Introduction
Lie theory is the study of continuous symmetries through Lie groups—smooth manifolds equipped with group structures—and their associated Lie algebras, which capture the infinitesimal structure of the group. Originating from Sophus Lie’s work in the 19th century, it provides a powerful framework linking geometry, algebra, and analysis, with applications ranging from differential equations to modern physics. Central to the theory is the correspondence between Lie groups and their Lie algebras, enabling local analysis of global group properties. Lie theory plays a foundational role in understanding symmetry in mathematics and theoretical physics, particularly in geometry, representation theory, and quantum field theory.
Contents
Lie Group and Lie Algebra Correspondence
Topological Groups
Lie Groups
Lie Algebra
The Exponential Map
Lie’s Theorems
Coalgebra and Hopf Algebra
Lie Group Cohomology
Properties of Lie Algebras
Ideals, Simple and Semisimple Lie Algebras
Engel’s Theorem
Jordan-Chevalley Decomposition
Killing Forms and Cartan’s Criteria
Structure Theorem for Semisimple Lie Algebras
Root Space Decomposition
Lie Algebra Representations
Universal Enveloping Algebra
Central Extensions of Lie Algebras
Lie Algebra Cohomology
Infinite Dimensional Lie Algebras
Some Matrix Lie Groups and Their Lie Algebras
Unitary Groups
General and Special Linear Groups
Symplectic Group