Subject: categories: Categories: Re: Extensions of Z+Z by Z

From: "Stephen Urban Chase" <[email-masked]>
Date: Fri, 30 Apr 2004 17:37:43 -0400 (EDT)

I've often come across such central extensions when studying various
topics in algebra.  For reasons which are obvious from the messages of
Ronnie Brown and John Baez, I've been calling such extensions "Heisenberg
extensions".

The general construction is the following: Let  A  and  C  be abelian
groups, and  f:AxA ----> C  be bilinear.  Then  f  is a  2-cocycle for the
trivial action of  A  on  C, and the corresponding central extension has
the form  E = CxA  with composition law given by the usual formula:
(c,a)(c',a') = (c+c'+f(a,a'), a+a').  The classes of such extensions form
a subgroup of the group of all classes of central extensions of  A  by  C,
but in general is not the whole group.  If  C  is uniquely divisible by
2,
then these extensions have the property that they split over every cyclic
subgroup of  A.

The analogous construction for finite group schemes plays a role in
certain questions in Galois theory and related matters; see, e.g., my
paper, "On a variant of the Witt and Brauer groups", in "Brauer Groups
(Evanston 1975)", Springer LNM 549 (pp 148-187).  Also, relatively free
groups for certain varieties of nilpotent groups of class  2  can be
constructed as Heisenberg extensions.  In particular, the integral
Heisenberg group is the  2-generator relatively free group for the variety
of all nilpotent groups of class 2.

Steve Chase

---------------------------- Original Message ----------------------------
Subject: categories: Extensions of Z+Z by Z
From:    "Colin McLarty" <[email-masked]>
Date:    Sun, April 25, 2004 10:58 pm
To:      [email-masked]
--------------------------------------------------------------------------

After calculating the group extensions of Z+Z by Z, with constant action,
I am curious whether the groups have any more natural form than I found.
I mean extension of Z+Z by Z in this sense, as a sequence of groups where
E need not be commutative:

    0 --> Z --> E --> Z+Z --> 0

and the kernel is in the center of E.

The form I found is parametrized by the integers this way:  For any
integer c, the group E_c has triples of integers (i,,j,k) as elements and
the multiplication rule is coordinate-wise addition plus an extra bit in
the first coordinate.

(i,,j,k).(q,r,s) =  ( (i+j+c.(kr)), j+r, k+s)

When c=0 this is commutative and is just the coproduct Z+Z+Z.  In any
group E_c, the element (c,0,0) is the commutator of (0,0,1) and (0,1,0).
The Baer sum of extensions corresponds to addition of the parameters c as
integers.  So I understand the group of extensions.  Of course I
understood it before I calculated it, since it is the second cohomology
group of the torus.  That is why I tried the algebraic calculation.

But is there a natural way to think about each group E_c, for non-zero
values of c?  Do these groups appear in any other natural way?

thanks, colin