Finite gradings of special linear Lie algebras

In this paper, we study gradings of simple classical Lie algebras with arbitrary Abelian groups and the interconnection of such gradings and automorphism groups of Lie algebras. We give a complete classification of gradings of special linear Lie algebras that are specified by inner automorphisms in the case of an algebraically closed field of zero characteristic.     

Classification of three-dimensional zeropotent algebras over an algebraically closed field

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.     

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.     

李群的解析对合自同构对的分类

本文给出复单李代数的对合自同构对的分类,并利用它研究单连通,单李群的解析对合自同构对的分类.     

Convex topological algebras via linear vector fields and Cuntz algebras

A realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is in analogue to the classical Jordan—Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schrödinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.     

Invariant complex structures on four-dimensional solvable real lie groups

In this paper, invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups are classified where the dimension of the commutator is less than three. The resulting complex surfaces corresponding to these structures are also determined. The classification problem is reduced to determining certain complex “structure” subalgebras of the complexifications of the four-dimensional, solvable real Lie algebras. Most of the eleven types of non-abelian solvable real Lie algebras do have complex structure subalgebras; three do not. Only three types of algebras have solvable complex structure subalgebras, and only one possesses both abelian and solvable complex structure subalgebras. Each of the possible homogeneous surfaces is represented in the list of resulting manifolds.     

代数表示论的某些新进展

代数表示理论是代数学的一个新的重要分支，在近二十五年的时间里，这一理论有很大的发展，关于代数表示的基础理论的介绍可参见文献（１０１），本文主要从Ｈａｌｌ代数和拟遗传代数两个方面介绍代数表示论的一些最新进展，第一章给出了Ｈａｌｌ代数的基本理论及其方法，并且着重指出了利用这一理论和方法通过代数表示论去实现Ｋａｃ－Ｍｏｏｄｙ李代数及相应的量子包络代数，第二章介绍了拟遗传代数及其表示理论，以及这一理论与复     

Classification of three-dimensional solvable <Emphasis Type="Italic">p</Emphasis>-adic Leibniz algebras

The present paper is devoted to the study of low dimensional Leibniz algebras over the field of p-adic numbers. The classification up to isomorphism of three-dimensional Lie algebras over the integer p-adic numbers is already known [8]. Here, we extend this classification to solvable Lie and non-Lie Leibniz algebras over the field of p-adic numbers.     

Spectra for solvable Lie algebras of bundle endomorphisms

The aim of the paper is to investigate spectral properties of the Lie algebras corresponding to the symmetry groups of certain flags of vector bundles over a compact space. Under natural hypotheses, such Lie algebras are solvable, being in general infinite dimensional. The spectral theory of finite-dimensional solvable Lie algebras of operators is extended to this natural class of infinite-dimensional solvable Lie algebras. The discussion uses the language of continuous fields of -algebras. The flag manifolds in -algebraic framework are naturally involved here, they providing the basic method for obtaining flags of vector bundles. Received: 8 October 2001 / Revised version: 4 February 2002 / Published online: 6 August 2002 Research supported from the contract ICA1–CT–2000–70022 with the European Commission.     

The solution to the embedding problem of a (differential) Lie algebra into its Wronskian envelope

Any commutative algebra equipped with a derivation may be turned into a Lie algebra under the Wronskian bracket. This provides an entirely new sort of a universal envelope for a Lie algebra, the Wronskian envelope. The main result of this paper is the characterization of those Lie algebras which embed into their Wronskian envelope as Lie algebras of vector fields on a line. As a consequence we show that, in contrast to the classical situation, free Lie algebras almost never embed into their Wronskian envelope.     

The classification of two step nilpotent complex Lie algebras of dimension 8

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.     

Restricted Lie algebras with bounded cohomology and related classes of algebras

We determine the structure of restricted Lie algebras with bounded cohomology over arbitrary fields of prime characteristic. As a byproduct a classification of the serial restricted Lie algebras and the restricted Lie algebras of finite representation type is obtained. In addition, we derive complete information on the finite dimensional indecomposable restricted modules of these algebras over algebraically closed fields.     

Foundations of the Theory of Dual Lie Algebras

In this paper we introduce and study dual Lie algebras, i.e., Lie algebras over the algebra of dual numbers. We establish some fundamental properties of such Lie algebras and compare them with the corresponding properties of real and complex Lie algebras. We discuss the question of classification of dual Lie algebras of small dimension and consider the connection of dual Lie algebras with approximate Lie algebras.     

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

The aim of this paper is to classify (lócally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ² according to P.J. Olver [7].     

Differential equations uniquely determined by algebras of point symmetries

We continue to investigate strongly and weakly Lie remarkable equations, which we defined in a recent paper. We consider some relevant algebras of vector fields on ℝ^k (such as the isometric, affine, projective, or conformal algebras) and characterize strongly Lie remarkable equations admitted by the considered Lie algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 486–494, June, 2007.     

Invariant complex structures on solvable¶real Lie groups

In this paper, we classify the invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups with commutator of dimension three. The resulting complex surfaces corresponding to these structures are also determined. The classification is based on the determination of certain complex subalgebras of the complexifications of the corresponding real Lie algebras. Received: 17 December 1999 / Revised version: 1 May 2000     

Construction and classification of complex simple Lie algebras via projective geometry

We construct the complex simple Lie algebras using elementary algebraic geometry. We use our construction to obtain a new proof of the classification of complex simple Lie algebras that does not appeal to the classification of root systems.     

Lie algebras ofH-projective motions of Kähler manifolds of constant holomorphic sectional curvature

In the paper, the Lie algebras of infinitesimalH-projective transformations with2n-dimensional Kähler manifolds of constant holomorphic sectional curvature are considered. It is proved that these algebras are isomorphic to the realification of the complex Lie algebra $sl(n, \mathbb{C})$ , and their local realization in the form of an algebra of vector fields on a manifold is described.