POISSON ALGEBRA STRUCTURES ON sp2l（^～CQ） WITH NULLITY m

Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this article,the noncommutative Poisson algebra structures on sp2l（^～CQ） are determined.     

POISSON ALGEBRA STRUCTURES ON sp_(2■)(■_Q)WITH NULLITY m

Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law.In this article, the noncommutative Poisson algebra structures on sp_(2■)(■_Q)are determined.     

NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sln(fCq) WITH NULLITY M

Non-commutative Poisson algebras are the algebras having both an associa-tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson alg...     

The Lie Algebras in which Every Subspace Is Its Subalgebra

In this paper, we study the Lie algebras in which every subspace is its subalgebra （denoted by HB Lie algebras）. We get that a nonabelian Lie algebra is an HB Lie algebra if and only if it is isomorphic to g＋Cidg, where g is an abelian Lie algebra. Moreover we show that the derivation algebra and the holomorph of a nonabelian HB Lie algebra are complete.     

Superderivations for a Family of Lie Superalgebras of Special Type

By means of generators, superderivations are completely determined for a family of Lie superalgebras of special type, the tensor products of the exterior algebras and the finite-dimensional special Lie algebras over a field of characteristic p〉3. In particular, the structure of the outer superderivation algebra is concretely formulated and the dimension of the first cohomology group is given.     

RESEARCH ANNOUNCEMENTS——Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics^[10，12，13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras. In this paper, we study solvable quadratic Lie algebras. In Section 1, we study quadratic solvable Lie algebras whose Cartan subalgebras consist of semi-simple elements. In Section 2,we present a procedure to construct a class of quadratic Lie algebras, and we can exhaust all solvable quadratic Lie algebras in such a way. All Lie algebras mentioned in this paper are finite dimensional Lie algebras over a field F of characteristic 0.     

A-扩张Lie Rinehart代数

The purpose of this paper is to give a brief introduction to the category of Lie Rinehart algebras and introduces the concept of smooth manifolds associated with a unitary, commutative,associative algebra A.It especially shows that the A-extended algebra as well as the action algebra can be realized as the space of A-left invariant vector fields on a Lie group,analogous to the well known relationship of Lie algebras and Lie groups.     

Central simple Poisson algebras

Poisson algebras are fundamental algebraic structures in physics and sym-plectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero. The Lie algebra structures of these Poisson algebras are in general not finitely-graded.     

A specialization of prinjective Ringel-Hall algebra and the associated Lie algebra

In the present paper we describe a specialization of prinjective Ringel-Hall algebra to 1, for prinjective modules over incidence algebras of posets of finite prinjective type, by generators and relations. This gives us a generalisation of Serre relations for semisimple Lie algebras. Connections of prinjective Ringel-Hall algebras with classical Lie algebras are also discussed.     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

关于李三代数的幂零性

In this paper, we mainly concerned about the nilpotence of Lie triple algebras. We give the definition of nilpotence of the Lie triple algebra and obtained that if Lie triple algebra is nilpotent, then its standard enveloping Lie algebra is nilpotent.     

Invariants for Automorphisms of the Underlying Algebras Relative to Lie Algebras of Cartan Type

Let X denote a finite or infinite dimensional Lie algebra of Cartan type W, S, H or K over a field of characteristic p 〉 3. In this paper it is proved that certain filtrations of the underlying algebras are invariant under the admissible groups relative to Lie algebras of Cartan type X.     

Two Kinds of Solvable Complete Lie Algebras

In this paper,we will give the definition of completable nilpotent Lie algebras,discuss its decomposition and prove that the heisenberg algebras and extensions of abelian quadratic Lie algebras are all completable nilpotent Lie algebras.     

Simplicities and Automorphisms of a Special Infinite Dimensional Lie Algebra

In this paper, a special infinite dimensional Lie algebra is studied. The infinite dimensional Lie algebra appears in the fields of conformal theory, mathematical physics, statistic mechanics and Hamilton operator. The infinite dimensional Lie algebras is pop ularized Virasoro-like Lie algebra. Isomorphisms, homomorphisms, ideals of the infinite dimensional Lie algebra are studied.     

Note on Co-split Lie Algebras

It is proved that,any finite dimensional complex Lie algebra L = [L,L],hence,over a field of characteristic zero,any finite dimensional Lie algebra L = [L,L] possessing a basis with complex structure constants,should be a weak co-split Lie algebra.A class of non-semi-simple co-split Lie algebras is given.     

DERIVATIONS ON DIFFERENTIAL OPERATOR ALGEBRA AND WEYL ALGEBRA

Let L be an n-dimensional nilpotent Lie algebra with a basis {x1,…,xn}, and every xiacts as a locally nilpotent derivation on algebra A. This paper shows that there exists a setof derivations {y1,…,yn} on U(L) such that (A#U(L))#k[yi,…,yn] is isomorphic to theWeyl algebra An(A). The author also uses the derivations to obtain a necessary and sufficientcondition for a finite dimensional Lie algebra to be nilpotent.     

RESEARCH ANNOUNCEMENTS Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras.     

扩展的可积模型及其约化与达布变换

In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras?H and?T, respectively. By using the loop algebra?H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra?T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra?H to obtain an integrable hierarchy with variable coefficients.     

A Class of Solvable Lie Algebras and Their Hom-Lie Algebra Structures

The finite-dimensional indecomposable solvable Lie algebras s with Q2n＋1 as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n＋1＋1. Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.     

L-o cto-algebras

L-octo-algebra with 8 operations as the Lie algebraic analogue of octo-algebra such that the sum of 8 operations is a Lie algebra is discussed. Any octo-algebra is an L-octo-algebra. The relationships among L-octo-algebras, L-quadri-algebras, L-dendriform algebras, pre-Lie algebras and Lie algebras are given. The close relationships between L-octo-algebras and some interesting structures like Rota-Baxter operators, classical Yang-Baxter equations and some bilinear forms satisfying certain conditions are given also.