Lie Groups
Now we come to one of the most broadly useful and fascinating structures on all of mathematics: Lie groups. These are objects which are both smooth manifolds and groups in a compatible way. The fancy way to say it is, of course, that a Lie group is a group object in the category of smooth manifolds.
To be a little more explicit, a Lie group 
is a smooth 
-dimensional manifold equipped with a multiplication 
and an inversion 
which satisfy all the usual group axioms (wow, it’s been a while since I wrote that stuff down) and are also smooth maps between manifolds. Of course, when we write 
we mean the product manifold.
We can use these to construct some other useful maps. For instance, if 
is any particular element we know that we have a smooth inclusion 
defined by 
. Composing this with the multiplication map we get a smooth map 
defined by 
, which we call “left-translation by 
“. Similarly we get a smooth right-translation 
.
Like this:
Like Loading...
June 6, 2011 -
Posted by John Armstrong |
Algebra, Differential Topology, Lie Groups, Topology
[...] a Lie group is a smooth manifold we know that the collection of vector fields form a Lie algebra. But this is [...]
Pingback by The Lie Algebra of a Lie Group « The Unapologetic Mathematician | June 8, 2011 |
[...] Lie groups are groups, they have representations — homomorphisms to the general linear group of some [...]
Pingback by The Adjoint Representation « The Unapologetic Mathematician | June 13, 2011 |