Restricted Lie 2-algebras

In this article, we introduce the notions of restricted Lie 2-algebras and crossed modules of restricted Lie algebras, and give a series of examples of restricted Lie 2-algebras. We also construct restricted Lie 2-algebras from A(m)-algebras, restricted Leibniz algebras, restricted right-symmetric algebras. Finally, we prove that there is a one-to-one correspondence between strict restricted Lie 2-algebras and crossed modules of restricted Lie algebras.     

Restricted Lie algebras all whose elements are semisimple

People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra.     

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b ₂-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.     

Criteria for solvability and supersolvability in Leibniz algebras

Leibniz algebras are certain generalization of Lie algebras. Recently, analyzing the structure of subalgebras, David Towers gave some criteria for the solvability and supersolvability of Lie algebras. In this paper we define analogues concepts for Leibniz algebras and extend some of these results on solvability and supersolvability to that of Leibniz algebras.     

L-quadri代数

Quadri代数是由Aguiar和Loday引入的一类著名的Loday代数.在本文中,我们引入具有4个运算的L-quadri代数的概念,它满足广义左对称性,其4个运算的和的换位运算是Lie代数,并且是quadri代数的Lie代数类似结构.任何quadri代数是L-quadri代数,并且L-quadri代数可以放在Lod...     

The Finite Dimension Property of the Irreducible Representations of Noetherian p-Lie Algebras

We prove that the irreducible representations are finite-dimensional of the almost solvable restricted Lie algebras with the ascending chain condition for p-subalgebras over a perfect field.     

The Jordan socle and finitary Lie algebras

In this paper we introduce the notion of Jordan socle for nondegenerate Lie algebras, which extends the definition of socle given in [A. Fernández López et al., 3-Graded Lie algebras with Jordan finiteness conditions, Comm. Algebra, in press] for 3-graded Lie algebras. Any nondegenerate Lie algebra with essential Jordan socle is an essential subdirect product of strongly prime ones having nonzero Jordan socle. These last algebras are described, up to exceptional cases, in terms of simple Lie algebras of finite rank operators and their algebras of derivations. When working with Lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0, the exceptions disappear and the algebras of derivations are computed.     

李超代数与Frobenius代数

利用限制李超代数的新定义,给出了李超代数的p-包络的一些相关结果,并将李代数中表示理论的一些结果推广到李超代数上,进而研究了限制李超代数与Frobenius代数的关系．     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

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.     

Completion of restricted Lie algebras

We study pro-‘finite dimensional finite exponent’ completions of restricted Lie algebras over finite fields of characteristicp. These compact Hausdorff topological restricted Lie algebras, called pro- restricted Lie algebras, are the restricted Lie-theoretic analogues of pro-p groups. A structure theory for pro- restricted Lie algebras with finite rank is developed. In particular, the centre of such a Lie algebra is shown to be open. As an application we examinep-adic analytic pro-p groups in terms of their associated pro- restricted Lie algebras. Supported by NSERC of Canada.     

素特征李代数概述

近２０年来，国内外在素特征李代数的研究中取得了许多突破性的进展，本文是近年来国内外在这一领域的研究成果的一个综述，第一部分对Ｃａｒｔａｎ型李代数的定义以及主要结构的回顾，然后，重点介绍了两个重的分类定理，即，素特征域上的有限维局限单李代数的分类定理以及素特征域上的有际维单李代的分类定理。由于后一分类是一前一分类的基础上完成的，所以，本文对第一个分类定理的证明作了一个简单的介绍，在第三部分中，对素特     

COMPLETE LIE ALGEBRAS WITH l-STEP NILPOTENT RADICALS

The authors first give a necessary and sufficient condition for some solvable Lie algebras with l-step nilpotent radicals to be complete, and then construct a new class of infinite dimensional complete Lie algebras by using the modules of simple Lie algebras. The quotient algebras of this new constructed Lie algebras are non-solvable complete Lie algebras with l-step nilpotent radicals.     

Poisson color algebras of arbitrary degree

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g₀, where g₀ is some element of the grading group G such that g₀ = 0 or 4g₀≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras.     

Some Properties On Isoclinism in n-Lie Algebras

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.     

Equations des algèbres Lie triple qui sont des algèbres train

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank $\le 4$ with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of $e$-stable power-associative algebras.We also prove that the algebra obtained by $2$-gametization process of a Lie triple algebra is a Lie triple one.     

关于局部顶点李代数的一些注记

本文研究局部顶点李代数与顶点代数之间的关系.利用由局部顶点李代数构造顶点代数的方法,定义局部顶点李代数之间的同态,证明了同态可以唯一诱导出由局部顶点李代数构造所得到的顶点代数之间的同态.     

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.     

Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.     

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.