Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.
Leave a CommentTag: Riemann
After a lengthy spring-break, let us continue with our course on noncommutative geometry and
and though these points are very special there are enough of them (technically, they form a Zariski dense subset of all representations). Our aim will be twofold : (1) when viewing a classical object as a representation of
- This
- dessin determines a 24-dimensional permutation representation (of
as well of ) which- decomposes as the direct sum of the trivial representation and a simple
- 23-dimensional representation. We will see that the noncommutative
- tangent space in a semi-simple representation of
is determined by a quiver (that is, an- oriented graph) on as many vertices as there are non-isomorphic simple
- components. In this special case we get the quiver on two points
- $\xymatrix{\vtx{} \ar@/^2ex/[rr] & & \vtx{} \ar@/^2ex/[ll]
- \ar@{=>}@(ur,dr)^{96} } $ with just one arrow in each direction
- between the vertices and 96 loops in the second vertex. To the
- experienced tangent space-reader this picture (and in particular that
- there is a unique cycle between the two vertices) tells the remarkable
- fact that there is **a distinguished one-parameter family of
- 24-dimensional simple modular representations degenerating to the
- permutation representation of the largest Mathieu-group**. Phrased
- differently, there is a specific noncommutative modular Riemann surface
- associated to
, which is a new object (at least as far - as I’m aware) associated to this most remarkable of sporadic groups.
- Conversely, from the matrix-representation of the 24-dimensional
- permutation representation of
we obtain representants - of all of this one-parameter family of simple
-representations to which we can then perform- noncommutative flow-tricks to get a Zariski dense set of all
- 24-dimensional simples lying in the same component. (Btw. there are
- also such noncommutative Riemann surfaces associated to the other
- sporadic Mathieu groups, though not to the other sporadics…) So this
- is what we will be doing in the upcoming posts (10) : explain what a
- noncommutative tangent space is and what it has to do with quivers (11)
- what is the noncommutative manifold of
? and what is its connection with the Kontsevich-Soibelman coalgebra? (12) - is there a noncommutative compactification of
? (and other arithmetical groups) (13) : how does one calculate the noncommutative curves associated to the Mathieu groups? (14) : whatever comes next… (if anything).
Today we will explain how curves defined over
of the carthographic groups. We have seen that any smooth projective
curve
closure
_Belyi map_
exactly
of
associate a _dessin d\’enfant_, a drawing on
pre-images of
edges (the preimages of the open unit-interval). Next, we look at
the preimages of
counter-clockwise around these preimages and recording the edges we
meet. We repeat this procedure for the preimages of
get another permutation
subgroup of the symmetric group
group of the covering
For example, the
dessin on the right is
associated to a degree
the preimages of
corresponding partitions are
and
is the alternating group
GAP ).
But wait! The map is also
ramified in
permutation
the dessin? (Note that all three partitions are needed if we want to
reconstruct
which order the sheets fit together around the preimages). Well,
the monodromy group of a
in three points is an epimorphic image of the fundamental
group of the sphere
minus three points
ending in a basepoint upto homotopy (that is, two such loops are the
same if they can be transformed into each other in a continuous way
while avoiding the three points).
This group is generated by loops
point, doing a counter-clockwise walk around it and going back to be
basepoint
Now,
these three generators are not independent. In fact, this fundamental
group is
To understand this, let us begin
with an easier case, that of the sphere minus one point. The fundamental group of the plane minus one point is
point. However, on the sphere the situation is different as we can make
our walk around the point longer and longer until the whole walk is done
at the backside of the sphere and then we can just contract our walk to
the basepoint. So, there is just one type of walk on a sphere minus one
point (upto homotopy) whence this fundamental group is trivial. Next,
let us consider the sphere minus two points
Repeat the foregoing to the walk
is, strech the upper part of the circular tour all over the backside of
the sphere and then we see that we can move it to fit with the walk
modified walk
trivial walk. So, this fundamental group is
we can modify the third walk
it becomes the walk
with the reversed orientation ! As
\’missing\’ permutation
corresponding to the fact that the dessin has two regions (remember we
should draw ths on the sphere) : the head and the outer-region. Hence,
the pre-images of
dessin on the curve
consider the degree 168 map
which is the modified orbit map for the action of
The corresponding dessin is the heptagonal construction of the Klein
quartic
Here, the pre-images of 1 correspond to the midpoints of the
84 edges of the polytope whereas the pre-images of 0 correspond to the
56 vertices. We can label the 168 half-edges by numbers such that
resp. a of the 168-dimensional regular representation (see the atlas
page ).
Calculating with GAP the element
consists of 24 cycles of length 7, so again, the pre-images of
Klein quartic. Now, we are in a position to relate curves defined
over
dessins to Grothendiecks carthographic groups
dessin gives a permutation representation of the monodromy group and
because the fundamental group of the sphere minus three
points
any dessin determines a permutation representation of the congruence
subgroup
post where we proved that this
group is free). A clean dessin is one for which one type of
vertex has all its valancies (the number of edges in the dessin meeting
the vertex) equal to one or two. (for example, the pre-images of 1 in
the Klein quartic-dessin or the pre-images of 1 in the monsieur Mathieu
example ) The corresponding
permutation
monodromy group gives a permutation representation of the free
product
quilt dessin if also the other type of vertex has all its valancies
equal to one or three (as in the Klein quartic or Mathieu examples).
Then, the corresponding permutation has order 3 and for these
quilt-dessins the monodromy group gives a permutation representation of
the free product
Grothendieck to his anabelian geometric approach to the absolute Galois
group.