Two approaches to lie algebras

[Hymne]

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.

 

[Matterwave]

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.

 

[morphism]

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.

 

[conquest]

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.

 

[Fangyang Tian]

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.

 

[conquest]

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.

 

[Fangyang Tian]

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??

 

[conquest]

http://en.wikipedia.org/wiki/Abstract_nonsense

Now how much do you trust wikipedia?

 

[Fangyang Tian]

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.

 

[conquest]

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!

 

[Fangyang Tian]

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.