Topological groups, Lie groups Definition

Lie group

Builds on Group Smooth manifold Smooth function

Definition

A real Lie group is a group $G$ that is also a finite-dimensional smooth manifold, such that:

the multiplication map $\mu \colon G \times G \to G$ , $\mu(g, h) = gh$ , is smooth;
the inversion map $\iota \colon G \to G$ , $\iota(g) = g^{-1}$ , is smooth.

A complex Lie group is defined in the same way with a complex manifold and holomorphic maps. A homomorphism of Lie groups is a group homomorphism that is a smooth map.

Remarks

Example: the circle $S^1 = \set{z \in \C : \abs{z} = 1}$ with complex multiplication. It is a smooth submanifold of $\C = \R^2$ , and in the angle coordinate the product and the inverse are $(\theta, \varphi) \mapsto \theta + \varphi$ and $\theta \mapsto -\theta$ , which are smooth.

Example: $GL_n(\R)$ is an open subset of $\R^{n^2}$ , and the product of matrices is polynomial in the entries. By Cramer's rule, the inverse is a rational function of the entries with nonvanishing denominator $\det$ .

Non-example: the additive group $\Q$ with the topology induced from $\R$ is a topological group but not a Lie group. It is not locally Euclidean, since no point of $\Q$ has a neighbourhood homeomorphic to an open set of some $\R^n$ .

The two clauses can be merged: $G$ is a Lie group exactly when the map $(g, h) \mapsto g h^{-1}$ is smooth. The dimension of $G$ is its dimension as a manifold.

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Group concept
Smooth function concept
Smooth manifold concept
Lie group definition

Tag	`K14ZBKYT81`
URI	`m2://K14ZBKYT81`
Kind	definition
Writing	W0 stub
MSC	22E15, 22E10
Defines	Lie group
Uses	Group, Smooth manifold, Smooth function
Changed	2 November 2026

Lie group

Definition

Remarks

Reading path

Further reading

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