The launch of this Advanced Lectures in Mathematics series is aimed at keeping mathematicians informed of the latest developments in mathematics, as well as to aid in the learning of new mathematical topics by students all over the world. Each volume consists of either an expository monograph or a collection of signifi- cant introductions to important topics. This series emphasizes the history and sources of motivation for the topics under discussion, and also gives an overview of the current status of research in each particular field. These volumes are the first source to which people will turn in order to learn new subjects and to dis- cover the latest results of many cutting-edge fields in mathematics.
目录
1 Introduction 1.1 Outline of the book 1.2 Suggestions for further reading 1.3 Notations, background, conventions 2 Bilinear Forms, Quadratic Forms and Their Isometry Groups. 2.1 Standard results on quadratic forms and reflections 2.1.1 Principal ideal domains (PIDs) 2.2 Linear algebra 2.2.1 Interpretation of nonsingularity 2.2.2 Extension of scalars 2.2.3 Cyclicity of the values of a rational bilinea.r form 2.2.4 Gram matrix 2.3 Discriminant group 2.4 Relations between a lattice and sublattices 2.5 Involutions on quadratic spaces 2.6 Standard results on quadratic forms and reflections, II 2.6.1 Involutions on lattices 2.7 Scaled isometries: norm doublers and triplers 3 General Results on Finite Groups and Invariant Lattices 3.1 Discreteness of rational lattices 3.2 Finiteness of the isometry group 3.3 Construction of a G-invariant bilinear form 3.4 Semidirect products and wreath products 3.5 Orthogonal decomposition of lattices 4 Root Lattices of Types A, D, E 4.1 Background from Lie theory 4.2 Root lattices, their duals and their isometry groups 4.2.1 Definition of the AN lattices ……
