Topological groups, Lie groups Lie groups: basic notions Theorem

Lie group–Lie algebra correspondence

Builds on Lie group Exponential map of a Lie group Lie subgroup Fundamental group Lie's third theorem

Statement

Let $G$ and $H$ be real Lie groups with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ , and exponential maps $\exp$ .

For every Lie group homomorphism $\varphi \colon G \to H$ , the differential $d\varphi_e \colon \mathfrak{g} \to \mathfrak{h}$ is a Lie algebra homomorphism, and $\varphi(\exp X) = \exp(d\varphi_e X)$ for all $X \in \mathfrak{g}$ .
If $G$ is connected, a Lie group homomorphism $\varphi \colon G \to H$ is determined by $d\varphi_e$ .
If $G$ is connected and simply connected, each Lie algebra homomorphism $\psi \colon \mathfrak{g} \to \mathfrak{h}$ is $d\varphi_e$ for a unique homomorphism $\varphi \colon G \to H$ .
The map $K \mapsto \operatorname{Lie}(K)$ is a bijection from the connected Lie subgroups $K$ of $G$ onto the Lie subalgebras of $\mathfrak{g}$ .
Every finite-dimensional real Lie algebra is the Lie algebra of a connected and simply connected Lie group, unique up to isomorphism.

Here simply connected means that the fundamental group is trivial.

Remarks

Part 1 follows from the chain rule, since $\varphi$ maps one-parameter subgroups to one-parameter subgroups. Part 2 holds because a connected Lie group is generated by any neighbourhood of $e$ , and $\exp$ covers a neighbourhood of $e$ . Part 3 is proved by integrating the graph of $\psi$ , a subalgebra of $\mathfrak{g} \oplus \mathfrak{h}$ , to a connected Lie subgroup of $G \times H$ by part 4. Its projection to $G$ is a covering map, hence an isomorphism since $G$ is simply connected. Part 4 rests on the Frobenius theorem applied to the left-invariant distribution spanned by a subalgebra. Part 5 is Lie's third theorem. Simple connectivity in part 3 is necessary. The identity map of $\R$ is the differential of no homomorphism from the circle group $\R / \Z$ to $\R$ , since the only such continuous homomorphism is trivial.

Going deeper

For a student

In practice one computes with Lie algebras and lifts the answers to groups. For example, a representation of $\mathfrak{su}(2)$ gives a representation of the simply connected group $SU(2)$ by part 3, while for $SO(3)$ only some of them lift. Parts 3 and 5 together say that $G \mapsto \mathfrak{g}$ is an equivalence of categories between connected simply connected real Lie groups and finite-dimensional real Lie algebras. A connected Lie group with Lie algebra $\mathfrak{g}$ is the quotient of the simply connected one by a discrete central subgroup.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Exponential map of a Lie group concept
Fundamental group concept
Left-invariant vector field concept
Lie group concept
Lie subgroup concept
Universal covering group concept
Lie's third theorem statement
Lie group–Lie algebra correspondence statement

Tag	`M13HAQSHEM`
URI	`m2://M13HAQSHEM`
Kind	statement, theorem
Writing	W0 stub
MSC	22E60, 22E15
Status	This copy of the site has no build data. Run `m2 status M13HAQSHEM`.
About	Lie group
Uses	Lie group, Exponential map of a Lie group, Lie subgroup, Fundamental group, Lie's third theorem
Same as	Wikidata Q18353697 Wikipedia Lie group–Lie algebra correspondence
Changed	2 November 2026