Here the
story of an idea to construct new examples of non-commutative compact
manifolds, the computational difficulties one runs into and, when they
are solved, the white noise one gets. But, perhaps, someone else can
spot a gem among all gibberish…
[Qurves](https://lievenlb.local/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasi-free algebras, aka formally smooth
algebras) are the \’affine\’ pieces of non-commutative manifolds. Basic
examples of qurves are : semi-simple algebras (e.g. group algebras of
finite groups), [path algebras of
quivers](http://www.lns.cornell.edu/spr/2001-06/msg0033251.html) and
coordinate rings of affine smooth curves. So, let us start with an
affine smooth curve
qurve. First, we bring in finite groups. Let
acting on
hereditary orders). A more pompous way to phrase this is that these are
precisely the [one-dimensional smooth Deligne-Mumford
stacks](http://www.math.lsa.umich.edu/~danielch/paper/stacks.pdf).
As the 21-st century will turn out to be the time we discovered the
importance of non-Noetherian algebras, let us make a jump into the
wilderness and consider the amalgamated free algebra product
again a qurve on which
on
let
sending
group](http://mathworld.wolfram.com/SimpleGroup.html)
simple group has an involution, we have an embedding
compatible with the involution on the affine line. To study the
corresponding non-commutative manifold, that is the Abelian category
to compute the [one quiver to rule them
all](https://lievenlb.local/master/coursenotes/onequiver.pdf) for
connected components. The direct sum of representations turns the set of
all these components into an Abelian semigroup and the vertices of the
\’one quiver\’ correspond to the generators of this semigroup whereas
the number of arrows between two such generators is given by the
dimension of
may seem hard to compute but it can be reduced to the study of another
quiver, the Zariski quiver associated to
with on the left the \’one quiver\’ for
correspond to the two simples of
right the \’one quiver\’ for
many verticers as there are simple representations for
the number of arrows from a left- to a right-vertex is the number of
make matters even more concrete, let us consider the easiest example
when
Zariski quiver then turns out to be
calculate the dimensions of the EXt-spaces giving the number of arrows
in the \’one quiver\’ for
generators of the component semigroup we have to find the minimal
integral solutions to the pair of equations saying that the number of
simple
equal to that one the right-vertices. In this case it is easy to see
that there are as many generators as simple
having the first two components on the left)
info to determine the \’one quiver\’ for
result. Instead one obtains a complete graph on all vertices with plenty
of arrows. More precisely one obtains as the one quiver for
with the number of arrows (in each direction) indicated. Not very
illuminating, I find. Still, as the one quiver is symmetric it follows
that all quotient varieties
structure. Clearly, the above method can be generalized easily and all
examples I did compute so far have this \’nearly complete graph\’
feature. One might hope that if one would start with very special
curves and groups, one might obtain something more interesting. Another
time I\’ll tell what I got starting from Klein\’s quartic (on which the
simple group
to the sporadic simple Mathieu group
Tag: simples
I
found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front
cover the list of the 26 _known_ sporadic groups together with a
starred added in proof saying
- added in
proof … the classification of finite simple groups is complete.
there are no other sporadic groups.
(click on the left picture to see a larger scanned image). In it is a
beautiful paper by John Conway “Monsters and moonshine” on the
classification project. Along the way he describes the simplest
non-trivial simple group
other interpretations as Lie groups over finite fields. He also gives a
nice introduction to representation theory and the properties of the
character table allowing to reconstruct
must be a simple group of order 60.
A more technical account
of the classification project (sketching the main steps in precise
formulations) can be found online in the paper by Ron Solomon On finite simple
groups and their classification. In addition to the posts by John Baez mentioned
in this
post he has a few more columns on Platonic solids and their relation to Lie
algebras, continued here.
Last time we have seen that in order to classify all
non-commutative
dimensional simple algebras having as their center a finite
dimensional field-extension of
classes of simple algebras with the same center
group, the
Brauer group. The calculation of Brauer groups
is best done using
Galois-cohomology. As an aside :
Evariste Galois was one of the more tragic figures in the history of
mathematics, he died at the age of 20 as a result of a duel. There is
a whole site the Evariste Galois archive dedicated to his
work.
But let us return to a simple algebra
field
matrices over a division algebra
maximal commutative subfields of
In this way one can view a simple algebra as a bag containing all
sorts of degree n extensions of its center. All these maximal
subfields are also splitting fields for
if you tensor
matrices
separable field but for a long time it was an open question
whether the collection of all maximal commutative subfields also
contains a Galois-extension of
one could describe the division algebra
product. It was known for some time that there is always a simple
algebra
corresponding to a different number n’), that is, all elements of
the Brauer group can be represented by crossed products. It came as a
surprise when S.A. Amitsur in 1972 came up with examples of
non-crossed product division algebras, that is, division algebras
such that none of its maximal commutative subfields is a Galois
extension of the center. His examples were generic
division algebras
nxn matrices, that is, nxn matrices A and B such that all its
entries are algebraically independent over
field
one can show that this subalgebra is a domain and inverting all its
central elements (which, again, is somewhat of a surprise that
there are lots of them apart from elements of
central polynomials) one obtains the division algebra
center
way, it is still unknown (apart from some low n cases) whether
is purely trancendental over
nature of
field of rational numbers,
example
One can then
ask whether any division algebra
crossed whenever n is squarefree. Even teh simplest case, when n is a
prime number is not known unless p=2 or 3. This shows how little we do
know about finite dimensional division algebras : nobody knows
whether a division algebra of dimension 25 contains a maximal
cyclic subfield (the main problem in deciding this type of
problems is that we know so few methods to construct division
algebras; either they are constructed quite explicitly as a crossed
product or otherwise they are constructed by some generic construction
but then it is very hard to make explicit calculations with
them).