An Introduction To Lie Theory Through Path Geometry

Contents

Introduction

Why Yet Another Exposition on Lie Groups and Lie Theory?

Reading Guide

Informal Definitions for the Lie Group Axioms

Lie Groups: What They Are

Part 1: The Local View

Chapter 1: Connected Lie Groups – The Grounding Axioms

Chapter 2: Some Examples of Connected Lie Groups

Chapter 3: The Lie Algebra of the Lie Group and Canonical Co-ordinates

Chapter 4: The Adjoint Representation

Chapter 5: The Exponential Map

Chapter 6: The Exponential Map 2

Chapter 7: The Lie Bracket and the Little Adjoint Representation

Chapter 8: The Campbell Baker Hausdorff Theorem and The Dynkin Formula

Chapter 9: The Lie Group Topology

Chapter 10: Lie Groups as Manifolds: The Conventional Lie Group Definition

Chapter 11: Lie Groups as Manifolds: The Conventional Lie Group Definition 2

Chapter 12: The Lie Correspondence: Lie’s Analogies With Galois Theory

Part 2: The Global View: Global Topology

Chapter 13: Lie’s Third Theorem and Differential Equations in a Less-Than-Full-Rank Lie Algebra

Chapter 14: Lie Group Homotopy and Global Topology

Chapter 15: How Unique is the Lie Group Structure for a Given Lie Group?

Chapter 16: The Killing Form: A Powerful Insight into Global Lie Group Topology and Abstract Group Structure

Lie Groups: What They Do

Part 1: Lie Theoretic Systems Theory

Chapter 17: Parallel Parking a Car

Chapter 18: Of Cats and Their Most Wonderful Righting Reflex

Chapter 19: Generalised Space Curves, Coupled Waveguides and Quantum State Preparation

Still to Come

Translation with Notes: B. van der Waerden, “Stetigkeitssätze für halbeinfache Liesche Gruppen”

Translation with Notes: H. Freudenthal, “Die Topologie der Lieschen Gruppen als algebraisches Phänomen”

Topological Groups and Hilbert’s Fifth Problem

Lie Groups: Who They Are

The Classical Lie Groups

Classification of All Lie Algebras

The Exceptional Lie Groups

The Lorentz Group, the Möbius Group, $SL(2,\,\mathbb{C})^+$, $PSL(2,\,\mathbb{C})^+$ and the Celestial Sphere

Universal Covers 2: Of Spinors, Tensors and a Profound Mystery of Physics: Why Things Take Up Space

$SL(2,\,\,\mathbb{R})$, Hyperbolic Geometry, The Smith Chart and Ray Optics

Lie Groups: What They Do

Part 2: Representation Theory

Basic Representation Theory and Schur’s Lemma

Unitary Representations and the Wigner Classification

The Standard Model of Particle Physics