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.     

可换环上辛代数与一般线性李代数之间的中间李代数

本文研究了含幺可换环上一般线性李代数的子代数结构.通过构造特殊矩阵并利用这些矩阵进行计算, 得到了任意含幺可换环上辛代数与一般线性李代数之间的所有中间李代数的形式.并且有利于研究可换环上相应的典型群的子群结构.     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].     

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.     

On functional identities in left ideals of prime rings

In the present paper we describe surjective Lie and Jordan maps onto left ideals of prime noncommutative rings. Further, we describe bijective linear maps of left ideals of centrally closed prime algebras preserving commutativity.     

算子李代数的一些性质

算子群作为群的推广,算子群在群论里有许多应用.类似地,作为算子群和李代数的推广,算子李代数将会有许多应用.给出了算子李代数的一些性质,得到了算子李代数半单性的充分必要条件.同时得到算子李代数半单性与非退化killing型的关系.     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

关于完备李群与完备李代数

孟道骥等对完备李代数作了系统的研究并已获得很多基本和重要的结果。本文给出完备李群与完备李代数的某些关系。     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Fixed points of automorphisms of Lie rings and locally finite groups

Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group C _P(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 706-723, November-December, 1995.Supported by RFFR grant No. 94-01-00048-a and by ISF grant NQ7000.     

Classification of Lie algebras of specific type in complexified Clifford algebras

We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These 16 Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical matrix Lie algebras in the cases of arbitrary dimension and signature. We present 16 Lie groups: one Lie group for each Lie algebra associated with this Lie group. We study connection between these groups and spin groups.     

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.     

Heisenberg Lie Color Algebras

In this article, we give a sufficient condition for a Lie color algebra to be complete. The color derivation algebra Der(?) and the holomorph L of finite dimensional Heisenberg Lie color algebra ? graded by a torsion-free abelian group over an algebraically closed field of characteristic zero are determined. We prove that Der(?) and Der(L) are simple complete Lie color algebras, but L is not a complete Lie color algebra.     

可换环上一些上三角矩阵李代数的导子

设R是任意含单位元的可换环,t是R上n×n上三角矩阵组成的李代数,b是R上迹为零的n×n上三角矩阵组成的李代数,本文明确给出了t和b的导子代数.     

New simple modular Lie superalgebras as generalized prolongs

Over algebraically closed fields of characteristic p > 2, —prolongations of simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. We discover several new simple Lie superalgebras, serial and exceptional, including super versions of Brown and Melikyan algebras, and thus corroborate the super analog of the Kostrikin-Shafarevich conjecture. Simple Lie superalgebras with 2 × 2 Cartan matrices are classified.     

Positive definite Minkowski Lie algebras and bi-invariant Finsler metrics on Lie groups

In this paper, we introduce the notion of a Minkowski Lie algebra, which is the natural generalization of the notion of a real quadratic Lie algebra (metric Lie algebra). We then study the positive definite Minkowski Lie algebras and obtain a complete classification of the simple ones. Finally, we present some applications of our results to Finsler geometry and give a classification of bi-invariant Finsler metrics on Lie groups. This work was supported by NSFC (No.10671096) and NCET of China.