Lie groups

Definition and some properties

A Lie group is a manifold $G$ together with two smooth operations satisfying the group definition.

Some interesting elementary properties:

Proposition

The identity component $G_0$ of a Lie group $G$ is a closed normal subgroup. $\blacksquare$

Proof

First, a connected component is always closed.

Second, given $g,h \in G$, $g$ can be seen as an homeomorphism

by left multiplication. But since $h$ and $e$ are in the same connected component, $gh$ and $g$ are in the same component (and the same that $e$). Therefore, $gh\in G_0$, and it is a subgroup.

And finally, since the map

is an homeomorphism of $G$ into $G$, and $e\in G_0 \cap \phi_g(G_0)$, then $\phi_g(G_0)=G_0$, so $G_0$ is normal.

$\blacksquare$

Two key results for the classification of Lie groups are:

Proposition

Every Lie group has connected components diffeomorphic to the normal subgroup of the identity component.

$\blacksquare$

Proposition

Every connected Lie group $G$ has a simply connected universal covering group $G^*$ (see section covering group).

$\blacksquare$

The following is a general result used in the context of covering groups:

Proposition

Let $G$ be a connected Lie group and $N$ a discrete normal subgroup of $G$. Then $N$ is contained in the center $Z(G)$.

$\blacksquare$

Proof

Let $g\in N$ and consider the map

such that $F_g(h)=h^{-1}g h$. Since $F_g$ is continuous and $G$ is connected, the image is connected, and since $N$ is discrete, $im(F_g)=\{g\}$.

And so $gh=hg$ for every $h\in G$.

$\blacksquare$

Lie algebra and Lie groups correspondence

For $g\in G$ we define:

and

called left and right multiplication respectively.

A vector field in $G$ is left-invariant if $d(L_g)_p (X(p))=X(g\cdot p)$. That is to say: $X(g)=d(L_g)_e(X(e))$ for all $g$, and being $e$ the identity of $G$. Left invariant vector fields are closed under the Lie bracket, so they constitute a Lie algebra $\mathfrak { g } \cong T _ { e } G$.

We have that:

Theorem(Lie's third theorem). Every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group. $\blacksquare$

And it can be shown that if two simply connected Lie groups have the same Lie algebra, they are isomorphic.

Given two groups with isomorphic Lie algebras you can only infer that they are isomorphic in some neighborhood of identity. Moreover, they would have the same universal covering group, so both are quotient of the same group by a discrete subgroup.

Proposition. A connected Lie group is abelian if and only if the Lie algebra is abelian (see this. $\blacksquare$

Proposition. A connected, one-dimensional Lie group $G$ is isomorphic to $\mathbb{R}$ or $S^1$ (see this. $\blacksquare$

Corollary. Every connected 1-dimensional Lie group is commutative.$\blacksquare$

Parallelizable manifolds

A Lie group is a parallelizable manifold. The tangent spaces at different points can be identified using left translations, so we have $TG\approx G\times T_e G$. This is related to the Maurer-Cartan form.

Examples

• $(\mathbb{R}^n, +)$

They all are abelian groups. They are translations.

• $(\mathbb{R}^n-\{0\},\cdot)$ for $n=1,2,4$

They are lie groups only in this three cases. They are the real numbers, the complex numbers and quaternions respectively.

• $GL(n)$, $GL(n,\mathbb{R})$, $GL(n,\mathbb{C})$, general linear group.

• $SL(2)$. Special linear group.

Matrices of rank 2 whose determinant is 1. Inside it we have three subgroups:

It can be shown that every element of $SL(2)$ split as product of the elements of these groups. That is,

This is called Iwasawa decomposition. From here can be concluded that $SL(2)$ is homeomorphic to the interior of a solid torus. See Decomposing SL(2,R) in Calibre.

• $SL(n)$. Special linear group.

Matrices of rank $n$ whose determinant is 1. It is a manifold of dimension $n^2-1$. Geometrically, it corresponds with the volume and orientations preserving transformations of $\mathbb{R}^n$.

• $Aff(\mathbb{R}^n)$, the affine group.

• $E(n)$, the Euclidean group.

• $O(n)$ and $SO(n)$, see orthogonal matrix.

• SO(3)

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es