我们研究η形变的AdS2×S2×T6超弦的Poisson-Lie对偶。 η变形的背景满足II型超重力方程的一般化。 我们针对（i）完整的psu1,12 $$ \ mathfrak {p} \ mathfrak {s} \ mathfrak {u} \ left（1，\ left.1 \ right | 2 \ 右）$$超代数，（ii）完整的玻色子代数和（iii）Cartan子代数，其相应的背景有望满足标准的II型超重力方程。 前两种情况的度量和B字段是相同的，并通过对AdS2×S2×T6上的λ变形模型的解析连续性给出，其中圆环未变形。 但是，RR通量和膨胀系数会有所不同。 着眼于第二种情况，我们显式地得出背景，并与已知的λ变形模型在II型超重力中AdS2×S2上的已知嵌入的解析继续一致。

Poisson-Lie对G / H对称空间sigma模型相对于简单Lie组G的η变形进行对偶化，从而推测出相关λ变形模型的解析连续性。 在本文中，我们研究了何时可以将η变形模型相对于G的子组G0进行对偶化。从对复杂化组的一阶作用开始，并整合与不同子代数相关的自由度，我们发现有可能 当G0关联到子Dynkin图时进行对偶。 也可以包括由其余的Cartan发电机生成的其他U1因子。 最终的构造在单个框架中统一了关于G的Poisson-Lie对偶和η变形的完全阿贝尔对偶，并且在两种情况下都采用了单模积分的代数。 我们推测将这些结果扩展到路径积分形式可以为为什么η形变的AdS5×S5超弦不是单环Weyl不变提供一个解释，也就是说，联轴器不能解决IIB型超重力方程，但其完全阿贝尔方程 对偶和λ变形模型。

李群公式在2d和3d空间中的推导，文档详细讲解了李群表示的空间变换如何用于机器人和视觉领域。

Lie.Groups .Lie.Algebras and Representations Brian.C..Hall.

CCD调用电脑自带摄像头（lie）.zip

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor- porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (1) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry. (3) The isomorphism theorem is proved first in an elementary way (Theorem 14.2), but later obtained again as a corollary of Serre's Theorem (18.3), which gives a presentation by generators and relations. (4) From the outset, the simple algebras of types A, B, C, D are empha- sized in the text and exercises. (5) Root systems are treated axiomatically (Chapter III), along with some of the theory of weights. (6) A conceptual approach to Weyl's character formula, based on Harish-chandra's theory of "characters" and independent of Freudenthal's multiplicity formula (22.3), is presented in 23 and 24. This is inspired by D.-N. Verma's thesis, and recent work of I. N. Bernstein, I. M. Gel'fand, S. I. Gel'fand. (7) The basic constructions in the theory of Chevalley groups are given in Chapter VII, following lecture notes of R. Steinberg. I have had to omit many standard topics (most of which I feel are better suited to a second course), e.g., cohomology, theorems of Levi and Mal'cev, theorems of Ado and Iwasawa, classification over non-algebraically closed fields, Lie algebras in prime characteristic. I hope the reader will be stirn u- lated to pursue these topics in the books and articles listed under References, especially Jacobson [1], Bourbaki [1], [2], Winter [1], Seligman [1].

Lie 群和Lie 代数的理论是近代数学中的一个重要分支是挪威数学家M.S.Lie 1842-1899 在十九世纪后期创建的由于受Lagrange Abel Galois 等学者用群论方法 研究代数方程求解问题得到巨大成功的启发Lie 提出了用变换群的方法来研究微分方程的 求解问题及用无穷小变换来研究变换群的方法近代的Lie 群与Lie 代数理论就是在Lie 的 开创性工作的基础上发展起来的群变换群的概念起源于对几何图像对称性的研究虽 然历史悠久但未成为一种解决问题的系统方法这一情况到了十八世纪后期才发生了本质 的变化法国数学家J.Lagrange(1736-1813)在研究代数方程求解问题时认识到根的排列与 置换理论是解代数方程的关键所在开创了用置换群的理论来研究代数方程求解问题的新阶 段在此基础上挪威数学家N.H.Abel(1802-1829)与法国数学家E.Galois(1811-1832)发展 和应用了群论的方法彻底解决了代数方程用代数方法求解问题关于这方面的进一步介绍 有兴趣的学者可以参看附录1 用根的置换理论解二三次代数方程 与代数方程有关的置换群是有限群即由有限个元素构成的群对这种群的研究纯属 代数问题而Lie 引进的与微分方程有关的变换群则是由有限个连续参数所确定的变换所构 成的无限群这种确定群的元素的连续变化的参数可以看成广义的坐标所以Lie 研究的变 换群除了群的结构外还具有流形的结构其元素可以看成是流形上的点关于流形的概 念可参看李世雄. 波动方程的高频近似与辛几何. 第四章因而Lie 群是代数几何与分 析的有机结合其理论和方法对近代数学的许多分支有重要的影响和作用

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