What prerequisites do I need before studying Lie algebras?

Linear algebra (matrix operations, eigenvalues, vector spaces, bases and dimensions) and basic group theory (group definition, subgroups, homomorphisms) are sufficient. Calculus is needed only in Chapter 6 for tangent spaces and the exponential map, and even then an intuitive understanding suffices to start. The content targets second- or third-year undergraduates and is suitable for physics students as well.

What is the relation between Lie algebras and Lie groups?

A Lie group is a differentiable manifold with a group structure, describing continuous symmetries such as rotations or Lorentz transformations. The Lie algebra is the tangent space at the identity of the Lie group, allowing us to study the group with the language of linear algebra. The exponential map exp: 𝔤 → G links the two, and locally the entire Lie-group structure is recovered from the Lie algebra alone (Lie's third theorem).

Why is the Lie bracket non-commutative?

The Lie bracket is defined as the commutator [X, Y] = XY - YX, which is obviously antisymmetric ([X, Y] = -[Y, X]). This non-commutativity is the essence of Lie theory: it captures the "order matters" nature of continuous symmetries like the non-commuting composition of rotations and translations. Commutative parts reduce to plain linear algebra, so Lie theory is essentially the study of genuinely non-commutative symmetries.

What are matrix Lie algebras?

Lie algebras whose elements are matrices. Canonical examples include 𝔤𝔩(n) (all n×n matrices), 𝔰𝔩(n) (trace zero), 𝔰𝔬(n) (skew-symmetric), and 𝔲(n) (skew-Hermitian), arising as the Lie algebras of the classical groups GL(n), SL(n), SO(n), U(n). They are ideal for a first course because Lie-theoretic concepts become concrete matrix computations, providing a bridge to the abstract theory.

Lie Algebra — Introduction

Definition, Lie Bracket, Matrix Lie Algebras, and the Relation to Lie Groups (Undergraduate years 2–3)

Learning Goals

Understand the definition and basic properties of Lie algebras
Grasp the meaning of the Lie bracket
Become familiar with concrete matrix Lie algebras
Learn the notions of subalgebras, ideals, and homomorphisms
Understand the relation between Lie groups and Lie algebras intuitively

Prerequisites

Linear algebra (matrix operations, eigenvalues, vector spaces)
Basic group theory (group definition, subgroups, homomorphisms)

Chapter Overview

Chapter 1: Definition of a Lie Algebra

What a Lie algebra is: a vector space equipped with a Lie bracket, with antisymmetry and the Jacobi identity.

Chapter 2: Lie Brackets and Commutators

The matrix commutator $[A, B] = AB - BA$ as the canonical example of a Lie bracket, with physics connections.

Chapter 3: Matrix Lie Algebras

Basic matrix Lie algebras: $\mathfrak{gl}(n)$, $\mathfrak{sl}(n)$, $\mathfrak{so}(n)$, $\mathfrak{u}(n)$, and their structures.

Chapter 4: Subalgebras and Ideals

Definitions of Lie subalgebras and ideals, and their analogy with subgroups and normal subgroups in group theory.

Chapter 5: Homomorphisms and Isomorphisms

Definitions and basic properties of Lie algebra homomorphisms and isomorphisms, with the homomorphism theorem.

Chapter 6: The Lie Group – Lie Algebra Relation

The Lie algebra as the tangent space at the identity of a Lie group. Exponential map and derivative, intuitively.

Chapter 7: Exercises

A problem set to consolidate the introductory material.

Overview

The notion of a Lie algebra was introduced in the 19th century by the Norwegian mathematician Sophus Lie in his study of continuous transformation groups (Lie groups). It provides a powerful tool for describing Lie groups in the language of linear algebra.

Concretely, a Lie algebra is a vector space equipped with an extra operation called the Lie bracket $[X, Y]$.

For matrices, the Lie bracket is defined as $[A, B] = AB - BA$. This operation is in general non-commutative: $[A, B] \neq [B, A]$ (in fact, $[A, B] = -[B, A]$).

In this introduction we first learn the definition of a Lie algebra, then deepen the understanding through concrete examples of matrix Lie algebras. Even abstract concepts become natural when approached through familiar matrix examples.