The purpose of this paper is to investigate the nonlinear partial differential equation, known as... more The purpose of this paper is to investigate the nonlinear partial differential equation, known as potential Korteweg-de Vries (p-KdV) equation. We have implemented the Harrison technique that makes use of differential forms and Lie derivatives as tools to find the point symmetry algebra for the p-KdV equation. This approach allows us to obtain five infinitesimal generators of point symmetries. Fixing each generator of symmetries that we have found, we construct a complete set of functionally independent invariants, corresponding to the new independent and dependent variables. Using these new variables, called “similarity variables”, the reduced equations have been constructed systematically, which leads to exact solutions that are group-invariant solutions for the p-KdV equation. The obtained solutions are of two types. The reduced equations from the generator of space and time translation groups are the first and the third order ordinary differential equations respectively and lead...
Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained b... more Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action of G on the dual space of its Lie algebra. We investigate the group of automorphisms of the Lie algebra of $T^*G$. More precisely, amongst other results, we fully characterize the space of all derivations of the Lie algebra of $T^*G$. As a byproduct, we also characterize some spaces of operators on G amongst which, the space J of bi-invariant tensors on G and prove that if G has a bi-invariant Riemannian or pseudo-Riemannian metric, then J is isomorphic to the space of linear maps from the Lie algebra of G to its dual space which are equivariant with respect to the adjoint and coadjoint actions, as well as that of bi-invariant bilinear forms on G. We discuss some open problems and possible applications.
Let M and N be two commuting square matrices of order n with entries in an algebraically closed f... more Let M and N be two commuting square matrices of order n with entries in an algebraically closed field K. Then the associative commutative K-algebra, they generate, is of dimension at most n. This result was proved by Murray Gerstenhaber in [Gerstenhaber, M.; On dominance and varieties of commuting matrices. Ann. of Math. (2) 73 (1961), 324-348]. Although the analog of this property for three commuting matrices is still an open problem, its version for a higher number of commuting matrices is not true in general. In the present paper, we give a sufficient condition for this property to be satisfied, for any number of commuting matrices, for any arbitrary field K. Such a result is derived from a discussion on the structure of 2-step solvable Frobenius Lie algebras and a complete characterization of their associated left symmetric algebra (LSA) structure.
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and i... more Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie algebras and compact Lie algebras we recover by simple methods interesting co-homological known results. The Lie algebra of prederivations encompasses the one of derivations as a subalgebra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) of semi-simple Lie groups have the property that all their prederivations are derivations. This result is an extension of a well known result due to M\"uller. The structure of the Lie algebra of prederivations of Lie algebras of cotangent bundles of Lie groups is explored and we have shown that the Lie algebra of prederivations of Lie algebras of cotangent bundle of Lie groups are reductive Lie ...
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solut... more We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equation.
In this paper we introduce color Hom-Akivis algebras and prove that the commutator of any color n... more In this paper we introduce color Hom-Akivis algebras and prove that the commutator of any color non-associative Hom-algebra structure map leads to a color Hom-akivis algebra. We give various constructions of color Hom-Akivis algebras. Next we study flexible and alternative color Hom-Akivis algebras. Likewise color Hom-Akivis algebras, we introduce non-commutative color Hom-Leibniz-Poisson algebras and presente several constructions. Moreover we give the relationship between Hom-dialgebras and Hom-Leibniz-Poisson algebras; i.e. a Hom-dialgebras give rise to a Hom-Leibniz-Poisson algebra. Finally we show that twisting a color Hom-Leibniz module structure map by a color Hom-Leibniz algebra endomorphism, we get another one.
We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any ... more We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra structure can be embedded as a subalgebra of some sl(m,K), for K = R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m,K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring examples of Frobenius Lie algebras that are subalgebras of sl(m,K), but yet have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenval...
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solut... more We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equation.
We investigate Lie point symmetries of a system of four nonlinear secondorder ordinary differenti... more We investigate Lie point symmetries of a system of four nonlinear secondorder ordinary differential equations (ODEs), appropriated to the geodesics of a Drinfel’d double Lie group of the affine Lie group of ℝ. The first integrals associated with Lie point symmetries are obtained by utilizing the constructive method due to Wafo Soh and Mahomed [16]. This method deals with integrability when the symmetry vector fields and the operator associated to the system are unconnected. In certain cases we obtain the explicit expressions of the geodesics. We also show that the geodesics are not complete.
The purpose of this paper is to investigate the nonlinear partial differential equation, known as... more The purpose of this paper is to investigate the nonlinear partial differential equation, known as potential Korteweg-de Vries (p-KdV) equation. We have implemented the Harrison technique that makes use of differential forms and Lie derivatives as tools to find the point symmetry algebra for the p-KdV equation. This approach allows us to obtain five infinitesimal generators of point symmetries. Fixing each generator of symmetries that we have found, we construct a complete set of functionally independent invariants, corresponding to the new independent and dependent variables. Using these new variables, called “similarity variables”, the reduced equations have been constructed systematically, which leads to exact solutions that are group-invariant solutions for the p-KdV equation. The obtained solutions are of two types. The reduced equations from the generator of space and time translation groups are the first and the third order ordinary differential equations respectively and lead...
Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained b... more Let G be a Lie group, $T^*G$ its cotangent bundle with its natural Lie group structure obtained by performing a left trivialization of T^*G and endowing the resulting trivial bundle with the semi-direct product, using the coadjoint action of G on the dual space of its Lie algebra. We investigate the group of automorphisms of the Lie algebra of $T^*G$. More precisely, amongst other results, we fully characterize the space of all derivations of the Lie algebra of $T^*G$. As a byproduct, we also characterize some spaces of operators on G amongst which, the space J of bi-invariant tensors on G and prove that if G has a bi-invariant Riemannian or pseudo-Riemannian metric, then J is isomorphic to the space of linear maps from the Lie algebra of G to its dual space which are equivariant with respect to the adjoint and coadjoint actions, as well as that of bi-invariant bilinear forms on G. We discuss some open problems and possible applications.
Let M and N be two commuting square matrices of order n with entries in an algebraically closed f... more Let M and N be two commuting square matrices of order n with entries in an algebraically closed field K. Then the associative commutative K-algebra, they generate, is of dimension at most n. This result was proved by Murray Gerstenhaber in [Gerstenhaber, M.; On dominance and varieties of commuting matrices. Ann. of Math. (2) 73 (1961), 324-348]. Although the analog of this property for three commuting matrices is still an open problem, its version for a higher number of commuting matrices is not true in general. In the present paper, we give a sufficient condition for this property to be satisfied, for any number of commuting matrices, for any arbitrary field K. Such a result is derived from a discussion on the structure of 2-step solvable Frobenius Lie algebras and a complete characterization of their associated left symmetric algebra (LSA) structure.
Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and i... more Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie algebras and compact Lie algebras we recover by simple methods interesting co-homological known results. The Lie algebra of prederivations encompasses the one of derivations as a subalgebra. We find out that Lie algebras of cotangent Lie groups (which are not semi-simple) of semi-simple Lie groups have the property that all their prederivations are derivations. This result is an extension of a well known result due to M\"uller. The structure of the Lie algebra of prederivations of Lie algebras of cotangent bundles of Lie groups is explored and we have shown that the Lie algebra of prederivations of Lie algebras of cotangent bundle of Lie groups are reductive Lie ...
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solut... more We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equation.
In this paper we introduce color Hom-Akivis algebras and prove that the commutator of any color n... more In this paper we introduce color Hom-Akivis algebras and prove that the commutator of any color non-associative Hom-algebra structure map leads to a color Hom-akivis algebra. We give various constructions of color Hom-Akivis algebras. Next we study flexible and alternative color Hom-Akivis algebras. Likewise color Hom-Akivis algebras, we introduce non-commutative color Hom-Leibniz-Poisson algebras and presente several constructions. Moreover we give the relationship between Hom-dialgebras and Hom-Leibniz-Poisson algebras; i.e. a Hom-dialgebras give rise to a Hom-Leibniz-Poisson algebra. Finally we show that twisting a color Hom-Leibniz module structure map by a color Hom-Leibniz algebra endomorphism, we get another one.
We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any ... more We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra structure can be embedded as a subalgebra of some sl(m,K), for K = R or C. Hence, the work of Belavin and Drinfeld on solutions of the Classical Yang-Baxter Equation on simple Lie algebras, applied to the particular case of sl(m,K) alone, paves the way to the complete classification of Frobenius and more generally quasi-Frobenius Lie algebras. We prove that, if a Frobenius Lie algebra has the property that every derivation is an inner derivation, then every principal element is semisimple. As an important case, we prove that in the Lie algebra of the group of affine motions of the Euclidean space of finite dimension, every derivation is inner. We also bring examples of Frobenius Lie algebras that are subalgebras of sl(m,K), but yet have nonsemisimple principal elements as well as some with semisimple principal elements having nonrational eigenval...
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solut... more We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equation.
We investigate Lie point symmetries of a system of four nonlinear secondorder ordinary differenti... more We investigate Lie point symmetries of a system of four nonlinear secondorder ordinary differential equations (ODEs), appropriated to the geodesics of a Drinfel’d double Lie group of the affine Lie group of ℝ. The first integrals associated with Lie point symmetries are obtained by utilizing the constructive method due to Wafo Soh and Mahomed [16]. This method deals with integrability when the symmetry vector fields and the operator associated to the system are unconnected. In certain cases we obtain the explicit expressions of the geodesics. We also show that the geodesics are not complete.
Uploads
Papers