The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the p...The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the present paper, we obtain a matrix representation for each of these Lie algebras. We are able to find such representations by exploiting properties of the radical, principally, when it has a trivial center, in which case we can obtain such a representation by restricting the adjoint representation. Another important subclass of algebras is where the radical has a codimension one abelian nilradical and for which a representation can readily be found. In general, finding matrix representations for abstract Lie algebras is difficult and there is no algorithmic process, nor is it at all easy to program by computer, even for algebras of low dimension. The present paper represents another step in our efforts to find linear representations for all the low dimensional abstract Lie algebras.展开更多
In this paper, a better asymptotic order of Fourier transform on SL(2,R) is obtained by using classical analysis and Lie analysis comparing with that of [5],[6] ,and the Plancherel theorem on C2i(SL(2,R)) is also obta...In this paper, a better asymptotic order of Fourier transform on SL(2,R) is obtained by using classical analysis and Lie analysis comparing with that of [5],[6] ,and the Plancherel theorem on C2i(SL(2,R)) is also obtained as an application.展开更多
In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form assoc...In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form associated with the r-Jacobi algebra A.展开更多
Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are revi...Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are reviewed. A new symplectic approach to constructing nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and based upon the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is described. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix method [1,2] is analyzed in detail. A new modified differential-algebraic approach to analyzing the Lax integrability of generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is suggested and the corresponding Lax representations are constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of these generalized Riemann type hierarchies are discussed by means of the symplectic, gradientholonomic and geometric methods.展开更多
The key point of the method of calculating the normal form of ordinary differential equations based on the representation theory of sl (2, R) is to find generators of KerL_M and Kerad_M. In this paper, we give the nec...The key point of the method of calculating the normal form of ordinary differential equations based on the representation theory of sl (2, R) is to find generators of KerL_M and Kerad_M. In this paper, we give the necessary and sufficient conditions for generators of KerL_M and Ker ad_M and prove that any vector-valued polynomial in Ker ad_M can be constructed from polynomials in KerL_M in a certain way.展开更多
文摘The authors have recently completed a partial classification of the ten-dimensional real Lie algebras that have the non-trivial Levi decomposition, namely, for such algebras whose semi-simple factor is so(3). In the present paper, we obtain a matrix representation for each of these Lie algebras. We are able to find such representations by exploiting properties of the radical, principally, when it has a trivial center, in which case we can obtain such a representation by restricting the adjoint representation. Another important subclass of algebras is where the radical has a codimension one abelian nilradical and for which a representation can readily be found. In general, finding matrix representations for abstract Lie algebras is difficult and there is no algorithmic process, nor is it at all easy to program by computer, even for algebras of low dimension. The present paper represents another step in our efforts to find linear representations for all the low dimensional abstract Lie algebras.
文摘In this paper, a better asymptotic order of Fourier transform on SL(2,R) is obtained by using classical analysis and Lie analysis comparing with that of [5],[6] ,and the Plancherel theorem on C2i(SL(2,R)) is also obtained as an application.
文摘In this paper, we denote by A a commutative and unitary algebra over a commutative field K of characteristic 0 and r an integer ≥1. We define the notion of r-Jacobi algebra A and we construct the canonical form associated with the r-Jacobi algebra A.
文摘Recently devised new symplectic and differential-algebraic approaches to studying hidden symmetry properties of nonlinear dynamical systems on functional manifolds and their relationships to Lax integrability are reviewed. A new symplectic approach to constructing nonlinear Lax integrable dynamical systems by means of Lie-algebraic tools and based upon the Marsden-Weinstein reduction method on canonically symplectic manifolds with group symmetry, is described. Its natural relationship with the well-known Adler-Kostant-Souriau-Berezin-Kirillov method and the associated R-matrix method [1,2] is analyzed in detail. A new modified differential-algebraic approach to analyzing the Lax integrability of generalized Riemann and Ostrovsky-Vakhnenko type hydrodynamic equations is suggested and the corresponding Lax representations are constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of these generalized Riemann type hierarchies are discussed by means of the symplectic, gradientholonomic and geometric methods.
基金Project supported by the National Natural Science Foundation of China.
文摘The key point of the method of calculating the normal form of ordinary differential equations based on the representation theory of sl (2, R) is to find generators of KerL_M and Kerad_M. In this paper, we give the necessary and sufficient conditions for generators of KerL_M and Ker ad_M and prove that any vector-valued polynomial in Ker ad_M can be constructed from polynomials in KerL_M in a certain way.
