Let *G* be a semisimple algebraic group.


Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor  
 *G* : *Rings* → *Groups* by the second algebraic *K*-theory functor.


Plugging in ℂ((t)) into those functors, we get the well known central extension $\widetilde{G\big(\mathbb C((t))}\big)$ of
the loop group *G*(ℂ((t))) by the multiplicative group ℂ\*=*K*2(ℂ((t))).  

It is interesting to note that the above group comes from an algebraic 
group defined over the subfield ℂ of ℂ((t)). Namely, $\widetilde{G\big(\mathbb C((t))}\big)$ = $\widetilde{LG}(\mathbb C)$.   
  



Doing all this with ℚp instead of ℂ((t)),
we get a central extension $\widetilde{G(\mathbb Q\_p)}$ of *G*(ℚp) by the group *K*2(ℚp) = **F**p\*.
Now, here's an idea: maybe that central extension is defined over... the subfield 
**F**1 of ℚp?...


**My questions:**  

• Has this been considered before?  

• If yes, among all the exitsing notion of "*defined over* **F**1", 
which one(s) make this possible?  

• If no: is my heuristic argument is convincing?




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**References:**  

[1] Matsumoto,
"Sur les sous-groupes arithmétiques des groupes semi-simples déployés".  

[2] Brylinski, Deligne, "Central extensions of reductive groups by $K\_2$".  




Are centrally extended p-adic groups defined over F_1?
First a small thing. I am pretty sure we don't have $K\_2(\mathbb C((t)))=\mathbb
C^\*$, we have a surjective residue homomorphism $K\_2(\mathbb C((t)))\rightarrow
\mathbb C^\*$ but, I believe, with a non-trivial kernel. In any case, we can look
at the induced central extension and then the rest of what you say is
OK. Similarly, we have a surjective map $K\_2(\mathbb Q\_p)$.


Disrergarding this, there is a much simpler analogy between the two cases which on the
one hand, I think, makes the analogy that you want less likely and on the other
hand can be proven... To begin with it is not quite true that even $G(\mathbb
C((t)))$ is defined over $\mathbb C$ at least not as a group scheme. What
happens is that $G(\mathbb C[[t]])$ is a group scheme, it is the inverse limit
of the $G(\mathbb C[t]/(t^n))$ and these have a natural structure of algebraic
group over $\mathbb C$ (through the Greenberg functor). then $G(\mathbb C[[t]])$
as the inverse limit of algebraic groups is a group scheme (it is not of finite
type hence convention forces us to call it a group scheme rather than algebraic
group). Now, if we try to pass to $G(\mathbb
C((t)))$ we get into trouble. It is an infinite union of schemes (bound the
valuations of the entries of the elements of $G(\mathbb
C((t)))$ in some faithful linear representation of $G$) but an infinite union of
schemes does in general not have a scheme structure. There are ways of extending
the scheme notion to cover this case and what we get is what is called an
ind-group scheme over $\mathbb C$. Also the loop group type extension of $G(\mathbb
C((t)))$ by $\mathbb C^\*$ has such an extension (as does every Kac-Moody type
group).


The situation for $G(\mathbb Q\_p)$ is almost identical; $G(\mathbb Z/p^n)$ are
the $\mathbb Z/p$-points of a $\mathbb Z/p$-algebraic group, $G(\mathbb Z\_p)$
are the $\mathbb Z/p$-points of a group scheme over $\mathbb Z/p$ and $G(\mathbb
Q\_p)$ are the $\mathbb Z/p$-points of an ind-group scheme over $\mathbb Z/p$. I
think that the same thing is true for the central extension. The upshot is that
there is a close analogy to the $\mathbb C$ case but in that analogy $\mathbb C$
is replaced by $\mathbb F\_p$ not by $\mathbb F\_1$.


Note that in the Connes-Consani version of $\mathbb F\_1$ $G$ is defined over $\mathbb F\_{1^2}$ so perhaps that is the place to look for a version of the Brylinski-Deligne result.


**Addendum**: Just to add even more concreteness to George's answer about the explicit form of Greenberg's functor for $\mathbb G\_m$. We have that $W\_n(B)$ is just $B^n$ with a funny multiplication and addition. They are however given by polynomials (which are independent of $B$). The units in this ring are the tuples of the form $B^\ast\times B^{n-1}$ and multiplication is given by polynomials. This means that the algebraic group associated to this is just $\mathbb G\_m\times\mathbb A^{n-1}$ as scheme but with a funny product structure. In particular its $\mathbb F\_p$-points are just $(\mathbb Z/p^n)^\ast$.


