# Lie Groups Lecture 3

This lecture (Lie3) was given on September 3rd 2009.

### Excerpts

Looking ahead to the future, given a lie group $G$, there is an associated lie algebra $g$.  The central series of the lie algebra is $g_1 = [g,g], g_2, [g, g_1], ...$.  The group $G$ is called nilpotent if its associated central series terminates.  The term solvable is used when the commutator series $g^1 = [g,g], g^2 = [g^1,g^1], ...$ terminates.

The Heisenberg group $GL(3, \mathbb{R})$ is an example of a nilpotent lie group.

The Euclidean group is the group generated by rotations and translations in $\mathbb{R}^n$, it is a linear subgroup of $GL(n+1, \mathbb{R})$.  All elements can be written as $T = T_xR$; i.e. a rotation follows by a translation.  The Poincare group is define similarly, except rotations are replaced by elements of $O(n,1)$.

The lie group $O(n,1)$ has 4 connected components; this group can be interpreted as transformations which preserve a hyperbaloid.  A hyperbaloid has two components, so the group splits into the elements that switch the two components of the hyperbaloid and those the dont.  Then there is also a splitting based on whether the determinant is positive or negative.

The group $GL(n, \mathbb{C})$ is connected because it is the complement of a hypersurface, so has complex codimension 1, hence real codimension 2.  Using this fact one can show various other matrix lie groups are connected by finding retracts to them from $GL(n, \mathbb{C})$.

In particular $SU(2)$ is connected.  It can be seen in various ways that the action of conjugation gives a representation of $SU(2)$ as rotations in $\mathbb{R}^3$ and this gives $SU(2)$ as a double cover of $SO(3)$.  Consequently, $SU(2)$ is the universal covering space of $SO(3)$.  From $SU(2)/\mathbb{Z}_2 \cong SO(3)$ it follows that $\pi_1(SO(3)) = \mathbb{Z}_2$.