Central Extension and Cohomology Groups

In summary, the conversation discusses the concept of central extensions in mathematics, specifically in terms of Lie groups and Lie algebras. It is explained that central extensions arise when a group is not simply connected, and they can be understood through group cohomology. The example of a spin group as a central extension of SO(3) is given, and it is noted that central extensions can also be seen in terms of algebras. The connection between central extensions and topology is discussed, and it is mentioned that covers are a topological property while central extensions are an algebraic one. The conversation also touches on the motivation for considering central extensions in physics, specifically in relation to supersymmetry and two-dimensional conformal quantum field theory.
  • #1
104
6
TL;DR Summary
Wikipedia explanation about cohomology is very obscure to me and I'm wondering whether I can find here help to translate it in simpler terms.
Wikipedia says that a general projective representation cannot be lifted to a linear representation and the obstruction to this lifting can be understood via group cohomology.

For example, I see that a spin group is a central extension of SO(3) by Z/2.

More generally I can follow the reasoning that central extensions of Lie groups by discrete groups are covering groups and all projective representations of G are linear representations of the universal cover, hence no central charges occur.

But while I can easily admit that the discrete central group above happens to be (isomorphic to) the fundamental group of the Lie group G, I can't really grasp how the homotopy and cohomology enter here by their definitions ? How the second cohomology group is in one-to-one correspondence with the set of central extensions?
 
Physics news on Phys.org
  • #2
You can find the full answer e.g. here:
http://www.mathematik.uni-regensburg.de/loeh/teaching/grouphom_ss19/lecture_notes.pdf
Section 1.5.2 pages 32ff.

It is impossible to explain all the 40 pages here. But if you write down the coboundary and cocyclic groups in terms of the coboundary operator, then you may see where the central extensions come into play. Means: ##H^2= Z^2/B^2=\operatorname{ker}d/\operatorname{im}d##

It is easier to consider the algebras instead of the groups. The keyword then is the Chevalley-Eilenberg complex.
https://en.wikipedia.org/wiki/Lie_algebra_cohomology
You can also find the definition of the coboundary operator ##d## there.
 
  • Like
Likes sysprog and giulio_hep
  • #3
For example, in case of SO(3) can we say that as well as the Spin group also SU(2) is a double cover of SO (3) hence SU(2) is a central extension? Or which is the difference in terms of the coboundary operator?

Never mind: finally found it written explicitly: "Spin(3) is none other than SU(2)" and also here "SU(2) is a central extension of SO(3) (there is a relation with spin in physics).". Still funny, afaics, that it's always used the term Spin when it is described as central extension.
 
Last edited:
  • #5
In conclusion, I wanted also to ask a more philosophical question, an explanation without all the details but high level: what is the fact of being also a central extension adding on top of a double cover? My understanding is that it is adding a structure of morphisms, maybe a tensor product? Which is the obstruction given by cohomology groups? A topological obstruction for writing SU(2) as a product SO(3) ×Z2 is given by an element in the first Čech cohomology and IMHO this seems more relevant than the second - trivial - cohomology group, doesn't it?
 
  • #6
giulio_hep said:
In conclusion, I wanted also to ask a more philosophical question, an explanation without all the details but high level: what is the fact of being also a central extension adding on top of a double cover? My understanding is that it is adding a structure of morphisms, maybe a tensor product? Which is the obstruction given by cohomology groups? A topological obstruction for writing SU(2) as a product SO(3) ×Z2 is given by an element in the first Čech cohomology and IMHO this seems more relevant than the second - trivial - cohomology group, doesn't it?
This connection is more incidentally than it is some hidden truth. The centers are multiples of the identity. If the determinant is restricted to ##1## as in your example, then we simply have ##\pm 1## in even dimensions. Since there is no path from ##+1## to ##-1## within the group, it has to be a double cover. Hence the center determines the cover in this example. However, I do not see a way back. Covers are a topological property, central extensions an algebraic one. You have to add the entire group property to the topological property in order to arrive at a central extension. But then you have added so much algebra to the topology, that you can't say what implicated what anymore.

Here is a list of all the bijections in the realm of your example:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/
 
  • Like
Likes giulio_hep
  • #7
Ok, maybe I took a too trivial example just to start and indeed you're right, here's more a matter of algebra than topology, in fact my goal is to understand Lie algebra extensions more in general.

On the opposite side of complexity, I would rephrase my question with the example of Virasoro algebra as a central extension of the Witt algebra. In this - much more complicated case than su(2) - what is the central extension adding on top of the base algebra? The concept of a central charge? Again, aside from the nitty gritty details of the computation, I mean: what is the motivation instead, why is it useful to extend, let's say, the Witt into Virasoro algebra?
 
  • #8
giulio_hep said:
Ok, maybe I took a too trivial example just to start and indeed you're right, here's more a matter of algebra than topology, in fact my goal is to understand Lie algebra extensions more in general.

On the opposite side of complexity, I would rephrase my question with the example of Virasoro algebra as a central extension of the Witt algebra. In this - much more complicated case than su(2) - what is the central extension adding on top of the base algebra? The concept of a central charge? Again, aside from the nitty gritty details of the computation, I mean: what is the motivation instead, why is it useful to extend, let's say, the Witt into Virasoro algebra?
Not sure what you mean, or if that isn't a question better asked in the QM forum. The Virasoro algebra is defined as a one-dimensional central extension. So I read your question as to why we consider Virasoro algebras at all in physics. My (math) book says in its introduction:
Supersymmetry is a symmetry between bosons and fermions. The Virasoro algebra occurs in connection with statistics in two dimensions (two-dimensional conformal quantum field theory), which are neither Bose nor Fermi statistics, but so-called braid group statistics.
and further down the road:
If we examine the Virasoro algebra on a complex vector space ##V##, we may consider ##C## [the central element] as a complex multiple of the identical mapping of ##V.## ##C## is therefore described as the central charge of the Virasoro algebra.
This looks to me as if it mainly allows a ##1## in the algebra, which makes it more convenient to handle, especially if topological tools like the partition of unity are used (as in this case). On the other hand, as a central element, it doesn't harm anything else.

However, I'm no physicist, so this is a purely mathematical point of view.
(Quotations from https://www.amazon.com/dp/3519020874/)
 
Last edited:
  • Like
Likes giulio_hep

1. What is the definition of central extension?

Central extension is a mathematical concept in group theory where a group is extended by adding a new element that commutes with all the elements in the original group.

2. How are central extensions related to cohomology groups?

Central extensions and cohomology groups are closely related, as cohomology groups can be used to classify central extensions. Specifically, the second cohomology group of a group with coefficients in its center is isomorphic to the set of isomorphism classes of central extensions of that group.

3. What is the importance of central extensions in group theory?

Central extensions are important in group theory as they provide a way to study and classify groups with certain properties. They also allow for the construction of new groups with desired properties by extending existing ones.

4. Can central extensions be used to study non-abelian groups?

Yes, central extensions can be used to study non-abelian groups. In fact, central extensions are particularly useful for studying non-abelian groups as they provide a way to construct new non-abelian groups from existing ones.

5. Are there any applications of central extensions in other fields of science?

Yes, central extensions have applications in various fields of science such as physics, where they are used to study symmetries and gauge theories. They also have applications in cryptography, where they are used to construct secure communication protocols.

Suggested for: Central Extension and Cohomology Groups

Replies
1
Views
833
Replies
3
Views
601
Replies
11
Views
1K
Replies
2
Views
685
Replies
1
Views
699
Replies
6
Views
1K
Back
Top