（n-3）-Filiform李代数的极大环面

求出了（n-3）-filiform李代数的极大环面，并证明了（n-3）-filiform李代数是可完备化的．     

Solvable Indecomposable Extensions of Two Nilpotent Lie Algebras

A pair of sequences of nilpotent Lie algebras denoted by N_{n, 7} and N_{n, 16} are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that N_{n, 7} and N_{n, 16} serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.     

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.     

A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations

Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques. We only consider finite-dimensional Lie algebras over a fields of characteristic zero.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

von Neumann代数中套子代数上的Lie导子

本文对因子von Neumann代数中套子代数上的线性映射L:alg_Mβ→M满足L(AB—BA)=L(A)B-BL(A)+AL(B)-L(B)A( A,B∈alg_Mβ)进行了刻划,证明了存在线性函数h:alg_Mβ→C;且对任意A,B∈alg_Mβ,有h(AB—BA)=0和算子T∈M,使得对任意X∈alg_Mβ,都有L(X)=XT-TX+h(X)I.     

Automorphisms and Derivations of Certain Solvable Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R with identity. Let ?(R) be the solvable subalgebra of L _R spanned by the basis elements of the maximal toral subalgebra and the root vectors associated with positive roots. In this article, we prove that under some conditions for R, any automorphism of ?(R) is uniquely decomposed as a product of a graph automorphism, a diagonal automorphism and an inner automorphism, and any derivation of ?(R) is uniquely decomposed as a sum of an inner derivation induced by root vectors and a diagonal derivation. Correspondingly, the automorphism group and the derivation algebra of ?(R) are determined.     

Depth Generated Simple Lie Algebras

A Lie module algebra for a Lie algebra L is an algebra and L-module A such that L acts on A by derivations. The depth Lie algebra of a Lie algebra L with Lie module algebra A acts on a corresponding depth Lie module algebra . This determines a depth functor from the category of Lie module algebra pairs to itself. Remarkably, this functor preserves central simplicity. It follows that the Lie algebras corresponding to faithful central simple Lie module algebra pairs (A,L) with A commutative are simple. Upon iteration at such (A,L), the Lie algebras are simple for all i ∈ ω. In particular, the (i ∈ ω) corresponding to central simple Jordan Lie algops (A,L) are simple Lie algebras. Presented by Don Passman.     

On Behavior of Solvable Ideals of Lie Algebras Under Outer Derivations

Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log₂ p.     

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.     

On Polynomial Representations of Lie Algebras

We study polynomial representations of finite dimensional (R or C) Lie algebras. As a total classification, we show that there are altogether three types of such nontrivial representations and give their subtle structures.     

Derivation Algebra of Quasi Rn-filiform Lie Algebra

In this paper we explicitly determine the derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Pn-filiform Lie algebra is a completable nilpotent Lie algebra.     

The Moduli Space of Six-Dimensional Two-Step Nilpotent Lie Algebras

We determine the moduli space of metric two-step nilpotent Lie algebras of dimension up to 6. This space is homeomorphic to a cone over a four-dimensional contractible simplicial complex. Moreover, we exhibit standard metric representatives of the seven isomorphism types of six-dimensional two-step nilpotent Lie algebras within our picture. Mathematics Subject Classifications (2000): Primary 22E25, 53C30, 22E60     

Infinite Rank Affine Lie Algebras

１．ＩｎｆｉｎｉｔｅＲａｎｋＡｆｉｎｅＬｉｅＡｌｇｅｂｒａｓｇ（Ｘ）ａｎｄｇ（Ｘ）ＷｅｒｅｃａｌｔｈｅｄｅｆｉｎｉｔｉｏｎｏｆｉｎｆｉｎｉｔｅｒａｎｋａｆｉｎｅＬｉｅａｌｇｅｂｒａｓａｎｄｔｈｅｉｒｆｕｎｄａｍｅｎｔａｌｓｔｒｕｃｔｕｒｅ．Ａｇｅｎｅ...     

Derivation Algebra of Quasi $R_n$-Filiform Lie Algebra

In this paper we explicitly determine the derivation algebra of a quasi $R_n$-filiform Lie algebra and prove that a quasi $R_n$-filiform Lie algebra is a completable nilpotent Lie algebra.     

Lie Algops and Simple Lie Algebras

ABSTRACT

Pairs (A, L) with A a commutative algebra and L a Lie algebra acting on A by derivations, called Lie algops, are studied as algebraic structures over arbitrary fields of arbitrary characteristic. Lie algops possess modules and tensor products—and are considered with respect to a central simple theory.

The simplicity problem of determining the faithful unital simple Lie algops ( A, L ) is of interest since the corresponding Lie algebras AL are usually simple (Jordan, 2000 Jordan , D. A. ( 2000 ). On the simplicity of Lie algebras of derivations of commutative algebras . J. Algebra 228 : 580 – 585 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). For locally finite Lie algops, and up to purely inseparable descent, this problem reduces by way of closures to the closed central simplicity problem of determining those which are closed central simple.

The simplicity and representation theories for locally nilpotent separably triangulable unital Lie algops are of particular interest because they relate to the problems of classifying simple Lie algebras of Witt type and their representations. Of these, the simplicity theory reduces to that of Jordan Lie algops.

The main Theorems 7.3 and 7.4 reduce the simplicity and representation theories for Jordan Lie algops to the simplicity and representation theories for simple nil and toral Lie algops.