Simple definition of Lie group

In summary: Lie Algebra is the mathematical structure that generalizes linear algebra. Lie groups can be seen as vector spaces with an additional structure, that of a Lie algebra.In 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
  • #1
joneall
Gold Member
67
14
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.
 
Physics news on Phys.org
  • #2
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.
 
  • #3
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.
 
  • #4
joneall said:
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".
 
  • #5
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.
 
  • #6
wrobel said:
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.
 
  • #7
fresh_42 said:
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?
 
  • #8
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}.##
 
  • #9
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.
 
  • #10
gds said:
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.
 
  • Like
Likes fresh_42
  • #11
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.
 
  • #12
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.
 

1. What is a Lie group?

A Lie group is a mathematical object that combines the concepts of a group and a smooth manifold. It is a set of elements that can be multiplied and inverted, and has a continuous structure that allows for the use of calculus and other mathematical tools.

2. What is the difference between a Lie group and a general group?

A general group is a set of elements that can be combined using a binary operation, such as multiplication or addition. A Lie group is a type of group that also has a smooth structure, meaning that it can be described using continuous functions and derivatives.

3. Can you provide an example of a Lie group?

One example of a Lie group is the special orthogonal group, denoted as SO(n). This group consists of all n-by-n orthogonal matrices with determinant 1, and can be used to represent rotations in n-dimensional space.

4. What is the significance of Lie groups in mathematics?

Lie groups are important in many areas of mathematics, including differential geometry, representation theory, and physics. They provide a way to study symmetries and transformations in a continuous and rigorous manner.

5. How are Lie groups related to Lie algebras?

Lie algebras are mathematical objects that are closely related to Lie groups. They are vector spaces equipped with a bilinear operation, called the Lie bracket, that satisfies certain properties. Lie groups and Lie algebras are connected through the concept of the exponential map, which allows for the translation of algebraic structures to geometric ones and vice versa.

Similar threads

Replies
4
Views
1K
  • Differential Geometry
Replies
14
Views
2K
  • Differential Geometry
Replies
10
Views
2K
  • Differential Geometry
Replies
3
Views
1K
Replies
14
Views
2K
  • Differential Geometry
Replies
1
Views
1K
Replies
3
Views
1K
Replies
5
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
5
Views
2K
  • Advanced Physics Homework Help
Replies
14
Views
1K
Back
Top