Category: noncommutative

On the Reality of Noncommutative Space

Published January 20, 2011 by lievenlb

In my book “Virtual Topology and Functorial Geometry” (Taylor and Francis, 2009) I proposed a noncommutative version of space-time ; it is a toy model, but mathematically correct and I included a few philosophical remarks about : “What if reality is noncommutative ?”.

I want to add a few ideas about how “strange” ideas in quantum mechanics all fit naturally in the noncommutative world. First let us talk about noncommutative geometry in an intuitive way.

Then noncommutative space may be thought of as a set of noncommutative places but these noncommutative places need not be sets, in particular they are not sets of points. There is a noncommutative join $\vee$ and a noncommutative intersection $\wedge$, and they satify the axioms (very natural ones) of a noncommutative topology.

The non-commutativity is characterized by the existence of non $\wedge$-idempotent places, i.e. places with a nontrivial self intersection. This allows the $\wedge$ to be noncommutative. From algebraic geometric it follows that one may be interested to let $\vee$ be an abelian operation (hence defining a virtual topology) so let us assume this from hereon.

The set of $\wedge$-idempotent noncommutative-places forms the “commutative shadow” of the noncommutative space; it has operations $\vee$ and $\mathop{\wedge}\limits_{\bullet}$ which are abelian and $\sigma \mathop{\wedge}\limits_{\bullet}\tau$ may be thought of as the largest $\wedge$-idempotent smaller than $\sigma$ and $\tau$ in the partial ordering of the noncommutative space.

The $\wedge$-idempotent noncommutative places are sets in a commutative topology and these are the observable places in the noncommutative space. In the book I present a dynamic (time !) model allowing further elaboration on the noncommutative space but for now let us stick to the intuitive model and assume that space is in fact noncommutative with commutative shadow built upon our space time of physics.

In fact all observations, measurings and predictions made in physics are not about reality but about our observations of reality, so it may be an eternal fact that our observations of reality are described in our brains by commutative methods. Nevertheless we can observe effect of objects existing at noncommutative places in “neighboring” $\wedge$-idempotents sets or observable places.

First if an object exists at a noncommutative place it also exists at all subplaces (a harmless assumption not really essential for the rest). So if there is a noncommutative place, where some object exists, parts of this object may be observed at idempotent subplaces of the noncommutative place. These may even be disjoint in the commutative shadow, not “too far apart” as one object exists on the total noncommutative space.

Since only a part of the noncommutative object is observed on the $\wedge$-idempotent subplace it is not clear that one may actually recognize the observations at different commutative places as belonging to the same noncommutative object. Once one observes one observable place that object seems to exist only on that (commutative) place. Hence a quantum particle can be thought of as existing on several “places” but once observed it looks like it only exists there. This is a first natural phenomenon reflecting “strange” quantum mechanical principles.

Secondly let us look at the double slit experiment. The slits correspond to commutative places $\sigma_1$ and $\sigma_2$ and $\sigma_1 \mathop{\wedge}\limits_{\bullet}\sigma_2=\emptyset$, however in the noncommutative world $\sigma_1\wedge\sigma_2$ need not be empty, only it has no $\wedge$-idempotent subplaces !

Therefore if a photon is defined on a noncommutative place with “light”-effect on observable places “near enough” to it (in a neighborhood small enough to have an observable effect say) then the photon may pass though both slits without splitting or without splitting reality (parallel universes) but just moving into the noncommutative space inside $\sigma_1$ and $\sigma_2$ !

The observable effect at the slits will appear in commutative places near enough (for example, intersecting) to $\sigma_1$ or to $\sigma_2$. As the photon moves on, observable effects will appear in commutative places intersecting the one near to $\sigma_1$ or the one near to $\sigma_2$ and these may themselves have nonempty intersections.

At the moment the effect via $\sigma_1$ interacts with the effect via $\sigma_2$. As the photon progresses in its observed direction other $\wedge$-idempotents showing observable effects may meet and so several interactions between observable effects (via $\sigma_1$ and $\sigma_2$) build a picture of interference.

The symmetry of this picture actually suggests a symmetric arrangement of commutative places around a noncommutative place. So the noncommutativity of space may explain this phenomenon without holographic principle or parallel universes.

In a similar way dark mass may well be mass existing in a non-observable noncommutative place (i.e. containing no observable places). If a lot of mass is in a non-observable noncommutative place its gravity may pull matter from surrounding observable places into the noncommutative place and this may explain black holes. All kinds of problems relating to black holes may have natural non commutative solutions, e.g. information may pass from observable places to noncommutative places and is not lost, only non-observable.

In fact is the definition of information not depending on the nature and capability of the recipient ? There are many philosophically interesting ramifications of these ideas, for example every chemical or neurochemical activity should also be placed in the noncommutative space.

In the book I mentioned how “free will” could be a noncommutative space aspect of the brain activity. I also mention a possible relations with string theory. I am not a specialist in all these things but now I reached the point that I “feel” noncommutative space is a better approximation of the reality and one should investigate it further.

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Seating the first few billion Knights

Published December 30, 2010 by lievenlb

The odd Knight of the round table problem asks for a consistent placement of the n-th Knight in the row at an odd root of unity, compatible with the two different realizations of the algebraic closure of the field with two elements.

The first identifies the multiplicative group of its non-zero elements with the group of all odd complex roots of unity, under complex multiplication. The second uses Conway’s ‘simplicity rules’ in ONAG to define an addition and multiplication on the set of all ordinal numbers.

Here’s the seating arrangement for the first 15 knights. Conway proved that all finite ordinals smaller than $2^{2^i} $ form a subfield of $\overline{\mathbb{F}}_2 $. The first non-trivial one being $\{ 0,1,2,3 \} $ with smallest multiplicative generator $2 $, whence we place Knight 2 at $e^{2 i \pi/3} $ and as $2^2=3 $ we know where to place the third Knight.

The next subfield is made of the numbers $\{ 0,1,2,\ldots,13,14,15 \} $ and its non-zero numbers form a cyclic group of order 15. Hence we need to find the smallest generator of this group satisfying the additional property of being compatible with the earlier seating, that is, its fifth power must equal to 2. Checking the multiplication table reproduced here one verifies that the wanted generator is 4 and so we place Knight 4 at $e^{\frac{2 \pi i}{15}} $ and as all the ordinals smaller than 16 are powers of 4 this tells us where to place the Knights until we get to the 15th in the row.

In february we were able to seat the first few thousand Knights by showing by hand that 32 is the smallest ordinal such that its 15-th power is equal to 4 and using SAGE that 1051 is the smallest ordinal whose 257-th power equals 32. These calculations enabled us to seat the Knights until we get to the 65536-th in the row.

Today I managed to show that 1361923 is the smallest ordinal such that its 65537-th power equals 1051. You can verify this statement in SAGE using the method explained in the february post

sage: R.< x,y,z,t,u >=GF(2)[]

sage: S.< a,b,c,d,e > =
R.quotient((x^2+x+1,y^2+y+x,z^2+z+x*y,t^2+t+x*y*z,u^2+u+x*y*z*t))

sage: (c*e+b*e+a*b*c*d+b*c*d+a*b*d+a+1)^65537
c^2 + b*d + a + 1

(It takes a bit longer to check minimality of 1361923). That is, we have to seat Knight 1361923 at $e^{\frac{2 \pi i}{4294967295}} $ and because all the numbers smaller than 4294967296 are powers of 1361923 we have seating arrangements for the first 4294967295 Knights!

I did try the same method in february but ran into time- and memory-problems on my 2.4Ghz 2Gb MacBook. Today I upgraded from Sage 3.3 to Sage 4.6 and this version is a lot faster (using the 64-bit architecture) and also appears to be much better at memory-management. Thank you, Sage-community!

Wishing you all a lot of mathematical (and other) fun in the prime-number year 2011.

atb :: lieven.

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Jason & David, the Ninja warriors of noncommutative geometry

Published December 23, 2010 by lievenlb

SocialMention gives a rather accurate picture of the web-buzz on a specific topic. For this reason I check it irregularly to know what’s going on in noncommutative geometry, at least web-wise.

Yesterday, I noticed two new kids on the block : Jason and David. Their blogs have (so far ) 44 resp. 27 posts, this month alone. My first reaction was: respect!, until I glanced at their content…

David of E-Infinity

Noncommutative geometry has a deplorable track record when it comes to personality-cults and making extra-ordinary claims, but this site beats everything I’ve seen before. Its main mission is to spread the gospel according to E.N.

A characteristic quote :

“It was no doubt the intention of those well known internet thugs and parasites to distract us from science and derail us from our road. This was the brief given to them by you know who. Never the less we will attempt to give here what can only amount to a summary of the summary of what E. N. considers to be the philosophical background to his theory.”

Jason of the E.N. watch

The blog’s mission statement is to expose the said prophet E.N. as a charlatan.

The language used brings us back to the good(?!) old string-war days.

“This is amusing because E. N.’s sockpuppets go on and on about E. N. being a genius polymath with an expert grasp of science, art, history, philosophy and politics. E. N. Watch readers of course know that his knowledge in all areas comes primarily from mass-market popularizations.”

As long as the Connes support-blog and the Rosenberg support-blog remain silent and the Jasons and Davids of this world run the online ncg-show, it is probably better to drop the topic here.

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Grothendieck’s folly

Published December 21, 2010 by lievenlb

Never a dull moment with Books Ngram Viewer. Pick your favorite topic(s) and try to explain and name valleys and peaks in the Ngram.

An example. I wanted to compare the relative impact of a couple of topics I love, algebraic geometry (blue), category theory (red), representation theory (green) and noncommutative geometry (the bit of yellow in the lower right hand corner…) from 1960 onwards.

I was surprised to find out that the first three topics were almost in the same impact-league, but then Ngram-viewing can be cruel when you’re biased …

Anyone having an explanation/name for the great depressions of 1982, 1993 and 1996?

On the positive side, what happened in 1988-89 or what caused the representation-peak in 1999, or the category-delirium in 2006?

So far, I’ve only been able to pinpoint a couple of events. My favorite being the red peak in 1973, which I’d like to christen “Grothendieck’s folly”.

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Langlands versus Connes

Published October 26, 2010 by lievenlb

This is a belated response to a Math-Overflow exchange between Thomas Riepe and Chandan Singh Dalawat asking for a possible connection between Connes’ noncommutative geometry approach to the Riemann hypothesis and the Langlands program.

Here’s the punchline : a large chunk of the Connes-Marcolli book Noncommutative Geometry, Quantum Fields and Motives can be read as an exploration of the noncommutative boundary to the Langlands program (at least for $GL_1 $ and $GL_2 $ over the rationals $\mathbb{Q} $).

Recall that Langlands for $GL_1 $ over the rationals is the correspondence, given by the Artin reciprocity law, between on the one hand the abelianized absolute Galois group

$Gal(\overline{\mathbb{Q}}/\mathbb{Q})^{ab} = Gal(\mathbb{Q}(\mu_{\infty})/\mathbb{Q}) \simeq \hat{\mathbb{Z}}^* $

and on the other hand the connected components of the idele classes

$\mathbb{A}^{\ast}_{\mathbb{Q}}/\mathbb{Q}^{\ast} = \mathbb{R}^{\ast}_{+} \times \hat{\mathbb{Z}}^{\ast} $

The locally compact Abelian group of idele classes can be viewed as the nice locus of the horrible quotient space of adele classes $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^{\ast} $. There is a well-defined map

$\mathbb{A}_{\mathbb{Q}}’/\mathbb{Q}^{\ast} \rightarrow \mathbb{R}_{+} \qquad (x_{\infty},x_2,x_3,\ldots) \mapsto | x_{\infty} | \prod | x_p |_p $

from the subset $\mathbb{A}_{\mathbb{Q}}’ $ consisting of adeles of which almost all terms belong to $\mathbb{Z}_p^{\ast} $. The inverse image of this map over $\mathbb{R}_+^{\ast} $ are precisely the idele classes $\mathbb{A}^{\ast}_{\mathbb{Q}}/\mathbb{Q}^{\ast} $. In this way one can view the adele classes as a closure, or ‘compactification’, of the idele classes.

This is somewhat reminiscent of extending the nice action of the modular group on the upper-half plane to its badly behaved action on the boundary as in the Manin-Marcolli cave post.

The topological properties of the fiber over zero, and indeed of the total space of adele classes, are horrible in the sense that the discrete group $\mathbb{Q}^* $ acts ergodically on it, due to the irrationality of $log(p_1)/log(p_2) $ for primes $p_i $. All this is explained well (in the semi-local case, that is using $\mathbb{A}_Q’ $ above) in the Connes-Marcolli book (section 2.7).

In much the same spirit as non-free actions of reductive groups on algebraic varieties are best handled using stacks, such ergodic actions are best handled by the tools of noncommutative geometry. That is, one tries to get at the geometry of $\mathbb{A}_{\mathbb{Q}}/\mathbb{Q}^{\ast} $ by studying an associated non-commutative algebra, the skew-ring extension of the group-ring of the adeles by the action of $\mathbb{Q}^* $ on it. This algebra is known to be Morita equivalent to the Bost-Connes algebra which is the algebra featuring in Connes’ approach to the Riemann hypothesis.

It shouldn’t thus come as a major surprise that one is able to recover the other side of the Langlands correspondence, that is the Galois group $Gal(\mathbb{Q}(\mu_{\infty})/\mathbb{Q}) $, from the Bost-Connes algebra as the symmetries of certain states.

In a similar vein one can read the Connes-Marcolli $GL_2 $-system (section 3.7 of their book) as an exploration of the noncommutative closure of the Langlands-space $GL_2(\mathbb{A}_{\mathbb{Q}})/GL_2(\mathbb{Q}) $.

At the moment I’m running a master-seminar noncommutative geometry trying to explain this connection in detail. But, we’re still in the early phases, struggling with the topology of ideles and adeles, reciprocity laws, L-functions and the lot. Still, if someone is interested I might attempt to post some lecture notes here.

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noncommutative space quiz

Published May 21, 2009 by lievenlb

Creating (or taking) an image and explaining how it depicts your mental picture of a noncommutative space is one thing. Ideally, the image should be strong enough so that other people familiar with it might have a reasonable guess what you attempt to depict.

But, is there already enough concordance in our views of noncommutative spaces? I doubt it, whence this experiment.
Below my attempt ((the image is taken from Cran’s fractal art )) to depict one of the most popular noncommutative spaces around :

Can you guess what space this is? How does it agree with (resp. differ from) your own mental image of it?

Further, if you know of links to other depictions of noncommutative spaces, please leave a comment, or, send me an email.

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Connes & Consani go categorical

Published March 12, 2009 by lievenlb

Today, Alain Connes and Caterina Consani arXived their new paper Schemes over $ \mathbb{F}_1$ and zeta functions. It is a follow-up to their paper On the notion of geometry over $ \mathbb{F}_1$, which I’ve tried to explain in a series of posts starting here.

As Javier noted already last week when they updated their first paper, the main point of the first 25 pages of the new paper is to repace abelian groups by abelian monoids in the definition, making it more in tune with other approaches, most notably that of Anton Deitmar. The novelty, if you want, is that they package the two functors $\mathbf{rings} \rightarrow \mathbf{sets} $ and $\mathbf{ab-monoid} \rightarrow \mathbf{sets} $ into one functor $\mathbf{ring-monoid} \rightarrow \mathbf{sets} $ by using the ‘glued category’ $\mathbf{ring-monoid} $ (an idea they attribute to Pierre Cartier).

In general, if you have two categories $\mathbf{cat} $ and $\mathbf{cat’} $ and a pair of adjoint functors between them, then one can form the glued-category $\mathbf{cat-cat’} $ by taking as its collection of objects the disjoint union of the objects of the two categories and by defining the hom-sets between two objects the hom-sets in either category (if both objects belong to the same category) or use the adjoint functors to define the new hom-set when they do not (the very definition of adjoint functors makes that this doesn’t depend on the choice).

Here, one uses the functor $\mathbf{ab-monoid} \rightarrow \mathbf{rings} $ assigning to a monoid $M $ its integral monoid-algebra $\mathbb{Z}[M] $, having as its adjoint the functor $\mathbf{rings} \rightarrow \mathbf{ab-monoid} $ forgetting the additive structure of the commutative ring.

In the second part of the paper, they first prove some nice results on zeta-functions of Noetherian $\mathbb{F}_1 $-schemes and extend them, somewhat surprisingly, to settings which do not (yet) fit into the $\mathbb{F}_1 $-framework, namely elliptic curves and the hypothetical $\mathbb{F}_1 $-curve $\overline{\mathbf{spec}(\mathbb{Z})} $.

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noncommutative F_un geometry (2)

Published October 17, 2008 by lievenlb

Last time we tried to generalize the Connes-Consani approach to commutative algebraic geometry over the field with one element $\mathbb{F}_1 $ to the noncommutative world by considering covariant functors

$N~:~\mathbf{groups} \rightarrow \mathbf{sets} $

which over $\mathbb{C} $ resp. $\mathbb{Z} $ become visible by a complex (resp. integral) algebra having suitable universal properties.

However, we didn’t specify what we meant by a complex noncommutative variety (resp. an integral noncommutative scheme). In particular, we claimed that the $\mathbb{F}_1 $-‘points’ associated to the functor

$D~:~\mathbf{groups} \rightarrow \mathbf{sets} \qquad G \mapsto G_2 \times G_3 $ (here $G_n $ denotes all elements of order $n $ of $G $)

were precisely the modular dessins d’enfants of Grothendieck, but didn’t give details. We’ll try to do this now.

For algebras over a field we follow the definition, due to Kontsevich and Soibelman, of so called “noncommutative thin schemes”. Actually, the thinness-condition is implicit in both Soule’s-approach as that of Connes and Consani : we do not consider R-points in general, but only those of rings R which are finite and flat over our basering (or field).

So, what is a noncommutative thin scheme anyway? Well, its a covariant functor (commuting with finite projective limits)

$\mathbb{X}~:~\mathbf{Alg}^{fd}_k \rightarrow \mathbf{sets} $

from finite-dimensional (possibly noncommutative) $k $-algebras to sets. Now, the usual dual-space operator gives an anti-equivalence of categories

$\mathbf{Alg}^{fd}_k \leftrightarrow \mathbf{Coalg}^{fd}_k \qquad A=C^* \leftrightarrow C=A^* $

so a thin scheme can also be viewed as a contra-variant functor (commuting with finite direct limits)

$\mathbb{X}~:~\mathbf{Coalg}^{fd}_k \rightarrow \mathbf{Sets} $

In particular, we are interested to associated to any {tex]k $-algebra $A $ its representation functor :

$\mathbf{rep}(A)~:~\mathbf{Coalg}^{fd}_k \rightarrow \mathbf{Sets} \qquad C \mapsto Alg_k(A,C^*) $

This may look strange at first sight, but $C^* $ is a finite dimensional algebra and any $n $-dimensional representation of $A $ is an algebra map $A \rightarrow M_n(k) $ and we take $C $ to be the dual coalgebra of this image.

Kontsevich and Soibelman proved that every noncommutative thin scheme $\mathbb{X} $ is representable by a $k $-coalgebra. That is, there exists a unique coalgebra $C_{\mathbb{X}} $ (which they call the coalgebra of ‘distributions’ of $\mathbb{X} $) such that for every finite dimensional $k $-algebra $B $ we have

$\mathbb{X}(B) = Coalg_k(B^*,C_{\mathbb{X}}) $

In the case of interest to us, that is for the functor $\mathbf{rep}(A) $ the coalgebra of distributions is Kostant’s dual coalgebra $A^o $. This is the not the full linear dual of $A $ but contains only those linear functionals on $A $ which factor through a finite dimensional quotient.

So? You’ve exchanged an algebra $A $ for some coalgebra $A^o $, but where’s the geometry in all this? Well, let’s look at the commutative case. Suppose $A= \mathbb{C}[X] $ is the coordinate ring of a smooth affine variety $X $, then its dual coalgebra looks like

$\mathbb{C}[X]^o = \oplus_{x \in X} U(T_x(X)) $

the direct sum of all universal (co)algebras of tangent spaces at points $x \in X $. But how do we get the variety out of this? Well, any coalgebra has a coradical (being the sun of all simple subcoalgebras) and in the case just mentioned we have

$corad(\mathbb{C}[X]^o) = \oplus_{x \in X} \mathbb{C} e_x $

so every point corresponds to a unique simple component of the coradical. In the general case, the coradical of the dual coalgebra $A^o $ is the direct sum of all simple finite dimensional representations of $A $. That is, the direct summands of the coalgebra give us a noncommutative variety whose points are the simple representations, and the remainder of the coalgebra of distributions accounts for infinitesimal information on these points (as do the tangent spaces in the commutative case).

In fact, it was a surprise to me that one can describe the dual coalgebra quite explicitly, and that $A_{\infty} $-structures make their appearance quite naturally. See this paper if you’re in for the details on this.

That settles the problem of what we mean by the noncommutative variety associated to a complex algebra. But what about the integral case? In the above, we used extensively the theory of Kostant-duality which works only for algebras over fields…

Well, not quite. In the case of $\mathbb{Z} $ (or more general, of Dedekind domains) one can repeat Kostant’s proof word for word provided one takes as the definition of the dual $\mathbb{Z} $-coalgebra
of an algebra (which is $\mathbb{Z} $-torsion free)

$A^o = { f~:~A \rightarrow \mathbb{Z}~:~A/Ker(f)~\text{is finitely generated and torsion free}~} $

(over general rings there may be also variants of this duality, as in Street’s book an Quantum groups). Probably lots of people have come up with this, but the only explicit reference I have is to the first paper I’ve ever written. So, also for algebras over $\mathbb{Z} $ we can define a suitable noncommutative integral scheme (the coradical approach accounts only for the maximal ideals rather than all primes, but somehow this is implicit in all approaches as we consider only thin schemes).

Fine! So, we can make sense of the noncommutative geometrical objects corresponding to the group-algebras $\mathbb{C} \Gamma $ and $\mathbb{Z} \Gamma $ where $\Gamma = PSL_2(\mathbb{Z}) $ is the modular group (the algebras corresponding to the $G \mapsto G_2 \times G_3 $-functor). But, what might be the points of the noncommutative scheme corresponding to $\mathbb{F}_1 \Gamma $???

Well, let’s continue the path cut out before. “Points” should correspond to finite dimensional “simple representations”. Hence, what are the finite dimensional simple $\mathbb{F}_1 $-representations of $\Gamma $? (Or, for that matter, of any group $G $)

Here we come back to Javier’s post on this : a finite dimensional $\mathbb{F}_1 $-vectorspace is a finite set. A $\Gamma $-representation on this set (of n-elements) is a group-morphism

$\Gamma \rightarrow GL_n(\mathbb{F}_1) = S_n $

hence it gives a permutation representation of $\Gamma $ on this set. But then, if finite dimensional $\mathbb{F}_1 $-representations of $\Gamma $ are the finite permutation representations, then the simple ones are the transitive permutation representations. That is, the points of the noncommutative scheme corresponding to $\mathbb{F}_1 \Gamma $ are the conjugacy classes of subgroups $H \subset \Gamma $ such that $\Gamma/H $ is finite. But these are exactly the modular dessins d’enfants introduced by Grothendieck as I explained a while back elsewhere (see for example this post and others in the same series).

noncommutative F_un geometry (1)

Published October 13, 2008 by lievenlb

It is perhaps surprising that Alain Connes and Katia Consani, two icons of noncommutative geometry, restrict themselves to define commutative algebraic geometry over $\mathbb{F}_1 $, the field with one element.

My guess of why they stop there is as good as anyone’s. Perhaps they felt that there is already enough noncommutativity in Soule’s gadget-approach (the algebra $\mathcal{A}_X $ as in this post may very well be noncommutative). Perhaps they were only interested in the Bost-Connes system which can be entirely encoded in their commutative $\mathbb{F}_1 $-geometry. Perhaps they felt unsure as to what the noncommutative scheme of an affine noncommutative algebra might be. Perhaps …

Remains the fact that their approach screams for a noncommutative extension. Their basic object is a covariant functor

$N~:~\mathbf{abelian} \rightarrow \mathbf{sets} \qquad A \mapsto N(A) $

from finite abelian groups to sets, together with additional data to the effect that there is a unique minimal integral scheme associated to $N $. In a series of posts on the Connes-Consani paper (starting here) I took some care of getting rid of all scheme-lingo and rephrasing everything entirely into algebras. But then, this set-up can be extended verbatim to noncommuative $\mathbb{F}_1 $-geometry, which should start from a covariant functor

$N~:~\mathbf{groups} \rightarrow \mathbf{sets} $

from all finite groups to sets. Let’s recall quickly what the additional info should be making this functor a noncommutative (affine) F_un scheme :

There should be a finitely generated $\mathbb{C} $-algebra $R $ together with a natural transformation (the ‘evaluation’)

$e~:~N \rightarrow \mathbf{maxi}(R) \qquad N(G) \mapsto Hom_{\mathbb{C}-alg}(R, \mathbb{C} G) $

(both $R $ and the group-algebra $\mathbb{C} G $ may be noncommutative). The pair $(N, \mathbf{maxi}(R)) $ is then called a gadget and there is an obvious notion of ‘morphism’ between gadgets.

The crucial extra ingredient is an affine $\mathbb{Z} $-algebra (possibly noncommutative) $S $
such that $N $ is a subfunctor of $\mathbf{mini}(S)~:~G \mapsto Hom_{\mathbb{Z}-alg}(S,\mathbb{Z} G) $ together with the following universal property :

any affine $\mathbb{Z} $-algebra $T $ having a gadget-morphism $~(N,\mathbf{maxi}(R)) \rightarrow (\mathbf{mini}(T),\mathbf{maxi}(T \otimes_{\mathbb{Z}} \mathbb{C})) $ comes from a $\mathbb{Z} $-algebra morphism $T \rightarrow S $. (If this sounds too cryptic for you, please read the series on C-C mentioned before).

So, there is no problem in defining noncommutative affine F_un-schemes. However, as with any generalization, this only makes sense provided (a) we get something new and (b) we have interesting examples, not covered by the restricted theory.

At first sight we do not get something new as in the only example we did in the C-C-series (the forgetful functor) it is easy to prove (using the same proof as given in this post) that the forgetful-functor $\mathbf{groups} \rightarrow \mathbf{sets} $ still has as its integral form the integral torus $\mathbb{Z}[x,x^{-1}] $. However, both theories quickly diverge beyond this example.

For example, consider the functor

$\mathbf{groups} \rightarrow \mathbf{sets} \qquad G \mapsto G \times G $

Then, if we restrict to abelian finite groups $\mathbf{abelian} $ it is easy to see (again by a similar argument) that the two-dimensional integer torus $\mathbb{Z}[x,y,x^{-1},y^{-1}] $ is the correct integral form. However, this algebra cannot be the correct form for the functor on the category of all finite groups as any $\mathbb{Z} $-algebra map $\phi~:~\mathbb{Z}[x,y,x^{-1},y^{-1}] \rightarrow \mathbb{Z} G $ determines (and is determined by) a pair of commuting units in $\mathbb{Z} G $, so the above functor can not be a subfunctor if we allow non-Abelian groups.

But then, perhaps there isn’t a minimal integral $\mathbb{Z} $-form for this functor? Well, yes there is. Take the free group in two letters (that is, all words in noncommuting $x,y,x^{-1} $ and $y^{-1} $ satisfying only the trivial cancellation laws between a letter and its inverse), then the corresponding integral group-algebra $\mathbb{Z} \mathcal{F}_2 $ does the trick.

Again, the proof-strategy is the same. Given a gadget-morphism we have an algebra map $f~:~T \mapsto \mathbb{C} \mathcal{F}_2 $ and we have to show, using the universal property that the image of $T $ is contained in the integral group-algebra $\mathbb{Z} \mathcal{F}_2 $. Take a generator
$z $ of $T $ then the degree of the image $f(z) $ is bounded say by $d $ and we can always find a subgroup $H \subset \mathcal{F}_2 $ such that $\mathcal{F}_2/H $ is a fnite group and the quotient map $\mathbb{C} \mathcal{F}_2 \rightarrow \mathbb{C} \mathcal{F}_2/H $ is injective on the subspace spanned by all words of degree strictly less than $d+1 $. Then, the usual diagram-chase finishes the proof.

What makes this work is that the free group $\mathcal{F}_2 $ has ‘enough’ subgroups of finite index, a property it shares with many interesting discrete groups. Whence the blurb-message :

if the integers $\mathbb{Z} $ see a discrete group $\Gamma $, then the field $\mathbb{F}_1 $ sees its profinite completion $\hat{\Gamma} = \underset{\leftarrow}{lim}~\Gamma/ H $

So, yes, we get something new by extending the Connes-Consani approach to the noncommutative world, but do we have interesting examples? As “interesting” is a subjective qualification, we’d better invoke the authority-argument.

Alexander Grothendieck (sitting on the right, manifestly not disputing a vacant chair with Jean-Pierre Serre, drinking on the left (a marvelous picture taken by F. Hirzebruch in 1958)) was pushing the idea that profinite completions of arithmetical groups were useful in the study of the absolute Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $, via his theory of dessins d’enfants (children;s drawings).

In a previous life, I’ve written a series of posts on dessins d’enfants, so I’ll restrict here to the basics. A smooth projective $\overline{\mathbb{Q}} $-curve $X $ has a Belyi-map $X \rightarrow \mathbb{P}^1_{\overline{\mathbb{Q}}} $ ramified only in three points ${ 0,1,\infty } $. The “drawing” corresponding to $X $ is a bipartite graph, drawn on the Riemann surface $X_{\mathbb{C}} $ obtained by lifting the unit interval $[0,1] $ to $X $. As the absolute Galois group acts on all such curves (and hence on their corresponding drawings), the action of it on these dessins d’enfants may give us a way into the multiple mysteries of the absolute Galois group.

In his “Esquisse d’un programme” (Sketch of a program if you prefer to read it in English) he writes :

“C’est ainsi que mon attention s’est portée vers ce que j’ai appelé depuis la “géométrie algêbrique anabélienne”, dont le point de départ est justement une étude (pour le moment limitée à la caractéristique zéro) de l’action de groupe de Galois “absolus” (notamment les groupes $Gal(\overline{K}/K) $, ou $K $ est une extension de type fini du corps premier) sur des groupes fondamentaux géométriques (profinis) de variétés algébriques (définies sur $K $), et plus particulièrement (rompant avec une tradition bien enracinée) des groupes fondamentaux qui sont trés éloignés des groupes abéliens (et que pour cette raison je nomme “anabéliens”). Parmi ces groupes, et trés proche du groupe $\hat{\pi}_{0,3} $, il y a le compactifié profini du groupe modulaire $SL_2(\mathbb{Z}) $, dont le quotient par le centre $\pm 1 $ contient le précédent comme sous-groupe de congruence mod 2, et peut s’interpréter d’ailleurs comme groupe “cartographique” orienté, savoir celui qui classifie les cartes orientées triangulées (i.e. celles dont les faces des triangles ou des monogones).”

and a bit further, he writes :

“L’élément de structure de $SL_2(\mathbb{Z}) $ qui me fascine avant tout, est bien sur l’action extérieure du groupe de Galois $Gal(\overline{\mathbb{Q}}/\mathbb{Q}) $ sur le compactifié profini. Par le théorème de Bielyi, prenant les compactifiés profinis de sous-groupes d’indice fini de $SL_2(\mathbb{Z}) $, et l’action extérieure induite (quitte à passer également à un sous-groupe overt de $Gal(\overline{\mathbb{Q}},\mathbb{Q}) $), on trouve essentiellement les groupes fondamentaux de toutes les courbes algébriques définis sur des corps de nombres $K $, et l’action extérieure de $Gal(\overline{K}/K) $ dessus.”

So, is there a noncommutative affine variety over $\mathbb{F}_1 $ of which the unique minimal integral model is the integral group algebra of the modular group $\mathbb{Z} \Gamma $ (with $\Gamma = PSL_2(\mathbb{Z}) $? Yes, here it is

$N_{\Gamma}~:~\mathbf{groups} \rightarrow \mathbf{sets} \qquad G \mapsto G_2 \times G_3 $

where $G_n $ is the set of all elements of order $n $ in $G $. The reason behind this is that the modular group is the free group product $C_2 \ast C_3 $.

Fine, you may say, but all this is just algebra. Where is the noncommutative complex variety or the noncommutative integral scheme in all this? Well, we can introduce them too but as this post is already 1300 words long, I’ll better leave this for another time. In case you cannot stop thinking about it, here’s the short answer.

The complex noncommutative variety has as its ‘points’ all finite dimensional simple complex representations of the modular group, and the ‘points’ of the noncommutative $\mathbb{F}_1 $-scheme are exactly the (modular) dessins d’enfants…

Monstrous frustrations

Published June 19, 2008 by lievenlb

Thanks for clicking through… I guess.

If nothing else, it shows that just as much as the stock market is fueled by greed, mathematical reasearch is driven by frustration (or the pleasure gained from knowing others to be frustrated).

I did spend the better part of the day doing a lengthy, if not laborious, calculation, I’ve been postponing for several years now. Partly, because I didn’t know how to start performing it (though the basic strategy was clear), partly, because I knew beforehand the final answer would probably offer me no further insight.

Still, it gives the final answer to a problem that may be of interest to anyone vaguely interested in Moonshine :

What does the Monster see of the modular group?

I know at least two of you, occasionally reading this blog, understand what I was trying to do and may now wonder how to repeat the straightforward calculation. Well the simple answer is : Google for the number 97239461142009186000 and, no doubt, you will be able to do the computation overnight.

One word of advice : don’t! Get some sleep instead, or make love to your partner, because all you’ll get is a quiver on nine vertices (which is pretty good for the Monster) but having an horrible amount of loops and arrows…

If someone wants the details on all of this, just ask. But, if you really want to get me exited : find a moonshine reason for one of the following two numbers :

$791616381395932409265430144165764500492= 2^2 * 11 * 293 * 61403690769153925633371869699485301 $

(the dimension of the monster-singularity upto smooth equivalence), or,

$1575918800531316887592467826675348205163= 523 * 1655089391 * 15982020053213 * 113914503502907 $

(the dimension of the moduli space).

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