Two approaches to lie algebras

In summary, the group theory approach is usually that the author introduce the generators and show that these fulfil the algebra. In differential geometry our lie group is a differential manifold for which the algebra is found by left invariant fields or the tangent space..
  • #1
Hymne
89
1
Hello! I've read some lie algebra in both group theory books and differential geometry books, and is confused about the different perspectives.
The group theory approach is usually that the author introduce the generators and show that these fulfil the algebra.
In differential geometry our lie group is a differential manifold for which the algebra is found by left invariant fields or the tangent space..

I.e. in group theory our algebra is fulfilled by matrices and in diff. geometry use the differential operators that span the tangentspace.

Are we talking about the same algebra? How do they relate? Does the generators describe the partial derivatives/ our basis in T_eM? Etc. :)


Please explain!

Thanks.
 
Physics news on Phys.org
  • #2
Yes, you can show that the definitions are equivalent for matrix Lie groups. There are non-matrix Lie groups that you can't simply use the matrix commutator for the Lie bracket. This is where the differential geometry definition is more general.
 
  • #3
As far as algebra is concerned, a Lie algebra is simply a vector space V together with some "special" bilinear map VxV->V that satisfies certain additional properties. If you take this as your definition of "Lie algebra" then it becomes a theorem that the space of left-invariant vector fields on a Lie group together with the Lie bracket (as the "special" bilinear map) is a Lie algebra.
 
Last edited:
  • #4
morphism said:
As far as algebra is concerned, a Lie algebra is simply a vector space V together with some "special" bilinear map VxV->V that satisfies certain additional properties. If you take this as your definition of "Lie algebra" then it becomes a theorem that the space of left-invariant vector fields on a Lie group together with the Lie bracket (as the "special" bilinear map) is a Lie algebra.


Exactly from that point of view just take the definition of algebra, cross out associativity if you have it and write down jacobi identity and add anti-commutativity.

For matrices and indeed any associative algebra you can define this special kind of mulitplication by taking as the special multiplication of a and b the commutator with the other (associative) multiplication. That's how you define the Lie algebra structure on matrices.

For Lie groups you can define this special kind of multiplication on the tangent space by using tangets to conjugation and the exponential map etc.
 
  • #5
First, the differential geometry method still works when we talk about lie groups with infinite dimensions, while we could not calculate infinite dimension matrix seriously. Matrix of infinite dimension is actually a operator and is usually discussed in functional analysis. This generalization is not abstract nonsense since it has application in quantum mechanics. (I'm not major in physics, so I connot verify this detail. I just hear this from my instructor.) It is also necessary in mathematics for calculating diffeomorphism groups is a heartcore of differential topology. Those groups are naturally with infinite dimensions.
Just aware of the difference between the phrase "Lie algebra" and "Lie algebra of a Lie group". One more thing, one version of the Ado Theorem states that a compact Lie Group is isomorphic (in the category of Lie group) to a subgroup of a matrix group, thus when you deal with the compact case, all will be the same.
 
  • #6
Fangyang Tian said:
This generalization is not abstract nonsense since it has application in quantum mechanics. (I'm not major in physics, so I connot verify this detail. I just hear this from my instructor.) It is also necessary in mathematics for calculating diffeomorphism groups is a heartcore of differential topology. Those groups are naturally with infinite dimensions.

I can verify this for you especially in condensed matter theory and QFT this plays a major role.
but try not to obscure the fact that in the finite dimensional case matrices can still be used to signify operators with respect to a certain basis.

also I think the term absract nonsense is usually used to signify things explained by category theories, right? Which is not the case here.
 
  • #7
conquest said:
I can verify this for you especially in condensed matter theory and QFT this plays a major role.
but try not to obscure the fact that in the finite dimensional case matrices can still be used to signify operators with respect to a certain basis.

also I think the term absract nonsense is usually used to signify things explained by category theories, right? Which is not the case here.

Well, what I mean by abstract nonsense is something that purely logic generaliztion without essential application. I've heard of the abstract homotopy theory has solved many essential problems in algebraic geometry, using category language. Thus, sometimes, we could hastely call category theory abstract nonsense, am I right??
 
  • #9
conquest said:
http://en.wikipedia.org/wiki/Abstract_nonsense

Now how much do you trust wikipedia?

Sorry, I did not read those proofs and I'm sure it is beyond my current mathematical level. Maybe it is the theory that wiki criticize not the language abstract homotopy uses. Thanks for the reference.
 
  • #10
Oh sorry I think you got the wrong idea. It just says that the terms 'abstract nonsense' or 'general nonsense' are used to signify that the proof or whatever concerned is formulated in a general category theoretical way. Because of that it doesn't need to mention the actual topic at hand (like Lie algebra, Topology or whatever you want). The article also says that it isn't at all meant to be a derogatory comment, but rather a compliment to the sophistication of the argument.

The idea being that the proof is thus an essential element of the framework of mathematics in general. The only drawback being that it often obscures the underlying intuition to the theorem as stated in the specific field of interest (i.e. again Lie algebra, Topology or whatever you want).

Also we are getting majorly off topic now. I hope the Lie algebra was already quite clear!
 
  • #11
conquest said:
Oh sorry I think you got the wrong idea. It just says that the terms 'abstract nonsense' or 'general nonsense' are used to signify that the proof or whatever concerned is formulated in a general category theoretical way. Because of that it doesn't need to mention the actual topic at hand (like Lie algebra, Topology or whatever you want). The article also says that it isn't at all meant to be a derogatory comment, but rather a compliment to the sophistication of the argument.

The idea being that the proof is thus an essential element of the framework of mathematics in general. The only drawback being that it often obscures the underlying intuition to the theorem as stated in the specific field of interest (i.e. again Lie algebra, Topology or whatever you want).

Also we are getting majorly off topic now. I hope the Lie algebra was already quite clear!

Thanks for correcting me.:blushing:
 

Related to Two approaches to lie algebras

1. What are the two approaches to lie algebras?

The two approaches to lie algebras are the abstract approach and the matrix approach. The abstract approach involves defining a lie algebra as a vector space with a bilinear operation called the Lie bracket, while the matrix approach involves representing the elements of a lie algebra as matrices and using matrix multiplication as the Lie bracket operation.

2. What is the Lie bracket operation?

The Lie bracket is a bilinear operation that takes two elements of a lie algebra and produces another element. It is denoted by [x,y] and is defined as the commutator of x and y, where [x,y] = xy - yx.

3. What is the significance of the structure constants in the matrix approach?

In the matrix approach, the structure constants represent the coefficients of the Lie bracket operation. They are used to define the algebraic structure of the lie algebra and determine the properties of the algebra.

4. What is the relationship between Lie algebras and Lie groups?

Lie algebras and Lie groups are closely related. A Lie group is a type of continuous group that can be described by a differentiable manifold. Lie algebras, on the other hand, are the tangent spaces of a Lie group at its identity element. So, every Lie group has a corresponding Lie algebra and vice versa.

5. How are Lie algebras used in physics?

Lie algebras have many applications in physics, particularly in the study of symmetries and conservation laws. They are used to describe the algebraic structure of physical systems and play a crucial role in theoretical physics, including quantum mechanics and relativity.

Similar threads

Replies
8
Views
3K
  • Differential Geometry
Replies
1
Views
2K
Replies
14
Views
2K
  • Science and Math Textbooks
Replies
2
Views
2K
Replies
6
Views
1K
Replies
7
Views
2K
  • Differential Geometry
Replies
9
Views
2K
  • Advanced Physics Homework Help
Replies
2
Views
957
  • Differential Geometry
Replies
5
Views
3K
  • Science and Math Textbooks
Replies
5
Views
1K
Back
Top