# The Unapologetic Mathematician

## 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 $R_h(g)=gh$.

## 2 Comments	»

1. [...] a Lie group is a smooth manifold we know that the collection of vector fields form a Lie algebra. But this is [...]

Pingback by | June 8, 2011 | Reply

2. [...] Lie groups are groups, they have representations — homomorphisms to the general linear group of some [...]

Pingback by | June 13, 2011 | Reply

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## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

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