What Is a Lie Group?

[joneall]

TL;DR Summary

Is this adequate?

I'm writing some notes for myself (to read in my rapidly approaching declining years) and I'm wondering if this statement is correct. I"m not sure I am posting this question in the right place.

"Summary: The matrix representations of isometric (distance-preserving) subgroups of the general linear group GL(V), acting on the n-dimensional vector space V, are the orthogonal or unitary matrices, and the Lorentz transformations – O(n), U(n) and O(1,3). The parameter n is the dimension of the vector space and of the group. Both O(n) and U(n) have subgroups characterized as “special”, meaning that they contain only those matrices whose determinant is +1."

I'm especially unsure about that "acting on". Thanks in advance for considering something so simple.

 

[fresh_42]

It doesn't define a Lie group, and you rush through several quite different concepts in only a few lines. I would concentrate on what you actually want to write about.

 

[joneall]

Good point, thanks. However, I am not trying to define a Lie group. I can do that but it's not my intent. I want to condense down to the real minimum of saying why we bother with them and how they can help us. Apparently, this is too minimal.

 

[martinbn]

Good point, thanks. However, I am not trying to define a Lie group. I can do that but it's not my intent. I want to condense down to the real minimum of saying why we bother with them and how they can help us. Apparently, this is too minimal.

Actually, what you have written doesn't say at all "why we bother and how they can help us".

 

[wrobel]

Loosely speaking a Lie group is a smooth manifold $M$ with a smooth mapping $f:M\times M\to M$ such that the operation $x\cdot y:=f(x,y)$ turns $M$ into a group.

 

[fresh_42]

Loosely speaking a Lie group is a smooth manifold $M$ with a smooth mapping $f:M\times M\to M$ such that the operation $x\cdot y:=f(x,y)$ turns $M$ into a group.

Inversion has to be smooth, too, at least locally.

 

[Infrared]

Inversion has to be smooth, too, at least locally.

This follows from the implicit function theorem when multiplication is smooth. Also what does "locally" smooth mean? Isn't smoothness already a local notion?

 

[fresh_42]

I wanted to say that we do not need smoothness on the entire manifold. An open connected neighborhood of $1$ is sufficient, e.g. a local Lie group on $|x|<1$ in comparison to its embedding in $\mathbb{R}.$

 

[gds]

The heading to your question is "Simple definition of Lie group" and no-one above has provided one. A Lie group is a smooth manifold which is also a group such that the group operations are smooth maps.

You then said you wanted to "condense down to the real minimum of saying why we bother with them and how they can help us." Well Lie groups are important in physics because they are used to represent symmetries.

 

[Infrared]

The heading to your question is "Simple definition of Lie group" and no-one above has provided one. A Lie group is a smooth manifold which is also a group such that the group operations are smooth maps.

I think wrobel gave this definition in post 5. You don't need to stipulate that inversion is smooth as I noted in post 7.

 

[gds]

A couple other comments:

• Your statement "The matrix representations of isometric (distance-preserving) subgroups of the general linear group GL(V)..." does not seem to make sense (the "isometric subgroups" part seems to imply that you are saying that these subgroups are "isometric"). It might be better to say something like "The matrix representations of the subgroups of the general linear group $GL(V)$ containing isometries (distance preserving maps)..." might read better.
• I think your statement "acting on the n-dimensional vector space V" is ok. In general, a left group action of a group $G$ on a set $V$ is a map $\ell:G\times V\rightarrow V$ satisfying $\ell(e,v)=v$, where $e\in G$ is the group identity, and $\ell(gh,v)=\ell\big(g,\ell(h,v)\big)$. In your case the group is some (matrix) subgroup of $GL(V)$ and it acts on the left of $V$ in the usual way.

 

[WWGD]

For a Lie group, you can "Do Calculus (Manifold part) on a group". Group operations and Manifold ones (e.g., doing Calculus) combine in a nice way. Groups do not , by default, come with a differential/manifold structure; manifolds do not have a default group operation. Lie Groups combine these two types of structure. Key concepts: Lie Algebra, Exponential map. Exponential map connects Lie group with its Lie Algebra.