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: A Lie group carries both structures — and the group operations are smooth

Group product, inverse Manifold differentiable Lie Group smooth operations (Q, +) with R's topology Diff(M) infinite-dim. groups ⋮ GL(n), O(n) U(n), SU(n) (R, +) ⋮ Sphere S² genus-2 surface Klein bottle ⋮

Finite and countable discrete groups count as $0$-dimensional manifolds, so they belong in the middle region — they are Lie groups (the $(\mathbb{Q},+)$ on the left of the figure carries the topology of $\mathbb{R}$; with the discrete topology it becomes a $0$-dimensional Lie group). So are $\mathbb{R}^n$ under addition and the torus $T^n$. Among spheres, only $S^0$, $S^1$ and $S^3$ admit a Lie group structure; $S^2$ does not.

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 local problem on the curved Lie group can be studied with linear-algebraic tools on the tangent space, together with the Lie bracket (the bracket is non-commutative, so the problem itself does not become linear).

* The figure depicts a tangent plane for intuition; in general one speaks of the tangent space $T_eG$, whose dimension equals that of the Lie group. 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] (Lie bracket) for matrices: [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 near the identity 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. Throughout, the Lie algebras are finite-dimensional and defined over $\mathbb{R}$ or $\mathbb{C}$ (the classification of semisimple Lie algebras and the theory of root systems assume this setting).

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 complex 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 of a complex semisimple Lie algebra

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

Weyl dimension formula for a finite-dimensional irreducible representation of a complex semisimple Lie algebra ($\lambda$ dominant integral)

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

Angular-momentum commutation

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

(as real Lie algebras $\mathfrak{so}(3) \cong \mathfrak{su}(2)$; as groups $SU(2)$ is a double cover of $SO(3)$)

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 general representation theory (representations, 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 $[X,Y]$ (given by the commutator $XY-YX$ for matrix Lie algebras), 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 $\exp: \mathfrak{g}\to G$ sends elements of the Lie algebra to elements of the group, and near the identity it gives a local correspondence between the two (it need not be surjective in general). 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 flavor symmetry and, in QCD, color symmetry; $\mathfrak{so}(3)$ underlies 3D rotations and rigid-body mechanics; $\mathfrak{su}(1,1)\simeq\mathfrak{sl}(2,\mathbb{R})$ appears in quantum optics (squeezing) and in some formulations of the harmonic oscillator. The dynamical symmetry of the bound states of the hydrogen atom is $\mathfrak{so}(4)$. The representation theory of flavor $SU(3)$ — its weights and roots — underlies the Eightfold Way classification of hadrons.