Lie group–Lie algebra correspondence

In mathematics, Lie group–Lie algebra correspondence is a special way to connect two important ideas: a Lie group and a Lie algebra. This helps mathematicians study these ideas by seeing how they relate.

When two Lie groups are isomorphic, meaning they have the same shape and properties, their matching Lie algebras are also isomorphic. But the reverse 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 is 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 property called being second countable, meaning they have a limited number of connected components.

Basics

The Lie algebra of a Lie group can be understood using special vector fields. These vector fields look the same even when the group moves in a certain way. All of these vector fields together make a structure called a Lie algebra. This helps us learn about the group's properties.

For matrix Lie groups—groups made of matrices—the Lie algebra is made of special matrices. When we use a process called exponentiation on these matrices, they stay inside the group. This link lets mathematicians change group properties into algebraic ones, which makes tough problems simpler to solve.

The correspondence

The correspondence between Lie groups and Lie algebras has 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 connects 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 its Lie algebra is also abelian. When a Lie group is abelian, a special map called the exponential map connects the Lie algebra to the group. This map helps us understand how these two structures are related.

There is also a way to see how different Lie groups can come from the same Lie algebra by looking at their basic group properties. This explains why some different groups can share the same Lie algebra.

Compact Lie groups

Main article: Compact Lie group

When we study special types of math groups called Lie groups, we sometimes look at ones that are "compact." This means they have a nice, limited size.

For these compact Lie groups, if they also have a finite center (meaning they don’t stretch out forever), then their "covering group" is also compact. This helps mathematicians understand these groups better by looking at their related structures, called Lie algebras. These connections show how closely the group's geometry relates to 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. We do this by looking at tiny pieces near the identity element of the group, called distributions. We study how these pieces combine. They form a structure called a Hopf algebra. The Lie algebra linked to the group comes from the simplest pieces in this structure.

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

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