Geometry Algorisms of Dynkin Diagrams in Lie Group Machine Learning

This paper uses the geometric method to describe Lie group machine learning(LML)based on the theoretical framework of LML,which gives the geometric algorithms of Dynkin diagrams in LML.It includes the basic conceptions of Dynkin diagrams in LML,the classification theorems of Dynkin diagrams in LML,the classification algorithm of Dynkin diagrams in LML and the verification of the classification algorithm with experimental results.

Root Systems Lie Algebras

A root system is any collection of vectors that has properties that satisfy the roots of a semi simple Lie algebra. If g is semi simple, then the root system A, (Q) can be described as a system of vectors in a Euclidean vector space that possesses some remarkable symmetries and completely defines the Lie algebra of g. The purpose of this paper is to show the essentiality of the root system on the Lie algebra. In addition, the paper will mention the connection between the root system and Ways chambers. In addition, we will show Dynkin diagrams, which are an integral part of the root system.

李代数D_n和无穷维李代数D_n的相因子的确定

运用邓金链图列举了Ｄｎ的各级非零根并讨论了这些根的一些特性．在Ｊｏ－ｃｏｂｉ恒等式和反对称关系εα，β＝－εβ，α的基础上找到了相因子的约束条件．适当选择ｅα系数和εαβ的位相得到满足约束条件的Ｄｎ的相因子．讨论了Ｄｎ与Ｄｎ的相因子的对应关系，从而相应地得到了无穷维李代数Ｄｎ的相因子．

李代数A_n和无穷维李代数A_n的相因子的确定

运用邓金链图列举了Ａｎ的各级非零根并讨论了这些根的一些特性．在Ｊｏ－ｃｏｂｉ恒等式和反对称关系εα，β＝－εβ，α的基础上找到了约束条件．适当选择ｅα系数和εαβ的位相得到满足约束条件的Ａｎ的相因子．构造出Ａ１与Ａ２的根图．借助根图讨论了Ａｎ与Ａｎ的相因子的对应关系。

高阶双曲型Kac-Moody代数的极小虚根

根据双曲型Kac-Moody代数的极小虚根的基本性质,结合双曲型Kac-Moody代数对应的广义Cartan矩阵的Dynkin图的特征,给出了n (5≤n≤10)阶双曲型Kac-Moody代数的全部极小虚根。

一类双曲型Kac—Moody代数的特殊虚根系

虚根是Ｋａｃ—Ｍｏｏｄｙ代数的重要概念。Ｂｅｎｎｅｔｔ［１］首先定义了所谓的特殊虚根，进而对其进行了深入的研究，最后对不可分解的不定型Ｋａｃ—Ｍｏｏｄｙ代数ｇ（Ａ）给出了特殊虚根的分类定理。作者计算了一类双曲型Ｋａｃ—Ｍｏｏｄｙ代数的所有基本特殊虚根，从而得到了此类代数的特殊虚根系。

亚双曲型广义Cartan矩阵的分类

本文解决了一类非双曲型的不定型（亚双曲型）广义Ｃａｒｔａｎ矩阵的分类问题．证明了亚双曲型矩阵的阶数范围，并做出了阶数为３，５，６的亚双曲型广义Ｃａｒｔａｎ矩阵的Ｄｙｎｋｉｎ图．

On Ellipsoids Attached to Root Systems

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.

Shuhan图分类及其应用

Shuhan图是一个图论、Kac-Moody代数和矩阵论等学科的交叉研究对象.首先给出了Shuhan图的定义,再利用图论的分类方法得到了所有连通的Shuhan图:A_(n)^((a))(n≥2)、D_(n)^((a))(n≥4)、E_(6)^((a))、E_(7)^((a))、E_(8)^((a)),这证明了连通的非扭单边仿射Lie代数对应的Dynkin图恰好是全部的最小点数为1的连通Shuhan图.进一步地,由Shuhan图的性质可知,任意Shuhan图都是由若干连通Shuhan图并在一起的,这就给出了Shuhan图的完整分类.

初等表示的权的标积

半单李代数的素根可以用适当的初等表示的权来表示,反过来,初等表示的权也可以用素根来表示。在适当地选取了半单李代数的根系之后,就可算出初等表示权的标积。本文利用Cartan矩阵给出了一种计算半单李代数初等表示的权的标积的方法,并由此算出了二次Casimir算符的本征值。

The Mutation Game,Coxeter-Dynkin Graphs,and Generalized Root Systems

We introduce the mutation game on a directed multigraph,which is dual to Mozes5 numbers game.This new game allows us to create geometric and combinatorial structure that allows generalization of root systems to more general graphs.We interpret Coxeter-Dynkin diagrams in this multigraph context and exhibit new geometric forms for the associated root systems.

Generalized McKay Quivers of Rank Three

For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi- definiteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL3 （C） are explicitly described and classified based on representation theory.

Injectivity Radius and Cartan Polyhedron for Simply Connected Symmetric Spaces

The author explores the relationship between the cut locus of an arbitrary simply connected and compact Riemannian symmetric space and the Cartan polyhedron of corresponding restricted root system,and computes the injectivity radius and diameter for every type of irreducible ones.