The Unapologetic Mathematician

Mathematics for the interested outsider

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 G is a smooth n-dimensional manifold equipped with a multiplication G\times G\to G and an inversion G\to G 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 G\times G we mean the product manifold.

We can use these to construct some other useful maps. For instance, if h\in G is any particular element we know that we have a smooth inclusion G\to G\times G defined by g\mapsto (h,g). Composing this with the multiplication map we get a smooth map L_h:G\to G defined by L_h(g)=hg, which we call “left-translation by h“. Similarly we get a smooth right-translation R_h(g)=gh.

June 6, 2011 Posted by | Algebra, Differential Topology, Lie Groups, Topology | 2 Comments