Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ? on 𝔤 is said to preserve solvability in both directions if ?, as well as ?^?1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ? on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.     

交换环上某些线性李超代数的理想

设Ｒ 是有１的交换环，２是Ｒ 的单位．设Ｌ 为环Ｒ 上的特殊线性李超代数或正交 辛李超代数．讨论了Ｌ 的理想与Ｒ 的理想的关系，证明了Ｌ 的所有理想都是标准的．     

Lie Regular Generators of General Linear Groups

In this article, we introduce the idea of Lie regular elements and study 2 × 2 Lie regular matrices. It is shown that the linear groups GL(2, ?_{2 ⁿ} ), GL(2, ? _{p ⁿ} ), and GL(2, ?_2p ) (where p is an odd prime) can be genrated by Lie regular matrices. Presentations of linear groups GL(2, ?₄), GL(2, ?₆), GL(2, ?₈), and GL(2, ?₁₀) are also given.     

整环上二阶线性李代数的自同构

设Ｒ是有１的交换环，２是Ｒ的单位．本文决定了Ｒ上李代数ｓｌ２（Ｒ）的理想．进而，若Ｒ是整环，本文决定了ｓｌ２（Ｒ）与ｇｌ２（Ｒ）的自同构形式．     

Vertex Representations of Toroidal Special Linear Lie Superalgebras

Based on the loop-algebraic presentation of 2-toroidal Lie superalgebras, a free field representation of toroidal Lie superalgebras of type A(m, n) is constructed using both vertex operators and bosonic fields.     

可换环上一些线性李代数的扩代数

设R是任意含单位元的可换环,gl(n,R)是R上n级一般线性李代数.t表示gl(n,R)中所有上三角矩阵组成的子代数,d表示gl(n,R)中所有对角矩阵组成的子代数.本文将分别确定t在gl(n,R)中的扩代数和d在t中的扩代数.     

套子代数上线性映射的Lie不变子空间

本文引入了Banach代数上线性映射的Lie不变子空间,给出了因子VonNeumann代数中套子代数上以导子空间为Lie不变子空间的线性映射的一般形式,研究了Lie导子与Lie自同构的概念及了Lie导子与Lie自同构半群的关系.     

超交换环上的一般线性李超代数的极大阶化子代数

In this paper, we determine all maximal graded subalgebras of the general linear Lie superalgebras containing the standard Cartan subalgebras over a unital supercommutative superring with 2 invertible.     

High Order Optimized Geometric Integrators for Linear Differential Equations

In this paper new integration algorithms based on the Magnus expansion for linear differential equations up to eighth order are obtained. These methods are optimal with respect to the number of commutators required. Starting from Magnus series, integration schemes based on the Cayley transform an the Fer factorization are also built in terms of univariate integrals. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is tested on some illustrative examples.     

线性中立型单延迟系统李代数稳定性判据

研究了线性中立型单延迟微分系统的稳定性.从矩阵李代数可解性角度,推导出新的简单的延迟独立稳定性判据.该新判据的重要意义和优越性在于首次突破了以往大多数相关文献的稳定性判据在应用上受条件‖C‖<1,ρ(|C|)<1或ρ(|N|)<1的限制,从而首次成功确定了在‖C‖≥,ρ(|C|)≥1和ρ(|N|)≥1的情形下中立型延迟微分系统渐近稳定性.最后,通过两个例子显示了新判据的优越性.     

Numerical Schemes for Linear and Non-Linear Enhancement of DW-MRI

We consider the linear and non-linear enhancement of diffusion weighted magnetic resonance images (DW-MRI) to use contextual information in denoising and inferring fiber crossings. We describe the space of DW-MRI images in a moving frame of reference, attached to fiber fragments which allows for convection-diffusion along the fibers. Because of this approach, our method is naturally able to handle crossings in data. We will perform experiments showing the ability of the enhancement to infer information about crossing structures, even in diffusion tensor images (DTI) which are incapable of representing crossings themselves. We will present a novel non-linear enhancement technique which performs better than linear methods in areas around ventricles, thereby eliminating the need for additional preprocessing steps to segment out the ventricles. We pay special attention to the details of implementation of the various numerical schemes.     

C*-Algebraic Intertwiners for Degenerate Principal Series of Special Linear Groups

The author constructs unitary intertwiners for degenerate C＊-algebraic uni- versal principal series of SL（n＋ 1） over a local field by explicitely normalizing standard intertwining integrals at the level of Hilbert modules.     

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.     

On the Implementation of the Method of Magnus Series for Linear Differential Equations

The method of Magnus series has recently been analysed by Iserles and Nørsett. It approximates the solution of linear differential equations y = a(t)y in the form y(t) = e ^(t) y ₀, solving a nonlinear differential equation for by means of an expansion in iterated integrals of commutators. An appealing feature of the method is that, whenever the exact solution evolves in a Lie group, so does the numerical solution.The subject matter of the present paper is practical implementation of the method of Magnus series. We commence by briefly reviewing the method and highlighting its connection with graph theory. This is followed by the derivation of error estimates, a task greatly assisted by the graph-theoretical connection. These error estimates have been incorporated into a variable-step fourth-order code. The concluding section of the paper is devoted to a number of computer experiments that highlight the promise of the proposed approach even in the absence of a Lie-group structure.     

Triple products on  n

Triple products in  _n whose related algebra is  _n itself or  _n are classified up to isomorphism. This classification is obtained using the intimate relation between triple products and Lie (super)algebra structures.     

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.     

A Sufficient and Necessary Condition for a Solvable Lie Algebra to Be Complete

In this paper,we give a sufficient and neccesary condition under which a solvable Lie algebra is complete.     

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.