Last time we tried to generalize the Connes-Consani approach to commutative algebraic geometry over the field with one element to the noncommutative world by considering covariant functors

which over resp. 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 -’points’ associated to the functor

(here denotes all elements of order of )

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)

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

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

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

This may look strange at first sight, but is a finite dimensional algebra and any -dimensional representation of is an algebra map and we take to be the dual coalgebra of this image.

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

In the case of interest to us, that is for the functor the coalgebra of distributions is Kostant’s dual coalgebra . This is the not the full linear dual of but contains only those linear functionals on which factor through a finite dimensional quotient.

So? You’ve exchanged an algebra for some coalgebra , but where’s the geometry in all this? Well, let’s look at the commutative case. Suppose is the coordinate ring of a smooth affine variety , then its dual coalgebra looks like

the direct sum of all universal (co)algebras of tangent spaces at points . 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

so every point corresponds to a unique simple component of the coradical. In the general case, the coradical of the dual coalgebra is the direct sum of all simple finite dimensional representations of . 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 -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 (or more general, of Dedekind domains) one can repeat Kostant’s proof word for word provided one takes as the definition of the dual -coalgebra of an algebra (which is -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 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 and where is the modular group (the algebras corresponding to the -functor). But, what might be the points of the noncommutative scheme corresponding to ???

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

Here we come back to Javier’s post on this : a finite dimensional -vectorspace is a finite set. A -representation on this set (of n-elements) is a group-morphism

hence it gives a permutation representation of on this set. But then, if finite dimensional -representations of are the finite permutation representations, then the simple ones are the transitive permutation representations. That is, the points of the noncommutative scheme corresponding to are the conjugacy classes of subgroups such that 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).

Amidst all LHC-noise, Yuri I. Manin arXived today an interesting paper Cyclotomy and analytic geometry over .

The paper gives a nice survey of the existent literature and focusses on the crucial role of roots of unity in the algebraic geometry over the non-existent field with one element (in French called ‘F-un’). I have tried to do a couple of posts on F-un some time ago but now realize, reading Manin’s paper, I may have given up way too soon…

At several places in the paper, Manin hints at a possible noncommutative geometry over :

This is the appropriate place to stress that in a wider context of Toen-Vaqui ‘Au-dessous de Spec Z’, or eventually in noncommutative -geometry, teh spectrum of loses its privileged position as a final object of a geometric category. For example, in noncommutative geometry, or in an appropriate category of stacks, the quotient of this spectrum modulo the trivial action of a group must lie below this spectrum.
Soule’s algebras are a very important element of the structure, in particular, because they form a bridge to Arakelov geometry. Soule uses concrete choices of them in order to produce ‘just right’ supply of morphisms, without a priori constraining these choices formally. In this work, we use these algebras and their version also to pave a way to the analytic (and possibly non-commutative) geometry over .

Back when I was writing the first batch of F-un posts, I briefly contemplated the possibility of a noncommutative geometry over , but quickly forgot about it because I thought it would be forced to reduce to commutative geometry.

Here is the quick argument : noncommutative geometry is really the study of coalgebras (see for example my paper or if you prefer more trustworthy sources the Kontsevich-Soibelman paper). Now, unless I made a mistake, I think all coalgebras over must be co-commutative (even group-like), so reducing to commutative geometry.

Surely, I’m missing something…

Here a list of saved pdf-files of previous NeverEndingBooks-posts on geometry in reverse chronological order.

(more…)

Here are my nominees for the 2006 paper of the year award in mathematics & mathematical physics : in math.RA : math.RA/0606241 : Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I by Maxim Kontsevich and Yan Soibelman. Here is the abstract :

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne’s conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product (this action is more or less equivalent to the structure of 2-dimensional Topological Field Theory associated with an “abstract” Calabi-Yau manifold).

why ? : Because this paper probably gives the correct geometric object associated to a non-commutative algebra (a huge coalgebra) and consequently the right definition of a map between noncommutative affine schemes. In a previous post (and its predecessors) I’ve tried to explain how this links up with my own interpretation and since then I’ve thought more about this, but that will have to wait for another time. in hep-th : hep-th/0611082 : Children’s Drawings From Seiberg-Witten Curves by Sujay K. Ashok, Freddy Cachazo, Eleonora Dell’Aquila. Here is the abstract :

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called “dessins d’enfants” or “children’s drawings” on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of N=1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck’s programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories.

why ? : Because this paper gives the best introduction I’ve seen to Grothendieck’s dessins d’enfants (slightly overdoing it by giving a crash course on elementary Galois theory in appendix A) and kept me thinking about dessins and their Galois invariants ever since (again, I’ll come back to this later).

Last time we saw that the coalgebra of distributions of a noncommutative manifold can be described as a coalgebra Takeuchi-equivalent to the path coalgebra of a huge quiver. This infinite quiver has as its vertices the isomorphism classes of finite dimensional simple representations of the qurve A (the coordinate ring of the noncommutative manifold) and there are as many directed arrows between the vertices corresponding to the simples S and T as is the dimension of .

The fact that this coalgebra of distributions is equivalent to the path coalgebra of some quiver is in the Kontsevich-Soibelman paper though it would have been nice if they had given reference for this fact to the paper Wedge Products and Cotensor Coalgebras in Monoidal Categories by Ardizzoni or to previous work by P. Jara, D. Llena, L. Merino and D. Stefan, “Hereditary and formally smooth coalgebras”, Algebr. Represent. Theory 8 (2005), 363-374. In those papers it is shown that a coalgebra with coseparable coradical is hereditary if and only if it is formally smooth if and only if it is a cotensor coalgebra of some bicomodule.

At first this looks just like the dual version of the classical result that a finite dimensional hereditary algebra is Morita equivalent to the path algebra of a quiver (which is indeed what the proof does) but again the condition that the coradical is coseparable does not require the coradical to be finite dimensional… In our case, the coradical is indeed coseparable being the direct sum over all matrix coalgebras corresponding to the simple representations. Hence, we can again recover the _points of our noncommutative manifold from the direct summands of the coradical. Fortunately, one can compute this huge coalgebra of distributions from a small quiver, the one quiver to rule them all, but as I’ve been babbling about all of this here numerous times I’ll let the interested find out for themselves how you use it (a) to get at the isoclasses of all simples (hint : morally they are the smooth points of the quotient varieties of n-dimensional representations and enough tools have been developed recently to spot some fake simples, that is smooth proper semi-simple points) and (b) to compute the ragball, that is the huge quiver with vertex set the simples and arows as described above. Over the years I’ve calculated several one-quivers for a variety of qurves (such as amalgamated free products of finite groups and smooth curves). If you are in for a puzzle, try to determine it for the qurve The answer is a mysterious hexagon