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noncommutative curves and their maniflds

Published March 17, 2007 by lievenlb

Last time we have
seen that the noncommutative manifold of a Riemann surface can be viewed
as that Riemann surface together with a loop in each point. The extra
loop-structure tells us that all finite dimensional representations of
the coordinate ring can be found by separating over points and those
living at just one point are classified by the isoclasses of nilpotent
matrices, that is are parametrized by the partitions (corresponding
to the sizes of the Jordan blocks). In addition, these loops tell us
that the Riemann surface locally looks like a Riemann sphere, so an
equivalent mental picture of the local structure of this
noncommutative manifold is given by the picture on teh left, where the surface is part of the Riemann surface
and a sphere is placed at every point. Today we will consider
genuine noncommutative curves and describe their corresponding
noncommutative manifolds.

Here, a mental picture of such a
_noncommutative sphere_ to keep in mind would be something
like the picture on the right. That is, in most points of the sphere we place as before again
a Riemann sphere but in a finite number of points a different phenomen
occurs : we get a cluster of infinitesimally nearby points. We
will explain this picture with an easy example. Consider the
complex plane $\mathbb{C} $, the points of which are just the
one-dimensional representations of the polynomial algebra in one
variable $\mathbb{C}[z] $ (any algebra map $\mathbb{C}[z] \rightarrow \mathbb{C} $ is fully determined by the image of z). On this plane we
have an automorphism of order two sending a complex number z to its
negative -z (so this automorphism can be seen as a point-reflexion
with center the zero element 0). This automorphism extends to
the polynomial algebra, again induced by sending z to -z. That
is, the image of a polynomial $f(z) \in \mathbb{C}[z] $ under this
automorphism is f(-z).

With this data we can form a noncommutative
algebra, the _skew-group algebra_ $\mathbb{C}[z] \ast C_2 $ the
elements of which are either of the form $f(z) \ast e $ or $g(z) \ast g $ where
$C_2 = \langle g : g^2=e \rangle $ is the cyclic group of order two
generated by the automorphism g and f(z),g(z) are arbitrary
polynomials in z.

The multiplication on this algebra is determined by
the following rules

$(g(z) \ast g)(f(z) \ast e) = g(z)f(-z) \astg $ whereas $(f(z) \ast e)(g(z) \ast g) = f(z)g(z) \ast g $

$(f(z) \ast e)(g(z) \ast e) = f(z)g(z) \ast e $ whereas $(f(z) \ast g)(g(z)\ast g) = f(z)g(-z) \ast e $

That is, multiplication in the
$\mathbb{C}[z] $ factor is the usual multiplication, multiplication in
the $C_2 $ factor is the usual group-multiplication but when we want
to get a polynomial from right to left over a group-element we have to
apply the corresponding automorphism to the polynomial (thats why we
call it a _skew_ group-algebra).

Alternatively, remark that as
a $\mathbb{C} $-algebra the skew-group algebra $\mathbb{C}[z] \ast C_2 $ is
an algebra with unit element 1 = 1\aste and is generated by
the elements $X = z \ast e $ and $Y = 1 \ast g $ and that the defining
relations of the multiplication are

$Y^2 = 1 $ and $Y.X =-X.Y $

hence another description would
be

$\mathbb{C}[z] \ast C_2 = \frac{\mathbb{C} \langle X,Y \rangle}{ (Y^2-1,XY+YX) } $

It can be shown that skew-group
algebras over the coordinate ring of smooth curves are _noncommutative
smooth algebras_ whence there is a noncommutative manifold associated
to them. Recall from last time the noncommutative manifold of a
smooth algebra A is a device to classify all finite dimensional
representations of A upto isomorphism Let us therefore try to
determine some of these representations, starting with the
one-dimensional ones, that is, algebra maps from

$\mathbb{C}[z] \ast C_2 = \frac{\mathbb{C} \langle X,Y \rangle}{ (Y^2-1,XY+YX) } \rightarrow \mathbb{C} $

Such a map is determined by the image of X and that of
Y. Now, as $Y^2=1 $ we have just two choices for the image of Y
namely +1 or -1. But then, as the image is a commutative algebra
and as XY+YX=0 we must have that the image of 2XY is zero whence the
image of X must be zero. That is, we have only
two one-dimensional representations, namely $S_+ : X \rightarrow 0, Y \rightarrow 1 $
and $S_- : X \rightarrow 0, Y \rightarrow -1 $

This is odd! Can
it be that our noncommutative manifold has just 2 points? Of course not.
In fact, these two points are the exceptional ones giving us a cluster
of nearby points (see below) whereas most points of our
noncommutative manifold will correspond to 2-dimensional
representations!

So, let’s hunt them down. The
center of $\mathbb{C}[z]\ast C_2 $ (that is, the elements commuting with
all others) consists of all elements of the form $f(z)\ast e $ with f an
_even_ polynomial, that is, f(z)=f(-z) (because it has to commute
with 1\ast g), so is equal to the subalgebra $\mathbb{C}[z^2]\ast e $.

The
manifold corresponding to this subring is again the complex plane
$\mathbb{C} $ of which the points correspond to all one-dimensional
representations of $\mathbb{C}[z^2]\ast e $ (determined by the image of
$z^2\ast e $).

We will now show that to each point of $\mathbb{C} – { 0 } $
corresponds a simple 2-dimensional representation of
$\mathbb{C}[z]\ast C_2 $.

If a is not zero, we will consider the
quotient of the skew-group algebra modulo the twosided ideal generated
by $z^2\ast e-a $. It turns out
that

$\frac{\mathbb{C}[z]\ast C_2}{(z^2\aste-a)} =
\frac{\mathbb{C}[z]}{(z^2-a)} \ast C_2 = (\frac{\mathbb{C}[z]}{(z-\sqrt{a})}
\oplus \frac{\mathbb{C}[z]}{(z+\sqrt{a})}) \ast C_2 = (\mathbb{C}
\oplus \mathbb{C}) \ast C_2 $

where the skew-group algebra on the
right is given by the automorphism g on $\mathbb{C} \oplus \mathbb{C} $ interchanging the two factors. If you want to
become more familiar with working in skew-group algebras work out the
details of the fact that there is an algebra-isomorphism between
$(\mathbb{C} \oplus \mathbb{C}) \ast C_2 $ and the algebra of $2 \times 2 $ matrices $M_2(\mathbb{C}) $. Here is the
identification

$~(1,0)\aste \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} $

$~(0,1)\aste \rightarrow \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $

$~(1,0)\astg \rightarrow \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $

$~(0,1)\astg \rightarrow \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} $

so you have to verify that multiplication
on the left hand side (that is in $(\mathbb{C} \oplus \mathbb{C}) \ast
C_2 $) coincides with matrix-multiplication of the associated
matrices.

Okay, this begins to look like what we are after. To
every point of the complex plane minus zero (or to every point of the
Riemann sphere minus the two points ${ 0,\infty } $) we have
associated a two-dimensional simple representation of the skew-group
algebra (btw. simple means that the matrices determined by the images
of X and Y generate the whole matrix-algebra).

In fact, we
now have already classified ‘most’ of the finite dimensional
representations of $\mathbb{C}[z]\ast C_2 $, namely those n-dimensional
representations

$\mathbb{C}[z]\ast C_2 =
\frac{\mathbb{C} \langle X,Y \rangle}{(Y^2-1,XY+YX)} \rightarrow M_n(\mathbb{C}) $

for which the image of X is an invertible $n \times n $ matrix. We can show that such representations only exist when
n is an even number, say n=2m and that any such representation is
again determined by the geometric/combinatorial data we found last time
for a Riemann surface.

That is, It is determined by a finite
number ${ P_1,\dots,P_k } $ of points from $\mathbb{C} – 0 $ where
k is at most m. For each index i we have a positive
number $a_i $ such that $a_1+\dots+a_k=m $ and finally for each i we
also have a partition of $a_i $.

That is our noncommutative
manifold looks like all points of $\mathbb{C}-0 $ with one loop in each
point. However, we have to remember that each point now determines a
simple 2-dimensional representation and that in order to get all
finite dimensional representations with det(X) non-zero we have to
scale up representations of $\mathbb{C}[z^2] $ by a factor two.
The technical term here is that of a Morita equivalence (or that the
noncommutative algebra is an Azumaya algebra over
$\mathbb{C}-0 $).

What about the remaining representations, that
is, those for which Det(X)=0? We have already seen that there are two
1-dimensional representations $S_+ $ and $S_- $ lying over 0, so how
do they fit in our noncommutative manifold? Should we consider them as
two points and draw also a loop in each of them or do we have to do
something different? Rememer that drawing a loop means in our
geometry -> representation dictionary that the representations
living at that point are classified in the same way as nilpotent
matrices.

Hence, drawing a loop in $S_+ $ would mean that we have a
2-dimensional representation of $\mathbb{C}[z]\ast C_2 $ (different from
$S_+ \oplus S_+ $) and any such representation must correspond to
matrices

$X \rightarrow \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $ and $Y \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $

But this is not possible as these matrices do
_not_ satisfy the relation XY+YX=0. Hence, there is no loop in $S_+ $
and similarly also no loop in $S_- $.

However, there are non
semi-simple two dimensional representations build out of the simples
$S_+ $ and $S_- $. For, consider the matrices

$X \rightarrow \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} $ and $Y \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $

then these
matrices _do_ satisfy XY+YX=0! (and there is another matrix-pair
interchanging $\pm 1 $ in the Y-matrix). In erudite terminology this
says that there is a _nontrivial extension_ between $S_+ $ and $S_- $
and one between $S_- $ and $S_+ $.

In our dictionary we will encode this
information by the picture

$\xymatrix{\vtx{}
\ar@/^2ex/[rr] & & \vtx{} \ar@/^2ex/[ll]} $

where the two
vertices correspond to the points $S_+ $ and $S_- $ and the arrows
represent the observed extensions. In fact, this data suffices to finish
our classification project of finite dimensional representations of
the noncommutative curve $\mathbb{C}[z] \ast C_2 $.

Those with Det(X)=0
are of the form : $R \oplus T $ where R is a representation with
invertible X-matrix (which we classified before) and T is a direct
sum of representations involving only the simple factors $S_+ $ and
$S_- $ and obtained by iterating the 2-dimensional idea. That is, for
each factor the Y-matrix has alternating $\pm 1 $ along the diagonal
and the X-matrix is the full nilpotent Jordan-matrix.

So
here is our picture of the noncommutative manifold of the
noncommutative curve $\mathbb{C}[z]\ast C_2 $ : the points are all points
of $\mathbb{C}-0 $ together with one loop in each of them together
with two points lying over 0 where we draw the above picture of arrows
between them. One should view these two points as lying
infinetesimally close to each other and the gluing
data

$\xymatrix{\vtx{} \ar@/^2ex/[rr] & & \vtx{}
\ar@/^2ex/[ll]} $

contains enough information to determine
that all other points of the noncommutative manifold in the vicinity of
this cluster should be two dimensional simples! The methods used
in this simple minded example are strong enough to determine the
structure of the noncommutative manifold of _any_ noncommutative curve.


So, let us look at a real-life example. Once again, take the
Kleinian quartic In a previous
course-post we recalled that
there is an action by automorphisms on the Klein quartic K by the
finite simple group $PSL_2(\mathbb{F}_7) $ of order 168. Hence, we
can form the noncommutative Klein-quartic $K \ast PSL_2(\mathbb{F}_7) $
(take affine pieces consisting of complements of orbits and do the
skew-group algebra construction on them and then glue these pieces
together again).

We have also seen that the orbits are classified
under a Belyi-map $K \rightarrow \mathbb{P}^1_{\mathbb{C}} $ and that this map
had the property that over any point of $\mathbb{P}^1_{\mathbb{C}}
– { 0,1,\infty } $ there is an orbit consisting of 168 points
whereas over 0 (resp. 1 and $\infty $) there is an orbit
consisting of 56 (resp. 84 and 24 points).

So what is
the noncommutative manifold associated to the noncommutative Kleinian?
Well, it looks like the picture we had at the start of this
post For all but three points of the Riemann sphere
$\mathbb{P}^1 – { 0,1,\infty } $ we have one point and one loop
(corresponding to a simple 168-dimensional representation of $K \ast
PSL_2(\mathbb{F}_7) $) together with clusters of infinitesimally nearby
points lying over 0,1 and $\infty $ (the cluster over 0
is depicted, the two others only indicated).

Over 0 we have
three points connected by the diagram

$\xymatrix{& \vtx{} \ar[ddl] & \\ & & \\ \vtx{} \ar[rr] & & \vtx{} \ar[uul]} $

where each of the vertices corresponds to a
simple 56-dimensional representation. Over 1 we have a cluster of
two points corresponding to 84-dimensional simples and connected by
the picture we had in the $\mathbb{C}[z]\ast C_2 $ example).

Finally,
over $\infty $ we have the most interesting cluster, consisting of the
seven dwarfs (each corresponding to a simple representation of dimension
24) and connected to each other via the
picture

$\xymatrix{& & \vtx{} \ar[dll] & & \\ \vtx{} \ar[d] & & & & \vtx{} \ar[ull] \\ \vtx{} \ar[dr] & & & & \vtx{} \ar[u] \\ & \vtx{} \ar[rr] & & \vtx{} \ar[ur] &} $

Again, this noncommutative manifold gives us
all information needed to give a complete classification of all finite
dimensional $K \ast PSL_2(\mathbb{F}_7) $-representations. One
can prove that all exceptional clusters of points for a noncommutative
curve are connected by a cyclic quiver as the ones above. However, these
examples are still pretty tame (in more than one sense) as these
noncommutative algebras are finite over their centers, are Noetherian
etc. The situation will become a lot wilder when we come to exotic
situations such as the noncommutative manifold of
$SL_2(\mathbb{Z}) $…

One Comment

the noncommutative manifold of a Riemann surface

Published March 16, 2007 by lievenlb

The
natural habitat of this lesson is a bit further down the course, but it
was called into existence by a comment/question by
Kea

I don’t yet quite see where the nc
manifolds are, but I guess that’s coming.

As
I’m enjoying telling about all sorts of sources of finite dimensional
representations of $SL_2(\mathbb{Z}) $ (and will carry on doing so for
some time), more people may begin to wonder where I’m heading. For this
reason I’ll do a couple of very elementary posts on simple examples of
noncommutative manifolds.

I realize it is ‘bon ton’ these days
to say that noncommutative manifolds are virtual objects associated to
noncommutative algebras and that the calculation of certain invariants
of these algebras gives insight into the topology and/or geometry of
these non-existent spaces. My own attitude to noncommutative geometry is
different : to me, noncommutative manifolds are genuine sets of points
equipped with a topology and other structures which I can use as a
mnemotechnic device to solve the problem of interest to me which is the
classification of all finite dimensional representations of a smooth
noncommutative algebra.

Hence, when I speak of the
‘noncommutative manifold of $SL_2(\mathbb{Z}) $’ Im after an object
containing enough information to allow me (at least in principle) to
classify the isomorphism classes of all finite dimensional
$SL_2(\mathbb{Z}) $-representations. The whole point of this course is
to show that such an object exists and that we can make explicit
calculations with it. But I’m running far ahead. Let us start with
an elementary question :

Riemann surfaces are examples of
noncommutative manifolds, so what is the noncommutative picture of
them?


I’ve browsed the Google-pictures a bit and a picture
coming close to my mental image of the noncommutative manifold of a
Riemann surface locally looks like the picture on the left. Here, the checkerboard-surface is part of the Riemann surface
and the extra structure consists in putting in each point of the Riemann
surface a sphere, reflecting the local structure of the Riemann surface
near the point. In fact, my picture is slightly different : I want to
draw a loop in each point of the Riemann surface, but Ill explain why
the two pictures are equivalent and why they present a solution to the
problem of classifying all finite dimensional representations of the
Riemann surface. After all why do we draw and study Riemann
surfaces? Because we are interested in the solutions to equations. For
example, the points of the _Kleinian quartic Riemann
surface_ give us all solutions tex \in
\mathbb{C}^3 $ to the equation $X^3Y+Y^3Z+Z^3X=0 $. If (a,b,c) is such
a solution, then so are all scalar multiples $(\lambda a,\lambda
b,\lambda c) $ so we may as well assume that the Z$coordinate is equal
to 1 and are then interested in finding the solutions tex \in
\mathbb{C}^2 $ to the equation $X^3Y+Y^3+X=0 $ which gives us an affine
patch of the Kleinian quartic (in fact, these solutions give us all
points except for two, corresponding to the _points at infinity_ needed
to make the picture compact so that we can hold it in our hand and look
at it from all sides. These points at infinity correspond to the trivial
solutions (1,0,0) and (0,1,0)).

What is the connection
between points on this Riemann surface and representations? Well, if
(a,b) is a solution to the equation $X^3Y+Y^3+X=0 $, then we have a
_one-dimensional representation_ of the affine _coordinate ring_
$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) $, that is, an algebra
morphism

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) \rightarrow \mathbb{C} $

defined by sending X to a and Y to b.
Conversely, any such one-dimensonal representation gives us a solution
(look at the images of X and Y and these will be the coordinates of
a solution). Thus, commutative algebraic geometry of smooth
curves (that is Riemann surfaces if you look at the ‘real’ picture)
can be seen as the study of one-dimensional representations of their
smooth coordinate algebras. In other words, the classical Riemann
surface gives us already the classifcation of all one-dimensional
representations, so now we are after the ‘other ones’.

In
noncommtative algebra it is not natural to restrict attention to algebra
maps to $\mathbb{C} $, at least we would also like to include algebra
maps to $n \times n $ matrices $M_n(\mathbb{C}) $. An n-dimensional
representation of the coordinate algebra of the Klein quartic is an
algebra map

$\mathbb{C}[X,Y]/(X^3Y+Y^3+X) \rightarrow M_n(\mathbb{C}) $

That is, we want to find all pairs of $n \times n $ matrices A and B satisfying the following
matrix-identities

$A.B=B.A $ and $A^3.B+B^3+A=0_n $

The
first equation tells us that the two matrices must commute (because we
took commuting variables X and Y) and the second equation really is
a set of $n^2 $-equations in the matrix-entries of A and
B.

There is a sneaky way to get lots of such matrix-couples
from a given solution (A,B), namely by _simultaneous conjugation_.
That is, if $C \in GL_n(\mathbb{C}) $ is any invertible $n \times n $
matrix, then also the matrix-couple $~(C^{-1}.A.C,C^{-1}.B.C) $
satisfies all the required equations (write the equations out and notice
that middle terms of the form $C.C^{-1} $ cancel out and check that one
then obtains the matrix-identities

$C^{-1} A B C = C^{-1} BA C $ and $C^{-1}(A^3B+B^3+A)C = 0_n $

which are satisfied because
(A,B) was supposed to be a solution). We then say that these two
n-dimensional representations are _isomorphic_ and naturally we are
only interested in classifying the isomorphism classes of all
representations.

Using classical commutative algebra theory of
Dedekind domains (such as the coordinate ring $\mathbb{C}[X,Y]/(X^3Y+Y^3+X) $)
allows us to give a complete solution to this problem. It says that any
n-dimensional representation is determined up to isomorphism by the
following geometric/combinatorial data

  • a finite set of points $P_1,P_2,\dots,P_k $ on the Riemann surface with $k \leq n $.
  • a set of positive integers $a_1,a_2,\dots,a_k $ associated to these pointssatisfying $a_1+a_2+\dots_a_k=n $.
  • for each $a_i $ a partition of $a_i $ (that is, a decreasing sequence of numbers with total sum
    $a_i $).

To encode this classification I’ll use the mental
picture of associating to every point of the Klein quartic a small
loop. $\xymatrix{\vtx{}
\ar@(ul,ur)} $ Don\’t get over-exited about this
noncommutative manifold picture of the Klein quartic, I do not mean to
represent something like closed strings emanating from all points of the
Riemann surface or any other fanshi-wanshi interpretation. Just as
Feynman-diagrams allow the initiated to calculate probabilities of
certain interactions, the noncommutative manifold allows the
initiated to classify finite dimensional representations.

Our
mental picture of the noncommutative manifold of the Klein quartic, that
is : the points of the Klein quartic together with a loop in each point,
will tell the initiated quite a few things, such as : The fact
that there are no arrows between distict points, tells us that the
classification problem splits into local problems in a finite number of
points. Technically, this encodes the fact that there are no nontrivial
extensions between different simples in the commutative case. This will
drastically change if we enter the noncommutative world…

The fact that there is one loop in each point, tells us that
the local classification problem in that point is the same as that of
classifying nilpotent matrices upto conjugation (which, by the Jordan
normal form result, are classified by partitions) Moreover,
the fact that there is one loop in each point tells us that the local
structure of simple representations near that point (that is, the points
on the Kleinian quartic lying nearby) are classified as the simple
representations of the polynmial algebra $\mathbb{C}[x] $ (which are the
points on the complex plane, giving the picture
of the Riemann sphere in each point reflecting the local
neighborhood of the point on the Klein quartic)

In general, the
noncommutative manifold associated to a noncommutative smooth algebra
will be of a similar geometric/combinatorial nature. Typically, it will
consist of a geometric collection of points and arrows and loops between
these points. This data will then allow us to reduce the classification
problem to that of _quiver-representations_ and will allow us to give
local descriptions of our noncommutative manifolds. Next time,
I’ll give the details in the first noncommutative example : the
skew-group algebra of a finite group of automorphisms on a Riemann
surface (such as the simple group $PSL_2(\mathbb{F}_7) $ acting on the
Klein quartic). Already in this case, some new phenomena will
appear…

ADDED : While writing this post
NetNewsWire informed me that over at Noncommutative Geometry they have a
post on a similar topic : What is a noncommutative space.

One Comment

the cartographers’ groups (2)

Published March 15, 2007 by lievenlb

Fortunately,
there is a drastic shortcut to the general tree-argument of last time, due to
Roger Alperin. Recall that the Moebius
transformations corresponding to u resp. v send z resp. to

$-\frac{1}{z} $ and $\frac{1}{1-z} $

whence the Moebius transformation
corresponding to $v^{-1} $ send z to $1-\frac{1}{z} $.

Consider
the set $\mathcal{P} $ of all positive irrational real numbers and the
set $\mathcal{N} $ of all negative irrational real numbers and observe
that

$u(\mathcal{P}) \subset \mathcal{N} $ and
$v^{\pm}(\mathcal{N}) \subset \mathcal{P} $

We have to show
that no alternating word $w=(u)v^{\pm}uv^{\pm}u \dots v^{\pm}(u) $ in
u and $v^{\pm} $ can be the identity in $PSL_2(\mathbb{Z}) $.

If the
length of w is odd then either $w(\mathcal{P}) \subset
\mathcal{N} $ or $w(\mathcal{N}) \subset
\mathcal{P} $ depending on whether w starts with a u or with
a $v^{\pm} $ term. Either way, this proves that no odd-length word can
be the identity element in $PSL_2(\mathbb{Z}) $.

If the length of
the word w is even we can assume that $w = v^{\pm}uv^{\pm}u \dots
v^{\pm}u $ (if necessary, after conjugating with u we get to this form).

There are two subcases, either $w=v^{-1}uv^{\pm}u \dots
v^{\pm}u $ in which case $w(\mathcal{P}) \subset v^{-1}(\mathcal{N}) $
and this latter set is contained in the set of all positive irrational
real numbers which are strictly larger than one .

Or, $w=vuv^{\pm}u \dots v^{\pm}u $ in which case
$w(\mathcal{P}) \subset v(\mathcal{N}) $ and this set is contained in
the set of all positive irrational real numbers strictly smaller than
one .

Either way, this shows that w cannot be the identity
morphism on $\mathcal{P} $ so cannot be the identity element in
$PSL_2(\mathbb{Z}) $.
Hence we have proved that

$PSL_2(\mathbb{Z}) = C_2 \ast C_3 = \langle u,v : u^2=1=v^3 \rangle $

A
description of $SL_2(\mathbb{Z}) $ in terms of generators and relations
follows

$SL_2(\mathbb{Z}) = \langle U,V : U^4=1=V^6, U^2=V^3 \rangle $

It is not true that $SL_2(\mathbb{Z}) $ is the free
product $C_4 \ast C_6 $ as there is the extra relation $U^2=V^3 $.

This relation says that the cyclic groups $C_4 = \langle U \rangle $
and $C_6 = \langle V \rangle $ share a common subgroup $C_2 = \langle
U^2=V^3 \rangle $ and this extra condition is expressed by saying that
$SL_2(\mathbb{Z}) $ is the amalgamated free product of $C_4 $ with
$C_6 $, amalgamated over the common subgroup $C_2 $ and denoted
as

$SL_2(\mathbb{Z}) = C_4 \ast_{C_2} C_6 $

More
generally, if G and H are finite groups, then the free product $G
\ast H $ consists of all words of the form $~(g_1)h_1g_2h_2g_3
\dots g_nh_n(g_{n-1}) $ (so alternating between non-identity
elements of G and H) and the group-law is induced by concatenation
of words (and group-laws in either G or H when end terms are
elements in the same group).

For example, take the dihedral groups $D_4
= \langle U,R : U^4=1=R^2,(RU)^2=1 \rangle $ and $D_6 = \langle V,S :
V^6=1=S^2,(SV)^2=1 \rangle $ then the free product can be expressed
as

$D_4 \ast D_6 = \langle U,V,R,S :
U^4=1=V^6=R^2=S^2=(RV)^2=(RU)^2 \rangle $

This almost fits in with
our obtained description of
$GL_2(\mathbb{Z}) $

$GL_2(\mathbb{Z}) = \langle U,V,R :
U^4=1=V^6=R^2=(RU)^2=(RV)^2, U^2=V^3 \rangle $

except for the
extra relations $R=S $ and $U^2=V^3 $ which express the fact that we
demand that $D_4 $ and $D_6 $ have the same subgroup

$D_2 = \langle U^2=V^3,S=R \rangle $

So, again we can express these relations by
saying that $GL_2(\mathbb{Z}) $ is the amalgamated free product of
the subgroups $D_4 = \langle U,R \rangle $ and $D_6 = \langle V,R
\rangle $, amalgamated over the common subgroup $D_2 = C_2 \times C_2 =
\langle U^2=V^3,R \rangle $. We write

$GL_2(\mathbb{Z}) = D_4
\ast_{D_2} D_6 $

Similarly (but a bit easier) for
$PGL_2(\mathbb{Z}) $ we have

$PGL_2(\mathbb{Z}) = \langle u,v,R
u^2=v^3=1=R^2 = (Ru)^2 = (Rv)^2 \rangle $

which can be seen as
the amalgamated free product of $D_2 = \langle u,R \rangle $ with $D_3
= \langle v,R \rangle $, amalgamated over the common subgroup $C_2 =
\langle R \rangle $ and therefore

$PGL_2(\mathbb{Z}) = D_2
\ast_{C_2} D_3 $

Now let us turn to congruence subgroups of
the modular group.
With $\Gamma(n) $ one denotes the kernel of the natural
surjection

$PSL_2(\mathbb{Z}) \rightarrow
PSL_2(\mathbb{Z}/n\mathbb{Z}) $

that is all elements represented by a matrix

$\begin{bmatrix} a
& b \\ c & d \end{bmatrix} $

such that a=d=1 (mod n) and b=c=0
(mod n). On the other hand $\Gamma_0(n) $ consists of elements
represented by matrices such that only c=0 (mod n). Both are finite
index subgroups of $PSL_2(\mathbb{Z}) $.

As we have seen that
$PSL_2(\mathbb{Z}) = C_2 \ast C_3 $ it follows from general facts
on free products that any finite index subgroup is of the
form

$C_2 \ast C_2 \ast \dots \ast C_2 \ast
C_3 \ast C_3 \ast \dots \ast C_3 \ast C_{\infty}
\ast C_{\infty} \dots \ast C_{\infty} $

that is the
free product of k copies of $C_2 $, l copies of $C_3 $ and m copies
of $C_{\infty} $ where it should be noted that k,l and m are allowed
to be zero. There is an elegant way to calculate explicit generators of
these factors for congruence subgroups, due to Ravi S. Kulkarni (An
Arithmetic-Geometric Method in the Study of the Subgroups of the Modular
Group , American Journal of Mathematics, Vol. 113, No. 6. (Dec.,
1991), pp. 1053-1133) which deserves another (non-course) post.

Using this method one finds that $\Gamma_0(2) $ is generated by
the Moebius transformations corresponding to the
matrices

$X=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ and
$Y=\begin{bmatrix} 1 & -1 \\ 2 & -1 \end{bmatrix} $

and that
generators for $\Gamma(2) $ are given by the
matrices

$A=\begin{bmatrix} 1 & 0 \\ -2 & 1 \end{bmatrix} $
and $B=\begin{bmatrix} 1 & -2 \\ 2 & -3 \end{bmatrix} $

Next,
one has to write these generators in terms of the generating matrices
u and v of $PSL_2(\mathbb{Z}) $ and as we know all relations between
u and v the relations of these congruence subgroups will follow.

We
will give the details for $\Gamma_0(2) $ and leave you to figure out
that $\Gamma(2) = C_{\infty} \ast C_{\infty} $ (that is that
there are no relations between the matrices A and
B).

Calculate that $X=v^2u $ and that $Y=vuv^2 $. Because the
only relations between u and v are $v^3=1=u^2 $ we see that Y is an
element of order two as $Y^2 = vuv^3uv^2= v^3 = 1 $ and that no power of
X can be the identity transformation.

But then also none of the
elements $~(Y)X^{i_1}YX^{i_2}Y \dots YX^{i_n}(Y) $ can be the identity
(write it out as a word in u and v) whence,
indeed

$\Gamma_0(2) = C_{\infty} \ast C_2 $

In fact,
the group $\Gamma_0(2) $ is staring you in the face whenever you come to
this site. I fear I’ve never added proper acknowledgements for the
beautiful header-picture

so I’d better do it now. The picture is due to Helena
Verrill and she has a
page with
more pictures. The header-picture depicts a way to get a fundamental
domain for the action of $\Gamma_0(2) $ on the upper half plane. Such a
fundamental domain consists of any choice of 6 tiles with different
colours (note that there are two shades of blue and green). Helena also
has a
Java-applet
to draw fundamental domains of more congruence subgroups.

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