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Anabelian vs. Noncommutative Geometry

Published December 12, 2007 by lieven

This is how my attention was drawn to what I have since termed
anabelian algebraic geometry, whose starting point was exactly a study
(limited for the moment to characteristic zero) of the action of absolute
Galois groups (particularly the groups $G a l (\overset{―}{K} / K)$ , where K is an extension of finite type of the prime field) on (profinite) geometric fundamental
groups of algebraic varieties (defined over K), and more particularly (breaking with a well-established tradition) fundamental groups which are very far
from abelian groups (and which for this reason I call anabelian). Among
these groups, and very close to the group ${\hat{π}}_{0, 3}$ , there is the profinite compactification of the modular group $S L_{2} (Z)$ , whose quotient by its centre
${\pm 1}$ contains the former as congruence subgroup mod 2, and can also be
interpreted as an oriented cartographic group, namely the one classifying triangulated oriented maps (i.e. those whose faces are all triangles or
monogons).

The above text is taken from Alexander Grothendieck‘s visionary text Sketch of a Programme. He was interested in the permutation representations of the modular group $Γ = P S L_{2} (Z)$ as they correspond via Belyi-maps and his own notion of dessins d’enfants to smooth projective curves defined over $\overset{―}{Q}$ . One can now study the action of the absolute Galois group $G a l (\overset{―}{Q} / Q)$ on these curves and their associated dessins. Because every permutation representation of $Γ$ factors over a finite quotient this gives an action of the absolute Galois group as automorphisms on the profinite compactification

$\hat{Γ} = \underset{\leftarrow}{l i m} Γ / N$

where the limit is taken over all finite index normal subgroups $N ◃ P S L_{2} (Z)$ . In this way one realizes the absolute Galois group as a subgroup of the outer automorphism group of the profinite group $\hat{Γ}$ . As a profinite group is a compact topological group one should study its continuous finite dimensional representations which are precisely those factoring through a finite quotient. In the case of $\hat{Γ}$ the simple continuous representations ${simp}_{c} \hat{Γ}$ are precisely the components of the permutation representations of the modular group. So in a sense, anabelian geometry is the study of these continuous simples together wirth the action of the absolute Galois group on it.

In noncommutative geometry we are interested in a related representation theoretic problem. We would love to know the simple finite dimensional representations $simp Γ$ of the modular group as this would give us all simples of the three string braid group $B_{3}$ . So a natural question presents itself : how are these two ‘geometrical’ objects ${simp}_{c} \hat{Γ}$ (anabelian) and $simp Γ$ (noncommutative) related and can we use one to get information about the other?

This is all rather vague so far, so let us work out a trivial case to get some intuition. Consider the profinite completion of the infinite Abelian group

$\hat{Z} = \underset{\leftarrow}{l i m} Z / n Z = \prod_{p} {\hat{Z}}_{p}$

As all simple representations of an Abelian group are one-dimensional and because all continuous ones factor through a finite quotient $Z / n Z$ we see that in this case

${simp}_{c} \hat{Z} = μ_{\infty}$

is the set of all roots of unity. On the other hand, the simple representations of $Z$ are also one-dimensional and are determined by the image of the generator so

$simp Z = C - 0 = C^{*}$

Clearly we have an embedding $μ_{\infty} \subset C^{*}$ and the roots of unity are even dense in the Zariski topology. This might look a bit strange at first because clearly all roots of unity lie on the unit circle which ‘should be’ their closure in the complex plane, but that’s because we have a real-analytic intuition. Remember that the Zariski topology of $C^{*}$ is just the cofinite topology, so any closed set containing the infinitely many roots of unity should be the whole space!

Let me give a pedantic alternative proof of this (but one which makes it almost trivial that a similar result should be true for most profinite completions…). If $c$ is the generator of $Z$ then the different conjugacy classes are precisely the singletons $c^{n}$ . Now suppose that there is a polynomial $a_{0} + a_{1} x + \dots + a_{m} x^{m}$ vanishing on all the continuous simples of $\hat{Z}$ then this means that the dimensions of the character-spaces of all finite quotients $Z / n Z$ should be bounded by $m$ (for consider $x$ as the character of $c$ ), which is clearly absurd.

Hence, whenever we have a finitely generated group $G$ for which there is no bound on the number of irreducibles for finite quotients, then morally the continuous simple space for the profinite completion

${simp}_{c} \hat{G} \subset simp G$

should be dense in the Zariski topology on the noncommutative space of simple finite dimensional representations of $G$ . In particular, this should be the case for the modular group $P S L_{2} (Z)$ .

There is just one tiny problem : unlike the case of $Z$ for which this space is an ordinary (ie. commutative) affine variety $C^{*}$ , what do we mean by the “Zariski topology” on the noncommutative space $simp P S L_{2} (Z)$ ? Next time we will clarify what this might be and show that indeed in this case the subset

${simp}_{c} \hat{Γ} \subset simp Γ$

will be a Zariski closed subset!

more iguanodons via kfarey.sage

Published December 11, 2007 by lieven

For what it is worth, Ive computed some more terms in the iguanodon series. Here they are

$L_{2} (7), M_{12}, A_{16}, M_{24}, A_{28}, A_{40}, A_{48}, A_{60}, A_{68}, A_{88}, A_{96}, A_{120}, A_{132}, A_{148}, A_{164}, A_{196}, \dots$

By construction, the n-th iguanodon group $I g_{n}$ (corresponding to the n-th Farey sequence) is a subgroup of the alternating group on its (half)legs. Hence to prove that all remaining iguanodons are alternating groups boils down to proving that they are sufficiently transitive, for example, by showing that there are permutations of certain cycle-types in the group. Im sure any grouptheorist can crack this problem over lunch, so if you did please drop a comment.

Clearly, I didnt do the calculations in the archaic way of the previous post (as depicted on the left) which consisted in adding a pair of new legs at the proper place in the spine for every new Farey number, write down the two generating permutations, giving them to GAP and check simplicity and the isomorphism type.

Instead I used a nice SAGE-package to compute with Farey-symbols written by Chris Kurth and available from his website. As this package is a good tool to experiment hunting for other dinosaur-series of simple groups coming from series of Farey-symbols, Ill include the details for $I g_{3}$ (the example used to outline the construction of the Iguanodon-series ).

First we need to have the n-th Farey-sequence $F (n)$ . There are several short Python programs around to do this, for example this one from the Python-Cookbook. Save it to your sage-directory and name it fareyseq.py and load it into sage via load fareyseq.py. Then typing farey(3) to the sage-prompt spits back

sage: farey(3)
[(1, 3), (1, 2), (2, 3)]

That is, 0 and 1 are not included and Farey-numbers are represented by numerator-denominator couples. The iguanodon-series uses the Fareys upto 1/2, identifies the edges connecting 0 and 1 to $\infty$ and makes all other intervals odd. That is, the corresponding Farey symbol for F(3) is

[tex]\xymatrix{\infty \ar@{-}[r]_{1} & 0 \ar@{-}[r]_{\bullet} & \frac{1}{3} \ar@{-}[r]_{\bullet} & \frac{1}{2} \ar@{-}[r]_{\bullet} & 1 \ar@{-}[r]_{1} & \infty}[/tex]

(to add to the confusion, I denote odd intervals by a black-bullet whereas in Kulkarni’s paper they are white…) Anyway, get Kurth’s kfarey-package and save the folder as kfarey in your sage-folder. Kurth uses the following notation for Farey-symbols

The Farey Symbol is a list [a,b,p] where
a is a list of numerators, b a list of denominators, and p the pairing
information. If x[i]=a[i]/b[i]:
inf x0 x1 x2 ... xn inf
  p0 p1 p2 ... pn pn+1
So p[i] is the pairing of the side between x[i-1] and x[i]. The p[i]’s can be
positive integers, indicating pairing between sides, or -2 or -3, meaning
an even or odd pairing respectively.

The above Farey-symbol is therefore represented as
[[0,1,1,1],[1,3,2,1],[1,-3,-3,-3,1]]. The kfarey-function LRCosetRep(F) returns two permutations L and R giving the permutation action of the two generators of the modular group $P S L_{2} (Z)$

$L = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] and R = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}]$

on the half-legs of the inguanodon (the dessin corresponding to the Farey-symbol). Here’s the sage transcript

sage: load kfarey/farey.sage
sage: load kfarey/conggroups.sage
sage: load kfarey/LR.sage
sage: ig3=[[0,1,1,1],[1,3,2,1],[1,-3,-3,-3,1]]
sage: LRCosetRep(ig3)
[(1,2,3,9,10,11,6,7,8,4,5)(12), (1,8,4,2,11,6,3,12,10,7,5)(9)]

Giving these two generators to GAP one verifies that they indeed generate $M_{12}$

gap> ig3:=Group((1,2,3,9,10,11,6,7,8,4,5)(12), (1,8,4,2,11,6,3,12,10,7,5)(9));
Group([ (1,2,3,9,10,11,6,7,8,4,5), (1,8,4,2,11,6,3,12,10,7,5) ])
gap> IsSimpleGroup(ig3);
true
gap> IsomorphismTypeInfoFiniteSimpleGroup(ig3);
rec( series := “Spor”, name := “M(12)” )

kfarey has plenty of other useful functions. One can even create an .eps file of the fundamental domain specified by the subgroup of the modular group encoded by the Farey symbol using MakeEpsFile(F). For the above example it returns the picture on the right. Not quite as nice as the one on the left, but surely a lot easier to create.

Segal’s formal neighbourhood result

Published December 8, 2007 by lieven

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3-fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, D-branes, Calabi-Yaus, superpotentials, all the pseudo-physics babble that spreads virally among the youngest generation of algebraists and geometers.

Fortunately, his talk did contain a general ringtheoretic gem. After a bit of polishing up this gem, contained in his paper The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus, can be stated as follows.

Let $A$ be a $C$ -algebra and let $M = S_{1} \oplus \dots \oplus S_{k}$ be a finite dimensional semi-simple representation with distinct simple components. Let $m$ be the kernel of the algebra epimorphism $A \to S$ to the semi-simple algebra $S = E n d (M)$ . Then, the $m$ -adic completion of $A$ is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the $A_{\infty}$ -structure on the Ext-algebra $E x t_{A}^{∙} (M, M)$ . More precisely, there is an isomorphism

${\hat{A}}_{m} ≃ \frac{{\hat{T}}_{S} (E x t_{A}^{1} (M, M)^{*})}{(I m (H M C)^{*})}$

where the homotopy Maurer-Cartan map comes from the $A_{\infty}$ structure maps

$H M C = \oplus_{i} m_{i} : T_{S} (E x t_{A}^{1} (M, M)) \to E x t_{A}^{2} (M, M)$

and hence the defining relations of the completion are given by the image of the dual of this map.

For ages, Ive known this result in the trivial case of formally smooth algebras (where $E x t_{A}^{2} (M, M) = 0$ and hence there are no relations to divide out) and where it is a consequence of a special case of the Cuntz-Quillen “tubular neighborhood” result. Completions of formally smooth algebras at semi-simples are Morita equivalent to completions of path algebras. This fact motivated all the local-quiver technology that was developed here in Antwerp over the last decade (see my book if you want to know the details).

Also for 3-dimensional Calabi-Yau algebras it states that the completions at semi-simples are Morita equivalent to completions of quotients of path algebras by the relations coming from a superpotential (aka a necklace) by taking partial noncommutative derivatives. Here the essential ingredient is that $E x t_{A}^{2} (M, M)^{*} ≃ E x t_{A}^{1} (M, M)$ in this case.

Category: featured

Anabelian vs. Noncommutative Geometry

more iguanodons via kfarey.sage

Segal’s formal neighbourhood result