n-Extended Lorentzian Kac–Moody algebras

Letters in Mathematical Physics - We investigate a class of Kac–Moody algebras previously not considered. We refer to them as n-extended Lorentzian Kac–Moody algebras defined by their...     

A class of Lorentzian Kac–Moody algebras

We consider a natural generalisation of the class of hyperbolic Kac–Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they possess a real principal so(1,2) subalgebra. Our class of algebras include the Lorentzian Kac–Moody algebras that have recently been proposed as symmetries of M-theory and the closed bosonic string.     

Geometric approach to Kac–Moody and Virasoro algebras

In this paper we show the existence of a group acting infinitesimally transitively on the moduli space of pointed-curves and vector bundles (with formal trivialization data) and whose Lie algebra is an algebra of differential operators. The central extension of this Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely a semidirect product of a Kac–Moody algebra and the Virasoro algebra. As an application of this geometric approach, we give a local Mumford-type formula in terms of the cocycle associated with this central extension. Finally, using the original Mumford formula we show that this local formula is an infinitesimal version of a general relation in the Picard group of the moduli of vector bundles on a family of curves (without any formal trivialization).     

Embeddings of Hyperbolic Kac–Moody Algebras into E 10

We show that the rank 10 hyperbolic Kac–Moody algebra E ₁₀ contains every simply laced hyperbolic Kac–Moody algebra as a Lie subalgebra. Our method is based on an extension of earlier work of Feingold and Nicolai.      

Double Bruhat Cells in Kac–Moody Groups and Integrable Systems

We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter elements in the affine Weyl group. In the process we extend various results on double Bruhat cells in simple algebraic groups to the setting of Kac–Moody groups. We also generalize some fundamental results in Poisson–Lie theory to the setting of ind-algebraic groups, which is of interest beyond our immediate applications to integrable systems.     

Real forms of extended Kac–Moody symmetries and higher spin gauge theories

We consider the relation between higher spin gauge fields and real Kac–Moody Lie algebras. These algebras are obtained by double and triple extensions of real forms \mathfrakg₀{\mathfrak{g}_0} of the finite-dimensional simple algebras \mathfrakg{\mathfrak{g}} arising in dimensional reductions of gravity and supergravity theories. Besides providing an exhaustive list of all such algebras, together with their associated involutions and restricted root diagrams, we are able to prove general properties of their spectrum of generators with respect to a decomposition of the triple extension of \mathfrakg₀{\mathfrak{g}_0} under its gravity subalgebra \mathfrakgl(D,\mathbb R){\mathfrak{gl}(D,\mathbb {R})} . These results are then combined with known consistent models of higher spin gauge theory to prove that all but finitely many generators correspond to non-propagating fields and there are no higher spin fields contained in the Kac–Moody algebra.     

The Principal SO(1,2) Subalgebra of a Hyperbolic Kac–Moody Algebra

The analog of the principal SO(3) subalgebra of a finite-dimensional simple Lie algebra can be defined for any hyperbolic Kac–Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact algebra SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2). With the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated exponents are complex and noninteger.     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

On the Endomorphisms of Weyl Modules over Affine Kac–Moody Algebras at the Critical Level

We present an independent short proof of the recently established result that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from our results about the shift of argument subalgebras.     

椭圆量子群极小表示的系数代数

给出了对应于A(1)n–1面模型的玻尔兹曼权的L矩阵及FelderAn系列椭圆量子群的极小表示(所有矩阵元均为c数)的系数代数.这个代数满足杨–Baxter方程.同时,对此代数还给出了一组PBW基.     

KAC-MOODY ALGEBRA FOR TWO DIMENSIONAL PRINCIPAL CHIRPL MODELS

A Darbou transformation depending on single continuous parameter t is constructed for a principal chiral field. The transformation forms a nonlinear representation of the group for any fixed value of t. Part of the kernel in the Riemann- Hilbert transform is shown to be-related to the Darboux trans- formation with its generators forming a Kac-Moody algebra. Conserved currents associated with the Kac-Moody algebra of the linearized equations and the Noether current for the group transformations with fixed value of t are obtained.     

一般情况下二维非线性σ模型的等时Kac-Moody代数

在本文中对一般的紧致单纯群和耦合常数λ₀的任意值证明了带Wess-Zumino项的二维非线性σ模型的等时流代数构成带中心荷的Kac-Moody代数。 关键词：     

THE ADJOINT REPRESENTATION OF QUANTUM ALGEBRA Uq(sl(2))

Starting from any representation of the Lie algebra on the finite dimensional vector space V we can construct the representation on the space Aut(V). These representations are of the type of ad. That is one of the reasons, why it is important to study the adjoint representation of the Lie algebra on the universal enveloping algebra U(). A similar situation is for the quantum groups U_q(). In this paper, we study the adjoint representation for the simplest quantum algebra U_q(sl(2)) in the case that q is not a root of unity.     

新的Lax可积发展方程族及其无限维双-哈密顿结构

基于一个新的具有三个位势函数(q,r,s)的等谱问题,获得了一族新的含有一个任意函数的Lax可积发展方程.特别地,当位势函数s取不同的函数时,这个方程族约化为若干类型的方程组,进一步利用迹恒等式,给出了这些方程组的双哈密顿结构,并且证明它们是Liouville可积的.此外,给出了守恒密度和对称. 关键词： 等谱问题 哈密顿结构 Lax可积 Liouville可积     

作用表示波动方程中与面有关项

这篇短文的内容是:(i)对於量子场论中的i(δψ[σ])/(δσ(x))=V(x,σ)ψ[σ] 如何由寻常的“曲面上的薛定谔方程”导出,作一个较严格的讨论,以及 (ii)讨论上式中的V(x,σ)在什么条件下不包含有σ。我们证明了所需的条件是 (?L^I)/(?φ_μ) (?L^I)/(?φ_ν)=(?²L)/(?φ_μ?φ_ν)F(φ,φ_ρ),式中L,L^I代表总拉格朗日及作用拉格朗日,φ代表场量,φ_μ代表φ/x^μ,F(φ,φ_ρ)代表φ及φ_μ的一个任意函数。     

热漫散射贡献的高能电子衍射吸收结构因子的Doyle-Turner解析表述

给出了一种数值计算高能电子衍射吸收结构因子的方法,并利用模拟退火结合保温相变以及线性最小二乘拟合等算法,对所计算出的吸收结构因子数值进行了含10个参数的Doyle-Turner解析表述的拟合,作为示例,给出Be,AlAs,Ag,Au等5个元素的拟合结果和精度,并作出Al的绝对偏差量和相对偏差量曲线     

GROUP PROPERTY AND ALGEBRA STRUCTURE FOR A KIND OF BÄCKLUND TRANSFORMATIONS

We explain further the group property and algebra struc-tore for a kind of Bäcklund transformations. If the Lax repre- sentation of a given nonlinear (evolution) system is dependent on a complex spectral parameter fractional Zinearly, the Back-land transformations with respect to the principal pseudopoten- tials (fundamental solutions) of the Lax representation can be identified with solutions of the regular matrix Riemann-Hilbert problem. It is shown that this kind of Riemann-HIbert probl em constructs an infinite dimensional Lie group and its infinitesi- mal form presents itself the infinite dimensional algebra of Kac-Moody type.     

Clifford代数很值得在物理学中推广

Ｃｌｉｆｆｏｒｄ代数（几何代数或多重矢运算）比通常数学工具诸如矢量、张量、旋量及外微分形式有很多优越性。作为介绍性章这里选取刚体转动及放置参考系作为例子，以展示此种数学工具的优越性。中基本的几何代数运算及两个例子很容易在一般力学课程中讲授。     

Dyson玻色子表示中解的讨论

本文对Dyson玻色子表示中的解进行了讨论。指出解的物理内容包括动力学对称性不依赖于具体表示,可通过将D的矩阵化为三角矩阵的方法求得所需的解。在某些具体情形下这种方法非常简便。 关键词：     

Zn对称统计模型的量子代数结构常数与杨-Baxter关系

本文对统计中的杨-Baxter方程的Z_n对称解进行了仔细的分析,自然地得到了相应的量子代数的结构常数,并对这个解给出了一个直接的证明。 关键词：