noncommutative curves and their maniflds

By lieven

Dessins d'enfants

Monsieur Mathieu
The best rejected proposal ever
The cartographers’ groups
the cartographers’ groups (2)
the noncommutative manifold of a Riemann surface
noncommutative curves and their maniflds
permutation representations of monodromy groups
anabelian geometry

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

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

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

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

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 $SL_2(\mathbb{Z})$ …

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Azumaya, geometry, Klein, noncommutative, representations, Riemann, simples

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Posted in geometry
Written on Sat, 17 March 2007 at 4:56 pm
Tags: Azumaya, geometry, Klein, noncommutative, representations, Riemann, simples
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noncommutative curves and their maniflds

Dessins d'enfants

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