The Monster is the largest of the 26 sporadic simple groups and has order

808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000

= 2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71.

It is not so much the size of its order that makes it hard to do actual calculations in the monster, but rather the dimensions of its smallest non-trivial irreducible representations (196 883 for the smallest, 21 296 876 for the next one, and so on).

In characteristic two there is an irreducible representation of one dimension less (196 882) which appears to be of great use to obtain information. For example, Robert Wilson used it to prove that The Monster is a Hurwitz group. This means that the Monster is generated by two elements g and h satisfying the relations

Geometrically, this implies that the Monster is the automorphism group of a Riemann surface of genus g satisfying the Hurwitz bound 84(g-1)=#Monster. That is,

g=9619255057077534236743570297163223297687552000000001=42151199 * 293998543 * 776222682603828537142813968452830193

Or, in analogy with the Klein quartic which can be constructed from 24 heptagons in the tiling of the hyperbolic plane, there is a finite region of the hyperbolic plane, tiled with heptagons, from which we can construct this monster curve by gluing the boundary is a specific way so that we get a Riemann surface with exactly 9619255057077534236743570297163223297687552000000001 holes. This finite part of the hyperbolic tiling (consisting of #Monster/7 heptagons) we’ll call the empire of the monster and we’d love to describe it in more detail.

Look at the half-edges of all the heptagons in the empire (the picture above learns that every edge in cut in two by a blue geodesic). They are exactly #Monster such half-edges and they form a dessin d’enfant for the monster-curve.

If we label these half-edges by the elements of the Monster, then multiplication by g in the monster interchanges the two half-edges making up a heptagonal edge in the empire and multiplication by h in the monster takes a half-edge to the one encountered first by going counter-clockwise in the vertex of the heptagonal tiling. Because g and h generated the Monster, the dessin of the empire is just a concrete realization of the monster.

Because g is of order two and h is of order three, the two permutations they determine on the dessin, gives a group epimorphism from the modular group onto the Monster-group.

In noncommutative geometry, the group-algebra of the modular group can be interpreted as the coordinate ring of a noncommutative manifold (because it is formally smooth in the sense of Kontsevich-Rosenberg or Cuntz-Quillen) and the group-algebra of the Monster itself corresponds in this picture to a finite collection of ‘points’ on the manifold. Using this geometric viewpoint we can now ask the question What does the Monster see of the modular group?

To make sense of this question, let us first consider the commutative equivalent : what does a point P see of a commutative variety X?

Evaluation of polynomial functions in P gives us an algebra epimorphism from the coordinate ring of the variety onto and the kernel of this map is the maximal ideal of consisting of all functions vanishing in P.

Equivalently, we can view the point as the scheme corresponding to the quotient . Call this the 0-th formal neighborhood of the point P.

This sounds pretty useless, but let us now consider higher-order formal neighborhoods. Call the affine scheme the n-th forml neighborhood of P, then the first neighborhood, that is with coordinate ring gives us tangent-information. Alternatively, it gives the best linear approximation of functions near P. The second neighborhood gives us the best quadratic approximation of function near P, etc. etc.

These successive quotients by powers of the maximal ideal form a system of algebra epimorphisms

and its inverse limit is the completion of the local ring in P and contains all the infinitesimal information (to any order) of the variety X in a neighborhood of P. That is, this completion contains all information that P can see of the variety X.

In case P is a smooth point of X, then X is a manifold in a neighborhood of P and then this completion is isomorphic to the algebra of formal power series where the form a local system of coordinates for the manifold X near P.

Right, after this lengthy recollection, back to our question what does the monster see of the modular group? Well, we have an algebra epimorphism

and in analogy with the commutative case, all information the Monster can gain from the modular group is contained in the -adic completion

where is the kernel of the epimorphism sending the two free generators of the modular group to the permutations g and h determined by the dessin of the pentagonal tiling of the Monster’s empire.

As it is a hopeless task to determine the Monster-empire explicitly, it seems even more hopeless to determine the kernel let alone the completed algebra… But, (surprise) we can compute as explicitly as in the commutative case we have for a point P on a manifold X.

Here the details : the quotient has a natural structure of -bimodule. The group-algebra of the monster is a semi-simple algebra, that is, a direct sum of full matrix-algebras of sizes corresponding to the dimensions of the irreducible monster-representations. That is,

with exactly 194 components (the number of irreducible Monster-representations). For any -bimodule one can form the tensor-algebra

and applying the formal neighborhood theorem for formally smooth algebras (such as ) due to Joachim Cuntz (left) and Daniel Quillen (right) we have an isomorphism of algebras

where the right-hand side is the completion of the tensor-algebra (at the unique graded maximal ideal) of the -bimodule , so we’d better describe this bimodule explicitly.

Okay, so what’s a bimodule over a semisimple algebra of the form ? Well, a simple S-bimodule must be either (1) a factor with all other factors acting trivially or (2) the full space of rectangular matrices with the factor acting on the left, acting on the right and all other factors acting trivially.

That is, any S-bimodule can be represented by a quiver (that is a directed graph) on k vertices (the number of matrix components) with a loop in vertex i corresponding to each simple factor of type (1) and a directed arrow from i to j corresponding to every simple factor of type (2).

That is, for the Monster, the bimodule is represented by a quiver on 194 vertices and now we only have to determine how many loops and arrows there are at or between vertices.

Using Morita equivalences and standard representation theory of quivers it isn’t exactly rocket science to determine that the number of arrows between the vertices corresponding to the irreducible Monster-representations and is equal to

Now, I’ve been wasting a lot of time already here explaining what representations of the modular group have to do with quivers (see for example here or some other posts in the same series) and for quiver-representations we all know how to compute Ext-dimensions in terms of the Euler-form applied to the dimension vectors.

Right, so for every Monster-irreducible we have to determine the corresponding dimension-vector for the quiver

Now the dimensions are the dimensions of the +/-1 eigenspaces for the order 2 element g in the representation and the are the dimensions of the eigenspaces for the order 3 element h. So, we have to determine to which conjugacy classes g and h belong, and from Wilson’s paper mentioned above these are classes 2B and 3B in standard Atlas notation.

So, for each of the 194 irreducible Monster-representations we look up the character values at 2B and 3B (see below for the first batch of those) and these together with the dimensions determine the dimension vector .

For example take the 196883-dimensional irreducible. Its 2B-character is 275 and the 3B-character is 53. So we are looking for a dimension vector such that and giving us for that representation the dimension vector of the quiver above .

Okay, so for each of the 194 irreducibles we have determined a dimension vector , then standard quiver-representation theory asserts that the number of loops in the vertex corresponding to is equal to

and that the number of arrows from vertex to vertex is equal to

This data then determines completely the -bimodule and hence the structure of the completion containing all information the Monster can gain from the modular group.

But then, one doesn’t have to go for the full regular representation of the Monster. Any faithful permutation representation will do, so we might as well go for the one of minimal dimension.

That one is known to correspond to the largest maximal subgroup of the Monster which is known to be a two-fold extension of the Baby-Monster. The corresponding permutation representation is of dimension 97239461142009186000 and decomposes into Monster-irreducibles

(in standard Atlas-ordering) and hence repeating the arguments above we get a quiver on just 9 vertices! The actual numbers of loops and arrows (I forgot to mention this, but the quivers obtained are actually symmetric) obtained were found after laborious computations mentioned in this post and the details I’ll make avalable here.

Anyone who can spot a relation between the numbers obtained and any other part of mathematics will obtain quantities of genuine (ie. non-Inbev) Belgian beer…

Last time we introduced the tangent quiver of an affine algebra A to be a quiver on the isoclasses of simple finite dimensional representations. When is the coordinate ring of an affine variety, these vertices are just the points of the variety and this set has the extra structure of being endowed with the Zariski topology. For a general, possibly noncommutative algebra, we would like to equip the vertices of also with a topology.

In the commutative case, the Zariski topology has as its closed sets the common zeroes of a set of polynomials on , so we need to generalize the notion of ‘functions’ the the noncommutative world. The NC-mantra states that we should view the algebra A as the ring of functions on a (usually virtual) noncommutative space. And, face it, for a commutative variety the algebra does indeed do the job. Still, this is a red herring.

Let’s consider the easiest noncommutative case, that of the group algebra of a finite group . In this case, the vertices of the tangent quiver are the irreducible representations of and no sane person would consider the full group algebra to be the algebra of functions on this set. However, we do have a good alternative in this case : characters which allow us to separate the irreducibles and are a lot more manageable than the full group algebra. For example, if is the monster group then the group algebra has dimension approx whereas there are just 194 characters to consider…

But, can we extend characters to arbitrary noncommutative algebras? and, more important, are there enough of these to separate the simple representations? The first question is easy enough to answer, after all characters are just traces so we can define for every element and any finite dimensional simple A-representation the character

where is the matrix describing the action of a on S. But, you might say, characters are then just linear functionals on the algebra A so it is natural to view A as the function algebra, right? Wrong! Traces have the nice property that and so they vanish on all commutators of A, so characters only carry information of the quotient space

where is the vectorspace spanned by all commutators (and not the ideal…). If one is too focussed on commutative geometry one misses this essential simplification as clearly for being a commutative algebra,

and therefore in this case

Ok, but are there enough characters (that is, linear functionals on , that is elements of the dual space ) to separate the simple representations? And, why do I (ab)use Lie-algebra notation to denote the vectorspace ???

Take an affine -algebra A (not necessarily commutative). We will assign to it a strange object called the tangent-quiver , compute it in a few examples and later show how it connects with existing theory and how it can be used. This series of posts can be seen as the promised notes of my talks at the GAMAP-workshop but in reverse order… If some of the LaTeX-pictures are not in the desired spots, please size and resize your browser-window and they will find their intended positions.

A vertex of corresponds to the isomorphism class of a finite dimensional simple A-representations and between any two such vertices, say and , the number of directed arrows from to is given by the dimension of the Ext-space

Recall that this Ext-space counts the equivalence classes of short exact sequences of A-representations

where two such sequences (say with middle terms V resp. W) are equivalent if there is an A-isomorphism making the diagram below commutative

The Ext-space measures how many non-split extensions there are between the two simples and is always a finite dimensional vectorspace. So the tangent quiver has the property that in all vertices there are at most finitely many loops and between any two vertices there are a finite number of directed arrows, but in principle a vertex may be the origin of arrows connecting it to infinitely many other vertices.

Right, now let us at least motivate the terminology. Let be a (commutative) affine variety with coordinate ring then what is in this case? To begin, as is commutative, all its finite dimensional simple representations are one-dimensional and there is one such for every point . Therefore, the vertices of correspond to the points of the affine variety . The simple A-representation corresponding to a point is just evaluating polynomials in . Moreover, if then there are no non-split extensions between and (a commutative semi-local algebra splits as a direct sum of locals), therefore in there can only be loops and no genuine arrows between different vertices. Finally, the number of loops in the vertex corresponding to the point can be computed using the fact that the self-extensions can be identified with the tangent space at , that is

That is, if is the coordinate ring of an affine variety , then the quiver is the set of points of having in each point as many loops as the dimension of the tangent space . So, in this case, the quiver contains all information about tangent spaces to the variety and that’s why we call it the tangent quiver.

Let’s go into the noncommutative wilderness. A first, quite trivial, example is the group algebra of a finite group , then the simple A-representations are just the irreducible G-representations and as the group algebra is semi-simple every short exact sequence splits so all Ext-spaces are zero. That is, in this case the tangent quiver in just a finite set of vertices (as many as there are irreducible G-representations) and no arrows nor loops.

Now you may ask whether there are examples of tangent quivers having arrows apart from loops. So, take another easy finite dimensional example : the path algebra of a finite quiver without oriented cycles. Recall that the path algebra is the vectorspace having as basis all vertices and all oriented paths in the quiver Q (and as there are no cycles, this basis is finite) and multiplication is induced by concatenation of paths. Here an easy example. Suppose the quiver Q looks like

then the path algebra is 6 dimensional as there are 3 vertices, 2 paths of length one (the arrows) and one path of length two (going from the leftmost to the rightmost vertex). The concatenation rule shows that the three vertices will give three idempotents in A and one easily verifies that the path algebra can be identified with upper-triangular matrices

where the diagonal components correspond to the vertices, the first offdiagonal components to the two arrows and the corner component corresponds to the unique path of length two. Right, for a general finite quiver without oriented cycles is the quite easy to see that all finite dimensional simples are one-dimensional and correspond to the vertex-idempotents, that is every simple is of the form where is the vertex idempotent. No doubt, you can guess what the tangent quiver will be, can’t you?

After a lengthy spring-break, let us continue with our course on noncommutative geometry and -representations. Last time, we have explained Grothendiecks mantra that all algebraic curves defined over number fields are contained in the profinite compactification
of the modular group and in the knowledge of a certain subgroup G of its group of outer automorphisms. In particular we have seen that many curves defined over the algebraic numbers correspond to permutation representations of . The profinite compactification is a continuous group, so it makes sense to consider its continuous n-dimensional representations Such representations are known to have a finite image in and therefore we get an embedding into all n-dimensional (semi-simple) representations of . We consider such semi-simple points as classical objects as they are determined by - curves defined over - representations of (sporadic) finite groups - modlart data of fusion rings in RCTF - etc… To get a feel for the distinction between these continuous representations of the cofinite completion and all representations, consider the case of . Its one-dimensional continuous representations are determined by roots of unity, whereas all one-dimensional (necessarily simple) representations of are determined by all elements of . Hence, the image of is contained in the unit circle

and though these points are very special there are enough of them (technically, they form a Zariski dense subset of all representations). Our aim will be twofold : (1) when viewing a classical object as a representation of we can define its modular content (which will be the noncommutative tangent space in this classical point to the noncommutative manifold of ). In this way we will associate noncommutative gadgets to our classical object (such as orders in central simple algebras, infinite dimensional Lie algebras, noncommutative potentials etc. etc.) which give us new tools to study these objects. (2) conversely, as we control the tangentspaces in these special points, they will allow us to determine other -representations and as we vary over all classical objects, we hope to get ALL finite dimensional modular representations. I agree this may all sound rather vague, so let me give one example we will work out in full detail later on. Remember that one can reconstruct the sporadic simple Mathieu group from the dessin d’enfant

This dessin determines a 24-dimensional permutation representation (of as well of ) which decomposes as the direct sum of the trivial representation and a simple 23-dimensional representation. We will see that the noncommutative tangent space in a semi-simple representation of is determined by a quiver (that is, an oriented graph) on as many vertices as there are non-isomorphic simple components. In this special case we get the quiver on two points with just one arrow in each direction between the vertices and 96 loops in the second vertex. To the experienced tangent space-reader this picture (and in particular that there is a unique cycle between the two vertices) tells the remarkable fact that there is a distinguished one-parameter family of 24-dimensional simple modular representations degenerating to the permutation representation of the largest Mathieu-group. Phrased differently, there is a specific noncommutative modular Riemann surface associated to , which is a new object (at least as far as I’m aware) associated to this most remarkable of sporadic groups. Conversely, from the matrix-representation of the 24-dimensional permutation representation of we obtain representants of all of this one-parameter family of simple -representations to which we can then perform noncommutative flow-tricks to get a Zariski dense set of all 24-dimensional simples lying in the same component. (Btw. there are also such noncommutative Riemann surfaces associated to the other sporadic Mathieu groups, though not to the other sporadics…) So this is what we will be doing in the upcoming posts (10) : explain what a noncommutative tangent space is and what it has to do with quivers (11) : what is the noncommutative manifold of ? and what is its connection with the Kontsevich-Soibelman coalgebra? (12) : is there a noncommutative compactification of ? (and other arithmetical groups) (13) : how does one calculate the noncommutative curves associated to the Mathieu groups? (14) : whatever comes next… (if anything).

According to a science article in the New York Times, archeologists have discovered “signs of advanced math” in medieval mosaics. An example of a quasi-crystalline Penrose pattern was found at the Darb-i Imam shrine in Isfahan, Iran.

A new study shows that the Islamic pattern-making process, far more intricate than the laying of one’s bathroom floor, appears to have involved an advanced math of quasi crystals, which was not understood by modern scientists until three decades ago. Two years ago, Peter J. Lu, a doctoral student in physics at Harvard University, was transfixed by the geometric pattern on a wall in Uzbekistan. It reminded him of what mathematicians call quasi-crystalline designs. These were demonstrated in the early 1970s by Roger Penrose, a mathematician and cosmologist at the University of Oxford. Mr. Lu set about examining pictures of other tile mosaics from Afghanistan, Iran, Iraq and Turkey, working with Paul J. Steinhardt, a Princeton cosmologist who is an authority on quasi crystals and had been Mr. Lu’s undergraduate adviser.

Penrose tilings are one of the motivating examples of Alain Connes’ book as there is a -algebra associated to it. In fact, the algebra is AF ( a limit of semi-simple finite dimensional algebras) so is even a formally smooth algebra in Kontsevichian noncommutative geometry (it is remarkable how quickly one gets used to silly terminology…). However, the Penrose algebra is simple, so rather useless from the point of view of finite dimensional representations… Still, Connesian noncommutative geometry may be a recent incarnation of the medieval Tehran program (pun intended). Thanks to easwaran for the link (via Technorati).

Added, March 1 : I haven’t looked at the Connes-Marcolli paper A walk in the noncommutative garden for a while but now that I do, I see that they mentioned the above already at the end of their section on Tilings (page 32). They also include clearer pictures.