Central Extension and Cohomology Groups

[giulio_hep]

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?

 

[fresh_42]

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.

 

[giulio_hep]

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.

 

[giulio_hep]

I've also found a 2 pages explanation from Yet another construction of the central extension of the loop group!

 

[giulio_hep]

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?

 

[fresh_42]

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/

 

[giulio_hep]

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?

 

[fresh_42]

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/)