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.

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.

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.