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 bilinearity, the alternating property (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.

Further Reading

The Dream of Sophus LieReading

Who is the “Lie” in Lie algebra? The life of the Norwegian mathematician Sophus Lie, who dreamed of a Galois theory for differential equations, and how his name became a whole field — told in a relaxed, story-driven way.

Overview

The notion of a Lie algebra grew out of the 19th-century study of continuous transformation groups (Lie groups) and infinitesimal transformations by the Norwegian mathematician Sophus Lie (the modern abstract definition took shape later). 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 operation called the Lie bracket $[X, Y]$ (a bilinear, alternating operation satisfying the Jacobi identity).

On $\mathfrak{gl}(n)$ and on subspaces of matrices closed under the commutator, the Lie bracket is given by $[A, B] = AB - BA$. This operation always satisfies $[A, B] = -[B, A]$, and for many pairs of matrices $[A, B] \neq 0$ (that is, $[A, B] \neq [B, A]$), so it measures how the order of multiplication matters.

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.

Frequently asked questions

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. Its Lie algebra is the tangent space at the identity equipped with the Lie bracket, which lets us study the group in the language of linear algebra. The exponential map $\exp: \mathfrak{g} \to G$ links the two: near the identity, the group product is determined by the Lie bracket (via the Baker–Campbell–Hausdorff formula). Lie groups with isomorphic Lie algebras are locally isomorphic, and a connected, simply connected Lie group is determined up to isomorphism by its Lie algebra. Moreover, every finite-dimensional real Lie algebra arises as the Lie algebra of some Lie group (Lie's third theorem).

Why is the Lie bracket non-commutative?

In matrix Lie algebras, the Lie bracket is given by the commutator $[X, Y] = XY - YX$, which always satisfies $[X, Y] = -[Y, X]$ (anticommutativity). In a general Lie algebra, it is defined as a bilinear, alternating ($[X, X] = 0$) operation satisfying the Jacobi identity. $[X, Y]$ measures the discrepancy when the order of two infinitesimal transformations is swapped, expressing non-commuting symmetries such as compositions of rotations. There are also commutative Lie algebras, in which all brackets vanish; they too are important basic examples.

What are matrix Lie algebras?

A matrix Lie algebra is a linear subspace of $\mathfrak{gl}(n)$ (all $n \times n$ matrices) that is closed under the commutator $[A, B] = AB - BA$. Canonical examples are $\mathfrak{gl}(n)$, $\mathfrak{sl}(n)$ (trace zero), $\mathfrak{so}(n)$ (skew-symmetric), and $\mathfrak{u}(n)$ (skew-Hermitian), arising naturally as the Lie algebras of the classical groups $GL(n)$, $SL(n)$, $SO(n)$, $U(n)$. They let you meet Lie algebras through concrete matrix computations, providing a bridge to the abstract theory.