Compatible Lie Bialgebras

A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket.We construct a bialgebra theory of compatible Lie algebras as an analogue of a Lie bialgebra.They can also be regarded as a "compatible version" of Lie bialgebras,that is,a pair of Lie bialgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra.Many properties of compatible Lie bialgebras as the "compatible version" of the corresponding properties of Lie bialgebras are presented.In particular,there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in Lie algebras.Furthermore,a notion of compatible pre-Lie algebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie algebras which leads to a construction of the solutions of the latter.As a byproduct,the compatible Lie bialgebras St into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.     

Generalized Lie bialgebroids and Jacobi structures

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

The supergroups of supersymmetry and supergravity as ordinary real Lie groups

The concept of Grassmannification of a Lie group, which is completely analogous to the concept of complexification of a Lie group, is introduced. Grassmannified Lie groups can also be viewed as ordinary real Lie groups. It is shown that every graded Lie algebra (= superalgebra) determines a subgroup (Kac-Berezin group, supergroup) in the Grassmannified full matrix group. On the other hand, it seems possible that not all supergroups can be found by a complete classification of all graded Lie algebras.     

COMMENTS ON THE PAPER WRITTEN BY ZHU ZHONG-YUAN

Mr.ZHU Zhong-yuan discussed the connection between Lie supergroups and Lie superalgebras in his paper entitled "On Three Theorem of Lie Supergroups and Lie Supperalgebras"^[1].     

Solutions of the Yang–Baxter equations on quadratic Lie groups: The case of oscillator groups

A Lie group is called quadratic if it carries a bi-invariant semi-Riemannian metric. Oscillator Lie groups constitute a subclass of the class of quadratic Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang–Baxter equation on a generic class of oscillator Lie algebras. Moreover, we show that any solution of the generalized classical Yang–Baxter equation (resp. classical Yang–Baxter equation) on a quadratic Lie group determines a left invariant locally symmetric (resp. flat) semi-Riemannian metric on the corresponding dual Lie groups.     

李超代数综述

本文将李超代数介绍给物理学工作者。讨论了阶化向量空间和李超代数的基本性质。给出经典李超代数的分类及基本经典李超代数的Kac-Dynkin图。特别是介绍了如何运用广义的不可约张量算符及Wigner-Eckart定理,求经典李超代数有限维不约可表示的方法。     

Contractions of Lie Bialgebras and Quantum Algebras

Contractions of Lie bialgebras and Hopf algebras are discussed with examples. Especially, it is shown that the Lie bialgebras associated with the compact simple Lie algebras and the quantum doubles associated with the complex simple Lie algebras can be contracted.     

EFFECTS OF CONSTRAINTS ON LIE SYMMETRIES AND CONSERVED QUANTITIES OF A BIRKHOFF SYSTEM

After a Birkhoff system is restricted by constraints, the determining equations, the Lie symmetries, the structure equation and the form of conserved quantities corresponding to the Lie symmetries will change. Some Lie symmetries will disappear and under certain conditions some Lie symmetries will still remain present. The condition under which Lie symmetries and conserved quantities of the system will remain is given.     

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

Chetaev型约束力学系统Appell方程的Lie对称性与守恒量

研究Chetaev型约束力学系统Appell方程的Lie对称性和Lie对称性直接导致的守恒量.分析Lagrange函数和A函数的关系;讨论Chetaev型约束力学系统Appell方程的Lie对称性导致的守恒量的一般研究方法;在群的无限小变换下,给出Appell方程Lie对称性的定义和判据;得到Lie对称性的结构方程以及Lie对称性直接导致的守恒量的表达式.举例说明结果的应用. 关键词： Appell方程 Chetaev 型约束力学系统 Lie对称性 守恒量     

Lie Algebras Associated with Group U(n)

Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra A1 are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

EFFECTS OF NON-CONSERVATIVE FORCES ON LIE SYMMETRIES AND CONSERVED QUANTITIES OF A LAGRANGE SYSTEM

Non-conservative forces are exerted on a Lagrange system. Their effects on Lie symmetries, structure equation and conserved quantities of the system are studied. It can be seen that some Lie symmetries disappear and some new Lie symmetries emerge. Under certain conditions, some Lie symmetries will still remain present.     

Lie Symmetrical Non-Noether Conserved Quantities of Poincarè-Chetaev Equations

In the present paper the Lie symmetrical non-Noether conserved quantity of the Poincarè-Chetaev equations under the general infinitesimal transformations of Lie groups is discussed. First, we establish the determining equations of Lie symmetry of the equations. Second, the Lie symmetrical non-Noether conserved quantity of the equations is deduced.     

Integration of Lie 2-Algebras and Their Morphisms

Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.     

Hypersymplectic Lie algebras

We characterize real Lie algebras carrying a hypersymplectic structure as bicrossproducts of two symplectic Lie algebras endowed with a compatible flat torsion-free connection. In particular, we obtain the classification of all hypersymplectic structures on 4-dimensional Lie algebras, and we describe the associated metrics on the corresponding Lie groups.     

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 (2006) 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.     

Invariant star-product on a Poisson-Lie group and h-deformation of the corresponding Lie algebra

In this paper we prove that a left-invariant star-product on a Poisson-Lie group leads to the quantum Lie algebra structure on the corresponding Lie algebra of the Lie group.