Lie Groups Lecture 3

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


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.


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