)-contact CR dimension. On these manifolds there exists an almost contact structure F which is naturally induced from the ambient space. Also, we study the condition h(F X, Y) − h(X, F Y) = g(F X, Y)ϕ, ϕ ∈ T M ⊥ , on the almost contact structure F and on the second fundament...
We study {\em right-invariant (resp., left-invariant) Poisson quasi-Nijenhuis structures} on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called {\em r-qn structures} on the corresponding Lie algebra $\mathfrak g$. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity...
In this presentation, using Cartan's equivalence method, we study the geometry associated with fourth order ODEs under point transformations.
We construct integrable Hamiltonian systems with Lie bialgebras $({\bf g} , {\bf \tilde{g}})$ of the bi-symplectic type for which the Poisson-Lie groups ${\bf G}$ play the role of the phase spaces, and their dual Lie groups ${\bf {\tilde {G}}}$ play the role of the symmetry groups of the systems. We give the new transformations to exchange the role...
We discuss bi-Hamiltonian structures for integrable and superintegrable Hamiltonian system on the list of symplectic four-dimensional real Lie groups are classified by G. Ovando. In addition, we creat corresponding control matrix for obtained bi-Hamiltonian structures.
We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two c...
We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that $r$-$n$ structures can be used to find compatible solutions of the classical Yang-Baxter equation. Conversely, two c...
Дана классификация всех четырехмерных действительных биалгебр Ли симплектического типа. Для этих биалгебр Ли получены классические $r$-матрицы и структуры Пуассона на всех соответствующих четырехмерных группах Пуассона-Ли. Получены некоторые новые интегрируемые модели, для которых группа Пуассона-Ли играет роль фазового пространства, а ее дуальная...
We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role...
We give a new method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltonian systems wi...
We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way, we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi's theorem we obtain new bi-Hamiltoni...
Let M be an (n + 1)-dimensional contact CR-submanifold of an odd-dimensional unit sphere S2m+1 of (n-q) contact CR-dimension. We study the condition h(FX; Y )+h(X; FY ) = 0 on the structure tensor F which is naturally induced from the almost contact structure φ of the ambient manifold and the second fundamental form h of the submanifold M. We obtai...
We classify all four dimensional real symplectic Lie bialgebras. Also the
classical r-matrices for these Lie bialgebras and Poisson structures on all of
the related four dimensional Poisson-Lie groups are obtained. As physical
applications, some new integrable models for which the Poisson-Lie group plays
the role of a phase space and its dual Lie g...
In 2001, A.V. Borisov, I. S.Mamaev, and V.V. Sokolov discovered a new integrable case on the Lie algebra so(4). This system coincides with the Poincaré equations on the Lie algebra so(4), which describe the motion of a body with cavities filled with an incompressible vortex fluid. Moreover, the Poincaré equations describe the motion of a four-dimen...
We define the bialgebra structure for the Leibniz algebras together with the
double and Manin triples and prove its correspondence with the Leibniz
bialgebras for different right or left cases for a Leibniz algebra and its
dual. We also define the classical r-matrices and Yang-Baxter equation for
Leibniz algebras. Finally, we give some simple examp...
We construct integrable and superintegrable Hamiltonian systems using the
realizations of four dimensional real Lie algebras as a symmetry of the system
with the phase space R4 and R6. Furthermore, we construct some integrable and
superintegrable Hamiltonian systems for which the symmetry Lie group is also
the phase space of the system.
In the present paper we obtain sharp estimates for the squared norm of the second fundamental form in terms of the mapping function for contact 3-structure CR-warped products isometrically immersed in Sasakian space form.
Several new integrable cases for Euler's equations on some six-dimensional Lie algebras were found by Sokolov in 2004. In this paper we study topological properties of one of these integrable cases on the Lie algebra so(4). In particular, for the system under consideration the bifurcation diagrams of the momentum mapping are constructed and all Fom...
We present a short overview of Hopf algebra theory. Starting with definitions of coalgebra.
bialgebra, Hopf algebra and these structures and pass to Hopf algebra symmetry and We introduce the notation, which is called Sweedler notation.
In this article we introduce the afine group scheme and circular group of a Hopf algebra H.Then pass to finite Ho...
The suggested method of investigating topology of isoenergy surfaces in so(4) is based on A. Oshemkov's idea. Using this method, the solution for one integrable case recently discovered by V. Sokolov is obtained. The problem is to describe topological type of three-dimensional manifolds Q3g,h at different values of parameters g and h.