# Lie Groups Lecture 3

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

### Excerpts

Looking ahead to the future, given a lie group , there is an associated lie algebra . The central series of the lie algebra is . The group is called nilpotent if its associated central series terminates. The term solvable is used when the commutator series terminates.

The Heisenberg group is an example of a nilpotent lie group.

The Euclidean group is the group generated by rotations and translations in , it is a linear subgroup of . All elements can be written as ; i.e. a rotation follows by a translation. The Poincare group is define similarly, except rotations are replaced by elements of .

The lie group 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 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 .

In particular is connected. It can be seen in various ways that the action of conjugation gives a representation of as rotations in and this gives as a double cover of . Consequently, is the universal covering space of . From it follows that .

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- Published:
- September 4, 2009 / 3:11 pm

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- Lie Groups (course)

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