Lie Group

Overview

A Lie group is simultaneously a group of transformations and a smooth space of parameters. Rotations illustrate both roles: they can be composed, and small rotations can be varied continuously and differentiated. The compatibility between these structures lets a local linear object, the Lie algebra, describe infinitesimal group behavior without encoding every global feature of the group.

Prerequisites

Use the group axioms, smooth maps between manifolds, tangent spaces, and vector fields. Matrix differentiation provides useful examples. The tangent space at the identity is a vector space; group multiplication is a different operation on the manifold itself.

Definition and conventions

A real Lie group is a group $G$ equipped with a finite-dimensional Hausdorff, second countable smooth manifold structure such that

are smooth. Manifolds here have no boundary. Neither compactness nor connectedness is assumed. Countable discrete groups qualify as zero-dimensional Lie groups; an uncountable discrete group fails the second-countability convention used here.

For each $g$, left translation $L_g(h)=gh$ is a diffeomorphism with inverse $L_{g^{-1}}$. The tangent space $\mathfrak g=T_eG$ becomes a Lie algebra by identifying $v\in\mathfrak g$ with the left-invariant vector field $X_v(g)=d(L_g)_e v$ and setting $[v,w]=[X_v,X_w]_e$. Choosing right-invariant fields instead changes the corresponding bracket sign.

Intuition

Translation carries information at the identity to every other point. Infinitesimal motions are vectors in $\mathfrak g$, and their bracket measures the leading failure of motions to commute. This does not require the group itself to be a vector space, or its product to be addition.

The distinction between local and global structure is visible in $\mathbb R$ and $S^1$. Their Lie algebras are both one-dimensional with zero bracket, but one group is noncompact and the other compact. The infinitesimal data do not record the circle's periodic identification.

Examples and non-examples

The invertible real matrices $\operatorname{GL}(n,\mathbb R)$ form an open subset of $\mathbb R^{n^2}$. Matrix multiplication is polynomial in the entries. Inversion is smooth because each inverse entry is a cofactor divided by the nonzero determinant. These observations verify the Lie group definition directly. Its Lie algebra is the space of all matrices with bracket $[A,B]=AB-BA$.

The circle $S^1=\{z\in\mathbb C:|z|=1\}$ is a Lie group under multiplication. The map $t\mapsto e^{it}$ is a homomorphism from the additive real line, with kernel $2\pi\mathbb Z$. It is locally invertible near zero but not globally injective.

For $\operatorname{SO}(n)$, differentiating $A(t)^TA(t)=I$ at $A(0)=I$ gives $A'(0)^T+A'(0)=0$. Thus its Lie algebra consists of skew-symmetric matrices. All square matrices together are not a group under multiplication: singular matrices lack inverses.

The additive rationals with their usual subspace topology are a topological group but not a Lie group. Every interval neighborhood contains nonrational gaps and is disconnected, so no positive-dimensional Euclidean chart is possible; the topology is also nondiscrete, excluding dimension zero.

Important properties

The exponential map sends $v\in\mathfrak g$ to the time-one point of the one-parameter subgroup generated by $X_v$. Left-invariance extends its locally defined flow for all time using translations. Since $d\exp_0$ is the identity, the inverse function theorem makes $\exp$ a diffeomorphism between neighborhoods of zero and $e$. This is a local assertion, as the circle example demonstrates.

Every Lie group has a trivial tangent bundle. Proof sketch: choose a basis of $T_eG$ and translate it by $dL_g$. These smooth vector fields form a basis at every point and hence give a global frame.

The closed subgroup theorem says a subgroup closed in a finite-dimensional Lie group is an embedded Lie subgroup. Closedness is a substantive hypothesis for this embedded conclusion; an abstract subgroup cannot automatically be treated as a submanifold with its subspace topology.

Connections and further reading

Lie groups act smoothly on manifolds and organize their symmetries. Their exponential map is defined from group structure; its agreement with a Riemannian exponential requires additional metric conditions. Meinrenken, §10 develops the inspected group, bracket, and exponential constructions.