Lie Groups — Mathematics Knowledge System

Overview

A Lie group is simultaneously a smooth manifold and a group, with the group operations (multiplication and inversion) being smooth maps. Lie groups are the mathematical model of continuous symmetry: the rotation group SO(3), the unitary group U(n), and the general linear group GL(n, ℝ) are all Lie groups. They are central to physics (gauge theories, quantum mechanics), geometry (isometry groups of Riemannian manifolds), and representation theory. Every Lie group has an associated Lie algebra — its tangent space at the identity — which encodes the infinitesimal structure of the group.

Intuition

A Lie group is a group that varies continuously. Think of the rotation group SO(3): you can continuously rotate a rigid body in 3D, and composing rotations gives another rotation. The group structure tells you how symmetries compose; the smooth structure tells you how they vary. The Lie algebra captures the infinitesimal rotations — the angular velocity vectors — and these form a vector space with a bracket operation encoding the non-commutativity of the group. The exponential map sends infinitesimal symmetries (Lie algebra elements) to actual group elements.

Formal Definition

Definition

A Lie group G is a smooth manifold equipped with a group structure such that the multiplication μ: G × G → G and inversion ι: G → G are smooth maps. The Lie algebra g = T_eG of G is the tangent space at the identity, equipped with the Lie bracket [·, ·]: g × g → g derived from left-invariant vector fields.

\mu: G \times G \to G,\; (g,h) \mapsto gh \text{ smooth}

Smooth multiplication

\iota: G \to G,\; g \mapsto g^{-1} \text{ smooth}

Smooth inversion

\mathfrak{g} = T_e G

Lie algebra as tangent space at identity

[X, Y] = X \circ Y - Y \circ X \text{ (as left-invariant vector fields)}

Lie bracket

\exp: \mathfrak{g} \to G,\; X \mapsto \gamma_X(1)

Exponential map (γ_X is the one-parameter subgroup with initial velocity X)

Properties

Left-invariant vector fields

X \in \mathfrak{X}(G) \text{ is left-invariant if } (L_g)_* X = X \text{ for all } g \in G

One-parameter subgroups

\text{Each } X \in \mathfrak{g} \text{ generates a one-parameter subgroup } \gamma_X: \mathbb{R} \to G, \; \gamma_X(s+t) = \gamma_X(s)\gamma_X(t)

Compact Lie groups are reductive

\text{Every finite-dimensional representation of a compact Lie group decomposes as a direct sum of irreducibles (Weyl's theorem).}

Theorems

Theorem 1: Lie's Third Theorem

\text{Every finite-dimensional Lie algebra is the Lie algebra of some simply connected Lie group.}

Theorem 2: Closed Subgroup Theorem

\text{Every closed subgroup of a Lie group is a Lie subgroup (with an induced smooth structure).}

Theorem 3: Cartan's Classification

\text{Simple complex Lie algebras are classified by Dynkin diagrams: } A_n, B_n, C_n, D_n \text{ and five exceptionals } G_2, F_4, E_6, E_7, E_8.

Worked Examples

1
U(1) ≅ S¹ is a smooth 1-manifold (the circle).
$U(1) \cong S^1$
2
Multiplication is z · w = zw (complex multiplication), which is smooth. Inversion is z ↦ 1/z = z̄ (for |z|=1), also smooth.
3
The Lie algebra is T_1 U(1) = iℝ — the imaginary axis, spanned by i.
$\mathfrak{u}(1) = T_1 U(1) = i\mathbb{R}$
4
The exponential map is exp(iθ) = e^{iθ}, the standard complex exponential.
$\exp: i\mathbb{R} \to U(1),\; it \mapsto e^{it}$

✓ Answer

U(1) is a 1-dimensional abelian Lie group with Lie algebra u(1) ≅ ℝ and exponential map t ↦ e^{it}.

Practice Problems

Mediumfree response

What is the Lie algebra of SU(2) and what is its relation to so(3)?

Hardproof writing

Prove that the exponential map exp: g → G is a local diffeomorphism near 0 ∈ g.

Common Mistakes

Common Mistake

All Lie groups are matrix groups

Every Lie group has a faithful representation by matrices (Ado's theorem for Lie algebras), but some Lie groups are not naturally matrix groups. The universal cover of SL(2, ℝ) is an example of a Lie group with no faithful finite-dimensional matrix representation.

Common Mistake

The exponential map is always surjective

exp is surjective for compact connected Lie groups and for the general linear group over ℂ. But for SL(2, ℝ), there are matrices not in the image of exp.

Quiz

Historical Background

Sophus Lie introduced Lie groups in the 1870s–1880s in his study of continuous transformation groups for solving differential equations, inspired by Galois theory. Lie's student Felix Klein connected Lie groups to geometry via the Erlangen programme (1872). Wilhelm Killing classified simple Lie algebras in 1888–1890 (with errors corrected by Cartan in his 1894 thesis). Hermann Weyl proved the complete reducibility of representations of compact Lie groups (1925–1926) and introduced the character formula. The modern treatment of Lie groups as smooth manifolds was systematised by Chevalley and others in the mid-twentieth century.

1872
Klein's Erlangen Programme: geometry as the study of invariants under a Lie group of transformations
Felix Klein
1874
Lie introduces continuous transformation groups, the precursor to Lie groups
Sophus Lie
1894
Cartan classifies simple complex Lie algebras, completing Killing's programme
Élie Cartan
1925
Weyl proves complete reducibility of representations of compact Lie groups
Hermann Weyl

Summary

A Lie group G is a smooth manifold with smooth group multiplication and inversion; its dimension is the dimension as a manifold.
The Lie algebra g = T_eG carries the Lie bracket [X,Y], encoding infinitesimal group structure.
The exponential map exp: g → G is a local diffeomorphism near 0; for matrix groups, exp is the matrix exponential.
Cartan's theorem classifies simple Lie algebras by Dynkin diagrams A_n, B_n, C_n, D_n and five exceptional types.
Compact Lie groups (e.g., SO(n), U(n), SU(n)) have complete representation theory; every representation is fully reducible.

References

BookHall, B. — Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, 2nd ed., Springer, 2015
BookBröcker, T. and tom Dieck, T. — Representations of Compact Lie Groups, Springer, 1985
WebsiteWikipedia — Lie group
WebsitenLab — Lie group