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双语推荐：Lie

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3-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of characteristic zero are provided.
3-Lie algebras have close relationships with many important fields in mathemat-ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char-acteristic zero are provided.

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In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras.Furthermore, we generalize Garland''s theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.
In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland’s theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.

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与三维Lie代数的Bianchi 13个分类相对应,欧氏空间R3上的Lie-Poisson结构也可分为13类.研究了三维Lie-Poisson结构的保结构线性变换,对应于每一类Lie-Poisson结构,获得了可逆线性变换是保结构变换的充要条件.
Based on the 13 Bianchi′s classification of all three dimensional Lie algebra,all structure matrices of three-dimensional Lie-Poisson brackets were classified into 13 different types. It was studied the linear structure-preserving transformations for three-dimensional Lie-Poisson brackets and presented the necessary and sufficient conditions under which a linear transformation was structure-preserving.

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研究完整系统Appell方程Lie对称性的共形不变性与Hojman守恒量.在时间不变的特殊无限小变换下,定义完整系统动力学方程的Lie对称性和共形不变性,给出该系统Lie对称性共形不变性的确定方程及系统的Hojman守恒量,并举例说明结果的应用.
The conformal invariance and Hojman conserved quantity of Lie symmetry for Appell equations in a holonomic system are studied. Under the special infinitesimal transformations in which the time is not variable, the Lie symmetry and conformal invariance of differential equations of motion for a holonomic system are defined, and the determining equations of the conformal invariance of Lie symmetry and the Hojman conserved quantity for the system are given. Finally, an example is presented to illustrate the application of the results.

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通过给出 Heisenberg Jordan-Lie 代数的定义，得到 Heisenberg Jordan-Lie 代数H 的自同构群Aut（H ）的一些子群，并在 H 为低维的情形下，讨论了自同构群 Aut （H ）的基本结构。
We introduced the notion of Heisenberg Jordan-Lie algebra so as to investigate some subgroups of the automorphism group Aut(H)of Heisenberg Jordan-Lie algebra H.Moreover,we discussed some basic structure of the automorphism group Aut (H ) in the case of H being low-dimensional.

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研究相对运动变质量完整系统Appell方程的广义Lie对称性及其直接导致的广义Hojman守恒量。在群的无限小变换下，给出相对运动变质量完整系统Appell方程广义Lie对称性的确定方程；得到相对运动变质量完整系统Appell方程广义Lie对称性直接导致的广义Hojman守恒量的表达式。最后，利用本文结果研究相对运动变质量完整约束的三自由度力学系统问题。
Generalized Lie symmetry and generalized Hojman conserved quantity of Appell equations for a variable mass holonomic system in relative motion are studied. The determining equation of generalized Lie symmetry of Appell equations for a variable mass holonomic system in relative motion under the infinitesimal transformations of groups is given. The expression of generalized Hojman conserved quantity deduced directly from generalized Lie symmetry for a variable mass holonomic system in relative motion is gained. Finally, the problem of dynamical system with three degree of freedom is studied by using the results of this paper.

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本文研究离散差分序列变质量 Hamilton 系统的 Lie 对称性与 Noether 守恒量。构建了离散差分序列变质量Hamilton 系统的差分动力学方程，给出了离散差分序列变质量 Hamilton 系统差分动力学方程在无限小变换群下的Lie 对称性的确定方程和定义，得到了离散力学系统 Lie 对称性导致 Noether 守恒量的条件及形式，举例说明结果的应用。
In this paper the Lie symmetry and Noether conserved quantity of a discrete difference sequence Hamilton system with variable mass are studied. Firstly, the difference dynamical equations of the discrete difference sequence Hamilton system with variable mass are built. Secondly, the determining equations and the definition of Lie symmetry for difference dynamical equations of the discrete difference sequence Hamilton system under infinitesimal transformation groups are given. Thirdly, the forms and conditions of Noether conserved quantities to which Lie symmetries will lead in a discrete mechanical system are obtained. Finally, an example is given to illustrate the application of the results.

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研究Chetaev型非完整系统Nielsen方程Lie对称性导致的一种守恒量,给出无限小群变换下Chetaev型非完整系统Nielsen方程Lie对称性的确定方程,得到Chetaev型非完整系统Nielsen方程Lie对称性直接导致的一种守恒量及其存在条件,并举例说明结果应用.
This paper studied a type of conserved quantity deduced by the Lie symmetry for nonholonomic system with Chetaev-type of the Nielsen equation.Firstly,the determining equations of the Lie symmetry for nonholo-nomic system with Chetaev-type of the Nielsen equation were given under the infinitesimal transformation of groups.Secondly,the conditions of the existence of the type of conserved quantity of the system as well as its forms were obtained.And finally,an example was given to illustrate the application of the results.

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在群的无限小变化下,研究奇异变质量单面非完整系统Nielsen方程的Noether-Lie对称性.建立系统运动微分方程的Nielsen形式,给出系统Nielsen方程的Noether-Lie对称性的定义、判据和命题,得到系统Nielsen方程的Noether-Lie对称性所导致的Noether守恒量和广义Hojman守恒量.最后给出说明性算例说明结果的应用.
Under the infinitesimal transformations of Lie group, Noether-Lie symmetry of Nielsen equations for a singular variable mass nonholonomic system with unilateral constraints is studied. Differential equations of motion for Nielsen equations of the system are established. The definition, criteria and propositions for Nielsen equations of the system are given. Noether conserved quantity as well as generalized Hojman quantity is obtained. An example is given to illustrate the application of the results.

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对一个与3＆#215;3的LAX对相联系的超AKNS方程族，通过遗传算子φ，找到了方程族的K对和τ对称。进一步，导出了超AKNS方程族中任一个方程的对称所构建的 Lie代数结构。通过比较，超 AKNS方程族和AKNS方程族的对称及其Lie代数结构，发现他们的一致性。
In this paper ,the K symmetry and τ symmetry for the super Dirac hierarchy ,which is associated with a 3 × 3 Lax pair ,are found by using the hereditary operator φ .Moreover ,the Lie algebra construction of the symmetries for any one equation of the hierarchy is de‐rived .Specially ,by contrast the symmetry and Lie algebra construction for the super AKNS hierarchy and AKNS hierarchy ,we found the super AKNS hierarchy and AKNS hierarchy have uniform symmetry and Lie algebra construction .

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