# Simple definition of Lie group

• A
Gold Member
TL;DR Summary
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.

## Answers and Replies

Mentor
2022 Award
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.

Gold Member
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.

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".

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.

Mentor
2022 Award
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.

Gold Member
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?

Mentor
2022 Award
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.