Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

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.     

李超代数的泛包络代数同构问题

李超代数的一个性质P叫做关于泛包络代数的不变量,如果对于任意李超代数L,H,只要L具有性质P,并且泛包络代数U(L)和U(H)作为结合超代数是同构的,那么H亦具有性质P.通过讨论李超代数关于泛包络代数的不变量证明了:如果L的幂零长度不超过2,那么L和H是同构的.     

泛阶化李超代数

对于给定的负阶化李超代数K-,本文定义了K-型泛阶化李超代数并证明了它的存在性．进而引出阶化Cartan型李超代数,并且证得阶化Cartan型李超代数 W(m,n),K(m,n,ωA),S(m,n)和H(m,n)分别可以用某种泛阶化李超代数来刻画．     

Lie comodules and the constructions of Lie bialgebras

In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.     

The Core of a Locally Extended Affine Lie Algebra

Locally extended affine Lie algebras are a general version of extended affine Lie algebras. In this article, we completely describe the structure of the core of a locally extended affine Lie algebra. We prove that the core of a locally extended affine Lie algebra is a direct limit of Lie tori.     

The Lie Algebras in which Every Subspace s 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.     

Generalized Fuzzy Lie Ideals of Lie Algebras

In this paper we introduce new generalized fuzzy Lie ideals of Lie algebras and study some of their important properties.We characterize these generalized Lie ideals of Lie algebras by their level subsets.Some characterization of the generalized fuzzy Lie ideals of Lie algebras are also established.     

有限维单Cartan型模李超代数HO

本文构造了一族有限维Cartan型模李超代数-奇Hamilton模李超代数HO, 并证明了其单性.通过建立Cartan型模李超代数W,H,K,S和HO的维数公式,讨 论了奇Hamilton模李超代数HO与Cartan型模李超代数W,S,H,K的同构关系.     

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 $g\dot{+}\mathbb{C}id_g$, 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.     

线状李超代数Ln,m上的Yang-Baxter 方程

在特征零的代数闭域上,首先做出Ln,m 的一个空间的直和分解,从而将Ln,m 上的Yang-Baxter 方程的解分为若干情形。然后分别在每种情形下对Yang-Baxter 方程进行求解,进而得到了Ln,m 上的所有的Yang-Baxter方程的解的矩阵形式。     

c-Graded filiform Lie algebras

In this paper we generalize naturally graded filiform Lie algebras as well as filiform Lie algebras admitting a connected gradation of maximal length, by introducing the concept of c-graded complex filiform Lie algebras. We deal with the particular case of 3-graded filiform Lie algebras and we obtain their classification in arbitrary dimension. We finally show a link among derived algebras, graded filiform and rigid solvable Lie algebras.     

Single-Valued Neutrosophic Lie Algebras

A single-valued neutrosophic (SVN) set is a powerful general formal framework that generalizes the concept of fuzzy set and intuitionistic fuzzy set. In SVN set, indeterminacy is quantified explicitly, and truth membership, indeterminacy membership, and falsity membership are independent. In this paper, we apply the notion of SVN sets to Lie algebras. We develop the concepts of SVN Lie subalgebras and SVN Lie ideals. We describe some interesting results of SVN Lie ideals.     

On the volume of orbifold quotients of symmetric spaces

Key to H. C. Wang's quantitative study of Zassenhaus neighbourhoods of non-compact semisimple Lie groups are two constants that depend on the root system of the corresponding Lie algebra. This article extends the list of values for Wang's constants to the exceptional Lie groups and also removes their dependence on dimension. The first application is an improved upper sectional curvature bound for a canonical left-invariant metric on a semisimple Lie group. The second application is an explicit uniform positive lower bound for arbitrary orbifold quotients of a given irreducible symmetric space of non-compact type.     

Isometries of Quaternionic Lie Groups

We study connected Lie groups whose Lie algebra is obtained as the tensor product of a real associative algebra and the algebra of quaternions. It is proved that they carry a natural integrable -structure. We endow such quaternionic Lie groups with a left-invariant Hermitian metric and study the identity connected component of their isometry groups. The determination of such identity connected component is illustrated with a family of examples.     

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.     

关于完满的Lie超代数

In this paper, some properties of perfect Lie superalgebras are investigated. We prove that the derivation superalgebra of a centerless perfect Lie superalgebra of arbitrary dimension over a field of arbitrary characteristic is complete and we obtain a necessary and sufficient condition for the holomorph of a centerless perfect Lie superalgebra to be complete. Finally, some properties of perfect restricted Lie superalgebras are given.     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Towers of derivation for Lie rings and some results on complete Lie rings

The well-known tower theorem of groups (resp. Lie algebras) shows that the tower of automorphism groups (resp. derivation algebras) of a finite group (resp. a finite dimensional Lie algebra) with trivial center terminates after finitely many steps. We generalize these results for Lie rings, and present some necessary and sufficient conditions for Lie rings to be complete.     

TOWERS OF DERIVATION FOR LIE RINGS AND SOME RESULTS ON COMPLETE LIE RINGS

The well-known tower theorem of groups （resp. Lie algebras） shows that the tower of automorphism groups （resp. derivation algebras） of a finite group （resp. a finite dimensional Lie algebra） with trivial center terminates after finitely many steps. We generalize these results for Lie rings, and present some necessary and sufficient conditions for Lie rings to be complete.