Lie Algebra

Introduction, Structure Theory, Root Systems, and Representation Theory

What is a Lie Algebra?

First, What is a Lie Group?

A Lie group is a group that is simultaneously a smooth manifold, with the group operations (product and inverse) being smooth maps. In other words, it is a group whose elements can be "moved continuously."

  • Rotation group $SO(3)$: all rotations in three-dimensional space, with rotation angles varied smoothly
  • General linear group $GL(n)$: all invertible $n \times n$ matrices, whose entries can vary continuously
  • Unitary group $U(n)$: the norm-preserving transformations appearing in quantum mechanics

Lie groups are often used in physics to describe symmetries (for example, rotational symmetry via $SO(3)$ or gauge symmetry via $SU(n)$). However, being a Lie group is not about "having a symmetry" per se — it is precisely about having a group structure compatible with a smooth manifold structure.

Figure 1: Lie Group = Group ∩ Manifold

Group product, inverse Manifold differentiable Lie Group smooth operations Finite group Sₙ Z/nZ Discrete group Zⁿ ⋮ GL(n), O(n) U(n), SU(n) (R, +) ⋮ Rⁿ Sphere Sⁿ Torus Tⁿ ⋮

Lie Algebra: The Linearization of a Lie Group

A Lie algebra extracts the "infinitesimal behaviour of a Lie group near the identity." As shown in Figure 2, one considers the tangent space at a special point on the curved Lie group (the identity element $e$) and equips it with the Lie bracket $[X, Y]$ defined below. This endows the geometric space with an algebraic structure.

Figure 2: Lie Group (Curved Surface) and Lie Algebra (Tangent Plane)

𝔤 (Lie algebra) e (identity) Lie group G Tangent plane 𝔤 is a (linear) vector space → amenable to linear algebra

A Lie group is a curved manifold, but its tangent plane* — the Lie algebra — is a flat vector space, which is much easier to handle. The great advantage is that a complicated non-linear problem can be "translated" into a tractable linear one.

* In four or more dimensions this is a tangent hyperplane; generally we call it the tangent space. Note that a Lie algebra captures only the information near the identity, and does not fully describe the global topology of a Lie group. In fact, different Lie groups can share the same Lie algebra.

Where It Is Used

In physics, Lie algebras appear in commutation relations of angular momentum, in gauge theories of particle physics, and in the symmetries of quantum mechanics. In pure mathematics they play a central role in representation theory, differential geometry, and algebraic geometry.

Figure 3: Relation between a Lie Group and Its Lie Algebra

Lie group G (smooth manifold) e (identity) log exp Lie algebra 𝔤 (tangent space + bracket) [X, Y] = XY - YX A Lie algebra is the tangent space at the identity equipped with the Lie bracket It lets us describe a Lie group in the language of linear algebra Physics: angular momentum, gauge theory | Math: representation theory, differential geometry * log/exp give a local correspondence near the identity (not globally one-to-one in general)

In this series we study Lie algebras systematically in four levels, from the fundamentals to the classification of semisimple Lie algebras and representation theory.

Content by Level

Introduction

Fundamentals of Lie algebras

  • Definition and examples of Lie algebras
  • Lie brackets and commutators
  • Matrix Lie algebras
  • Subalgebras and ideals
  • Homomorphisms and isomorphisms
  • The Lie group – Lie algebra connection (intuition)
Undergraduate (years 2-3)

Basic

Foundations of structure theory

  • Solvable Lie algebras
  • Nilpotent Lie algebras
  • Semisimple Lie algebras
  • Killing form
  • Cartan's criterion
  • Lie's and Engel's theorems
Undergraduate (years 3-4)

Intermediate

Structure of semisimple Lie algebras

  • Cartan subalgebras
  • Root systems
  • Weyl group
  • Dynkin diagrams
  • Classification of semisimple Lie algebras
  • Exceptional Lie algebras
Senior / Graduate

Advanced

Representation theory

  • Representations of Lie algebras
  • Weights and weight spaces
  • Highest weight representations
  • Weyl's character formula
  • Universal enveloping algebra
  • Applications to physics
Graduate

Key Concepts and Formulas

Properties of the Lie bracket

$$[X, Y] = -[Y, X]$$

$$[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0$$

Killing form

$$B(X, Y) = \mathrm{tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y)$$

Basis of $\mathfrak{sl}(2)$

$$[H, E] = 2E$$

$$[H, F] = -2F$$

$$[E, F] = H$$

Root-space decomposition

$$\mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha$$

Weyl dimension formula

$$\dim V_\lambda = \prod_{\alpha > 0}\dfrac{(\lambda + \rho, \alpha)}{(\rho, \alpha)}$$

Angular-momentum commutation

$$[L_x, L_y] = i\hbar L_z$$

($\mathfrak{so}(3) \cong \mathfrak{su}(2)$)

Prerequisites

  • Introduction: linear algebra (matrix operations, eigenvalues, vector spaces, bases and dimensions) and the basics of group theory (group definition, subgroups, homomorphisms)
  • Basic: the content of the Introduction, plus foundations of abstract algebra (rings, ideals, quotient structures)
  • Intermediate: the content of the Basic level, plus additional linear algebra (dual spaces, bilinear and quadratic forms, inner-product spaces)
  • Advanced: the content of the Intermediate level, plus the basics of representation theory (modules, irreducibility, Schur's lemma) and tensor products of vector spaces

Frequently Asked Questions

What is a Lie algebra?

A Lie algebra is a vector space equipped with a bilinear operation called the Lie bracket $[X,Y]$, satisfying anti-symmetry $[X,Y]=-[Y,X]$ and the Jacobi identity $[[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0$. Lie algebras capture the infinitesimal structure of Lie groups and are fundamental to the study of symmetry in mathematics and physics.

How are Lie algebras related to Lie groups?

The Lie algebra $\mathfrak{g}$ of a Lie group $G$ is the tangent space at the identity element, equipped with the Lie bracket derived from the group multiplication. The exponential map $\exp: \mathfrak{g}\to G$ recovers group elements from algebra elements. This connection lets physicists work with linear algebra (Lie algebras) rather than nonlinear group manifolds.

What are the most important Lie algebras in physics?

$\mathfrak{su}(2)$ governs spin angular momentum and isospin; $\mathfrak{su}(3)$ describes quark color symmetry in QCD; $\mathfrak{so}(3)$ underlies 3D rotations and rigid-body mechanics; $\mathfrak{sl}(2,\mathbb{R})$ appears in the hydrogen atom and harmonic oscillator. Root systems and the Weyl character formula classify elementary particle multiplets (the Eightfold Way).