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This section contains free e-books and guides on Lie Algebra, some of the resources in this section can be viewed online and some of them can be downloaded.

Lie Algebra Books
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Introduction to Lie Groups by Alistair Savage

PDF | 111 Pages | English
This note focus on the so-called matrix Lie groups since this allows us to cover the most common examples of Lie groups in the most direct manner and with the minimum amount of background knowledge. Topics covered includes: Matrix Lie groups, Topology of Lie groups, Maximal tori and centres, Lie algebras and the exponential map, Covering groups.

 
Lie Groups Representation Theory and Symmetric Spaces

PDF | 178 Pages | English
This note covers the following topics: Fundamentals of Lie Groups, A Potpourri of Examples, Basic Structure Theorems, Complex Semisimple Lie algebras, Representation Theory, Symmetric Spaces.

 
Lie groups and Lie algebras by Wilfried Schmid

PDF | 115 Pages | English
This note covers the following topics: Geometric preliminaries, The Lie algebra of a Lie group, Lie algebras, Geometry of Lie groups, The Universal Enveloping Algebra, Representations of Lie groups, Compact Lie groups, Root systems, Classificiation of compact Lie groups, Representations of compact Lie groups.

 
Lie Algebras and Representation Theory

PDF | 102 Pages | English
The aim of this note is to develop the basic general theory of Lie algebras to give a first insight into the basics of the structure theory and representation theory of semi simple Lie algebras. Topics covered includes: Group actions and group representations, General theory of Lie algebras, Structure theory of complex semisimple Lie algebras, Cartan subalgebras, Representation theory of complex semisimple Lie algebras, Tools for dealing with finite dimensional representations.

 
Introduction to Lie algebras

Online | NA Pages | English
In these lectures we will start from the beginning the theory of Lie algebras and their representations. Topics covered includes: General properties of Lie algebras, Jordan-Chevalley decomposition, semisimple Lie algebras, Classification of complex semisimple Lie algebras, Cartan subalgebras, classification of connected Coxeter graphs and complex semisimple Lie algebras, Poicare-Birkhoff-Witt theorem.

 
Introduction to Lie Groups and Lie Algebras

PDF | 177 Pages | English
This book covers the following topics: Lie Groups:Basic Definitions, Lie algebras, Representations of Lie Groups and Lie Algebras, Structure Theory of Lie Algebras, Complex Semisimple Lie Algebras, Root Systems, Representations of Semisimple Lie Algebras, Root Systems and Simple Lie Algebras.

 
Lecture Notes on Lie Algebras and Lie Groups

PDF | 150 Pages | English
This book covers the following topics: Elements of Group Theory, Lie Groups and Lie Algebras, Representation theory.

 
Semi Simple Lie Algebras and Their Representations

PDF | 164 Pages | English
The present volume is intended to meet the need of particle physicists for a book which is accessible to non-mathematicians. The focus is on the semi-simple Lie algebras, and especially on their representations since it is they, and not just the algebras themselves, which are of greatest interest to the physicist. Topics covered includes:The Killing Form, The Structure of Simple Lie Algebras, A Little about Representations, Structure of Simple Lie Algebras, Simple Roots and the Cartan Matrix, The Classical Lie Algebras, The Exceptional Lie Algebras, Casimir Operators and Freudenthal’s Formula, The Weyl Group, Weyl’s Dimension Formula, Reducing Product Representations, Subalgebras and Branching Rules.

 
Lecture notes in Lie Algebras

Online | NA Pages | English
This note covers the following topics: Universal envelopping algebras, Levi's theorem, Serre's theorem, Kac-Moody Lie algebra, The Kostant's form of the envelopping algebra and A beginning of a proof of the Chevalley's theorem.

 
Notes For Lie algebras

PDF | 106 Pages | English
This note covers the following topics: Ideals and homomorphism, Nilpotent and solvable Lie algebras , Jordan decomposition and Cartan's criterion, Semisimple Lie algebras and the Killing form, Abstract root systems, Weyl group and Weyl chambers, Classification of semisimple Lie algebras , Exceptional Lie algebras and automorphisms, Isomorphism Theorem, Conjugacy theorem.

 
Orbital Integrals on Reductive Lie Groups and Their Algebras

PDF | 194 Pages | English
This is an open source book written by Francisco Bulnes. The purpose of this book is to present a complete course on global analysis topics and establish some orbital applications of the integration on topological groups and their algebras to harmonic analysis and induced representations in representation theory.

 
Matrix Lie Groups And Control Theory

PDF | 60 Pages | English
This note covers the following topics: Matrix and Lie Groups, Dynamics and Control on Matrix Groups, Optimality and Riccati Equations, Geometric Control.

 
Modular Lie Algebras (PDF 74P)

PDF | 74 Pages | English
This note covers the following topics: Free algebras, Universal enveloping algebras , p th powers, Uniqueness of restricted structures, Existence of restricted structures , Schemes, Differential geometry of schemes, Generalised Witt algebra, Filtrations, Witt algebras are generalised Witt algebra, Differentials on a scheme, Lie algebras of Cartan type, Root systems, Chevalley theorem, Chevalley reduction, Simplicity of Chevalley reduction, Chevalley groups,  Abstract Chevalley groups, Engel Lie algebras and Lie algebra associated to a group .

 
Lectures on Lie Algebras (PDF 36P)

PDF | 36 Pages | English
This is a lecture note for beginners on representation theory of semisimple finite dimensional Lie algebras. It is shown how to use infinite dimensional representations to derive the Weyl character formula.

 
Lie algebras notes (PDF 34P)

PDF | 34 Pages | English
This note explains the following topics: Basic definitions and examples, Theorems of Engel and Lie, The Killing form and Cartan’s criteria, Cartan subalgebras, Semisimple Lie algebras, Root systems, Classification and examples of semisimple Lie algebras.

 
Theory of representations by Claudio Procesi

Online | NA Pages | English
This note explains the following topics: Lie groups, Lie algebra associated to a group, Correspondence between groups and algebras, classification of connected compact Lie groups, theory of Cartan Weyl.

 
Lie algebras by Shlomo Sternberg

PDF | 198 Pages | English
This note covers the following topics: The Campbell Baker Hausdorff Formula, sl(2) and its Representations, classical simple algebra, Engel-Lie-Cartan-Weyl, Conjugacy of Cartan sub algebras, simple finite dimensional algebras, Cyclic highest weight modules, Serre’s theorem, Clifford algebras and spin representations, The Kostant Dirac operator.

 
Notes on Lie Algebras

PDF | 172 Pages | English
This book presents a simple straightforward introduction, for the general mathematical reader, to the theory of Lie algebras, specifically to the structure and the (finite dimensional) representations of the semisimple Lie algebras.

 
An Introduction to Lie Groups and Symplectic Geometry

PDF | 170 Pages | English
The course note really was designed to be an introduction, aimed at an audience of students who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for Lie groups and symplectic geometry.

 
Lie methods

PDF | 148 Pages | English
This note covers the following topics: Numerical analysts in Plato’s temple, Theory and background, Runge–Kutta on manifolds and RK-MK, Magnus and Fer expansions, Quadrature and graded algebras, Alternative coordinates, Adjoint methods, Computation of exponentials, Stability and backward error analysis, Implementation, Applications.

 
F. Warner, Foundations of Differentiable Manifolds and Lie Groups (DJVU)
Lie Algebras Lecture Notes

PDF | 34 Pages | English
This note covers the following topics: Basic definitions and examples, Theorems of Engel and Lie, The Killing form and Cartan’s criteria, Cartan subalgebras, Semisimple Lie algebras, Root systems, Classification and examples of semisimple Lie algebras.

 
Lie Algebras by Shlomo Sternberg

Online | NA Pages | English
This note covers the following topics: Applications of the Cartan calculus, category of split orthogonal vector spaces, Super Poison algebras and Gerstenhaber algebras, Lie groupoids and Lie algebroids, Friedmann-Robertson-Walker metrics in general relativity, Clifford algebras.

 
Expository articles Computing rational points on curves, Elliptic curves
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