Abelian Lie group

Abelian Lie groups are a term from the mathematical theory of Lie groups and Lie algebras .

definition

A Lie group is called Abelian if its group multiplication is commutative .

For connected Lie groups this is equivalent to the fact that the Lie algebra of the Lie group is an Abelian Lie algebra , i.e. the Lie brackets are identically zero.

properties

For an Abelian Lie group and its Lie algebra , the exponential mapping is a homomorphism , so it is true ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle exp\colon {\mathfrak {g}}\to G}$

${\displaystyle exp(X+Y)=exp(X)exp(Y)}$

for everyone . This follows from the fact that the multiplication map has the differential and is a homomorphism for Abelian groups (and only these) , as well as from . ${\displaystyle X,Y\in {\mathfrak {g}}}$ ${\displaystyle m\colon G\times G\to G}$ ${\displaystyle (X,Y)\to X+Y}$ ${\displaystyle m}$ ${\displaystyle exp(Dm(X,Y))=m(exp(X),exp(Y))}$

Furthermore, for Abelian groups, the exponential mapping is surjective and has a discrete kernel .

Examples

The circle group is an Abelian Lie group. It is isomorphic to the special orthogonal group and to the unitary group . ${\displaystyle S^{1}}$ ${\displaystyle SO(2)}$ ${\displaystyle U(1)}$

The torus is also an Abelian Lie group. ${\displaystyle \mathbb {T} ^{2}=S^{1}\times S^{1}}$

classification

Every compact, connected, Abelian Lie group is a torus for a . ${\displaystyle n}$ ${\displaystyle \mathbb {T} ^{n}=\underbrace {\mathbb {S} ^{1}\times \cdots \times \mathbb {S} ^{1}} _{n\ {\text{mal}}}}$ ${\displaystyle n\in \mathbb {N} }$

Every connected, Abelian Lie group is isomorphic to for natural numbers . ${\displaystyle \mathbb {R} ^{k}\times \mathbb {T} ^{n}}$ ${\displaystyle k,n}$

Every Abelian Lie group is isomorphic to for a finite Abelian group and . ${\displaystyle F\times \mathbb {R} ^{k}\times \mathbb {T} ^{n}}$ ${\displaystyle F}$ ${\displaystyle k,n\in \mathbb {N} }$