Lie group–Lie algebra correspondence

In mathematics, Lie group–Lie algebra correspondence is a special way to match two important mathematical ideas: a Lie group and a Lie algebra. This helps mathematicians study these ideas by looking at how they relate to each other. When two Lie groups are isomorphic, meaning they have the same shape and properties, their matching Lie algebras are also isomorphic. However, the opposite is not always true. For example, real coordinate space and the circle group are different Lie groups, but their Lie algebras look the same.

For a special kind of Lie group called a simply connected Lie group, the matching between Lie groups and Lie algebras becomes very exact — it is one-to-one. This means each simply connected Lie group has a unique Lie algebra, and each Lie algebra matches exactly one simply connected Lie group.

This article focuses on real Lie groups. For other types, like complex Lie group or p-adic Lie group, different rules apply. Also, in this article, manifolds, including Lie groups, are assumed to have a certain property called being second countable, which means they have a limited number of connected components.

Basics

The Lie algebra of a Lie group can be understood using left-invariant vector fields. These are vector fields that stay the same when the group is shifted in a certain way. The collection of all such vector fields forms a structure called a Lie algebra, which helps us study the group's properties.

For matrix Lie groups—groups made of matrices—the Lie algebra consists of matrices that, when exponentiated (a process involving infinite sums), stay within the group. This connection lets mathematicians translate group properties into algebraic ones, making complex problems easier to handle.

The correspondence

The correspondence between Lie groups and Lie algebras includes three main results.

Lie's third theorem states that every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group. This means that for any such Lie algebra, we can find a Lie group that has it as its algebra.

The homomorphisms theorem says that if there is a Lie algebra homomorphism between the Lie algebras of two simply connected Lie groups, then there is a corresponding Lie group homomorphism between the groups themselves. This helps us understand how structures in Lie algebras reflect structures in Lie groups.

The subgroups–subalgebras theorem explains that if a Lie group has a Lie subalgebra, then there is a unique connected Lie subgroup of the original group that corresponds to this subalgebra.

These results show how closely tied Lie groups and Lie algebras are, allowing mathematicians to study one to understand the other.

Lie group representations

A special case of the Lie group–Lie algebra correspondence relates finite-dimensional representations of a Lie group to those of its Lie algebra. For example, the general linear group GLₙ(ℂ) is a Lie group, and any homomorphism from a Lie group G to GLₙ(ℂ) is called a representation of G. The differential of this homomorphism is a Lie algebra representation.

The adjoint representation is an important example. Each element in a Lie group defines an automorphism of the group by conjugation, and the differential of this automorphism is an automorphism of the Lie algebra. This leads to a representation called the adjoint representation, which helps determine the Lie bracket of the Lie algebra from the group law.

Abelian Lie groups

Main article: Abelian Lie group

In mathematics, a connected Lie group is abelian if and only if its Lie algebra is abelian. When a Lie group is abelian, a special map called the exponential map connects the Lie algebra to the group in a very neat way. This map is important because it helps us understand the relationship between these two structures.

There is also a special way to understand how different Lie groups can be built from the same Lie algebra by looking at their fundamental groups and central extensions. This helps explain why some groups that are not the same still share the same Lie algebra.

Compact Lie groups

Main article: Compact Lie group

When we talk about special types of groups in math, called Lie groups, we sometimes focus on ones that are "compact." This means they have a nice, finite size in a certain sense. For these compact Lie groups, several important properties line up perfectly.

If a Lie group is compact and has a finite center (which means it doesn't go on forever in any direction), then its "covering group" is also compact. This helps mathematicians study these groups using their related structures, called Lie algebras. These links show how deep the connections are between the geometry of the group and the algebra that represents it.

Compact Lie group	Complexification of associated Lie algebra	Root system
SU(n+1) = { A ∈ M n + 1 ( C ) ∣ A ¯ T A = I , det ( A ) = 1 } {\displaystyle =\left\{A\in M_{n+1}(\mathbb {C} )\mid {\overline {A}}^{\mathrm {T} }A=I,\det(A)=1\right\}}	s l ( n + 1 , C ) {\displaystyle {\mathfrak {sl}}(n+1,\mathbb {C} )} = { X ∈ M n + 1 ( C ) ∣ tr ⁡ X = 0 } {\displaystyle =\{X\in M_{n+1}(\mathbb {C} )\mid \operatorname {tr} X=0\}}	A_n
SO(2n+1) = { A ∈ M 2 n + 1 ( R ) ∣ A T A = I , det ( A ) = 1 } {\displaystyle =\left\{A\in M_{2n+1}(\mathbb {R} )\mid A^{\mathrm {T} }A=I,\det(A)=1\right\}}	s o ( 2 n + 1 , C ) {\displaystyle {\mathfrak {so}}(2n+1,\mathbb {C} )} = { X ∈ M 2 n + 1 ( C ) ∣ X T + X = 0 } {\displaystyle =\left\{X\in M_{2n+1}(\mathbb {C} )\mid X^{\mathrm {T} }+X=0\right\}}	B_n
Sp(n) = { A ∈ U ( 2 n ) ∣ A T J A = J } , J = [ 0 I n − I n 0 ] {\displaystyle =\left\{A\in U(2n)\mid A^{\mathrm {T} }JA=J\right\},\,J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\end{bmatrix}}}	s p ( n , C ) {\displaystyle {\mathfrak {sp}}(n,\mathbb {C} )} = { X ∈ M 2 n ( C ) ∣ X T J + J X = 0 } {\displaystyle =\left\{X\in M_{2n}(\mathbb {C} )\mid X^{\mathrm {T} }J+JX=0\right\}}	C_n
SO(2n) = { A ∈ M 2 n ( R ) ∣ A T A = I , det ( A ) = 1 } {\displaystyle =\left\{A\in M_{2n}(\mathbb {R} )\mid A^{\mathrm {T} }A=I,\det(A)=1\right\}}	s o ( 2 n , C ) {\displaystyle {\mathfrak {so}}(2n,\mathbb {C} )} = { X ∈ M 2 n ( C ) ∣ X T + X = 0 } {\displaystyle =\left\{X\in M_{2n}(\mathbb {C} )\mid X^{\mathrm {T} }+X=0\right\}}	D_n

Related constructions

When we have a Lie group, we can connect it to a special kind of algebra called a Lie algebra. One way to see this is by looking at small pieces near the identity element of the group, called distributions, and studying how they combine. These pieces form a structure called a Hopf algebra. The Lie algebra linked to the group is made from the simplest pieces in this structure.

There is a special relationship, shown by the Milnor–Moore theorem, between the Lie algebra and these small pieces near the identity. This helps us understand how Lie groups and Lie algebras are connected through algebra.

distributions convolution Hopf algebra primitive elements Milnor–Moore theorem universal enveloping algebra