Posts tagged representations

Pollock your own noncommutative space

May 19th

I really like Matilde Marcolli’s idea to use some of Jackson Pollock’s paintings as metaphors for noncommutative spaces. In her talk she used this painting

and refered to it (as did I in my post) as : Jackson Pollock “Untitled N.3”. Before someone writes a post ‘The Pollock noncommutative space hoax’ (similar to my own post) let me point out that I am well aware of the controversy surrounding this painting.

This painting is among 32 works recently discovered and initially attributed to Pollock. In fact, I’ve already told part of the story in Doodles worth millions (or not)? (thanks to PD1). The story involves the people on the right : from left to right, Jackson Pollock, his wife Lee Krasner, Mercedes Matter and her son Alex Matter.

Alex Matter, whose father, Herbert, and mother, Mercedes, were artists and friends of Jackson Pollock, discovered after his mother died a group of small drip paintings in a storage locker in Wainscott, N.Y. which he believed to be authentic Pollocks.

Read the post mentioned above if you want to know how mathematics screwed up his plan, or much better, reed the article Anatomy of the Jackson Pollock controversy by Stephen Litt.

So, perhaps the painting above was not the smartest choice, but we could take any other genuine Pollock ‘drip-painting’, a technique he taught himself towards the end of 1946 to make an image by splashing, pouring, sloshing colors onto the canvas. Typically, such a painting consists of blops of paint, connected via thin drip-lines.

What does this have to do with noncommutative geometry? Well, consider the blops as ‘points’. In commutative geometry, distinct points cannot share tangent information¹. In the noncommutative world though, they can!, or if you want to phrase it like this, noncommutative points ‘can talk to each other’. And, that’s what we cherish in those drip-lines.

But then, if two points share common tangent informations, they must be awfully close to each other… so one might imagine these Pollock-lines to be strings holding these points together. Hence, it would make more sense to consider the ‘Pollock-quotient-painting’, that is, the space one gets after dividing out the relation ‘connected by drip-lines’².

For this reason, my own mental picture of a genuinely noncommutative space ³ looks more like the picture below

The colored blops you see are really sets of points which you might view as, say, a FacebookGroup⁴. Some chatter may occur between two distinct FacebookGroups, the more chatter the thicker the connection depicted⁵. Now, there are some tiny isolated spots (say blue ones in the upper right-hand quadrant). These should really be looked at as remote clusters of noncommutative points (sharing no (tangent) information whatsoever with the blops in the foregound). If we would zoom into them beyond the Planck scale (if I’m allowed to say a bollock-word in a Pollock-post) they might reveal again a whole universe similar to the interconnected blops upfront.

The picture was produced using the fabulous Pollock engine. Just use your mouse to draw and click to change colors in order to produce your very own noncommutative space!

For the mathematicians still around, this may sound like a lot of Pollock-bollocks but can be made precise. See my note Noncommutative geometry and dual coalgebras for a very terse reading. Now that coalgebras are gaining popularity, I really should write a more readable account of it, including some fanshi-wanshi examples…

technically : a commutative semi-local ring splits as the direct sum of local rings and this does no longer hold for a noncommutative semi-local ring [↩]
my guess is that Matilde thinks of the lines as the action of a group on the points giving a topological horrible quotient space, and thats precisely where noncommutative geometry shines [↩]
that is, the variety corresponding to a huge noncommutative algebra such as free algebras, group algebras of arithmetic groups or fundamental groups [↩]
technically, think of them as the connected components of isomorphism classes of finite dimensional simple representations of your favorite noncommutative algebra [↩]
technically, the size of the connection is the dimension of the ext-group between generic simples in the components [↩]

Print Friendly

arxiv, coalgebras, geometry, groups, Marcolli, noncommutative, paintings, representations, simples

Conway’s big picture

May 2nd

Posted by lievenlb in groups

No comments

Moonshine and E(8)

the monster graph and McKay’s observation
Conway’s big picture
looking for the moonshine picture
E(8) from moonshine groups

Conway and Norton showed that there are exactly 171 moonshine functions and associated two arithmetic subgroups to them. We want a tool to describe these and here’s where Conway’s big picture comes in very handy. All moonshine groups are arithmetic groups, that is, they are commensurable with the modular group. Conway’s idea is to view several of these groups as point- or set-wise stabilizer subgroups of finite sets of (projective) commensurable 2-dimensional lattices.

Expanding (and partially explaining) the original moonshine observation of McKay and Thompson, John Conway and Simon Norton formulated monstrous moonshine :

To every cyclic subgroup $\langle m \rangle$ of the Monster $\mathbb{M}$ is associated a function

$f_m(\tau)=\frac{1}{q}+a_1q+a_2q^2+\hdots$ with $q=e^{2 \pi i \tau}$ and all coefficients $a_i \in \Z$ are characters at of a representation of $\mathbb{M}$ . These representations are the homogeneous components of the so called Moonshine module.

Each f_m is a principal modulus for a certain genus zero congruence group commensurable with the modular group $\Gamma = PSL_2(\Z)$ . These groups are called the moonshine groups.

Conway and Norton showed that there are exactly 171 different functions f_m and associated two arithmetic subgroups $F(m) \subset E(m) \subset PSL_2(\mathbb{R})$ to them (in most cases, but not all, these two groups coincide).

Whereas there is an extensive literature on subgroups of the modular group (see for instance the series of posts starting here), most moonshine groups are not contained in the modular group. So, we need a tool to describe them and here’s where Conway’s big picture comes in very handy.

All moonshine groups are arithmetic groups, that is, they are subgroups of $PSL_2(\mathbb{R})$ which are commensurable with the modular group $\Gamma = PSL_2(\Z)$ meaning that the intersection $G \cap \Gamma$ is of finite index in both and in $\Gamma$ . Conway’s idea is to view several of these groups as point- or set-wise stabilizer subgroups of finite sets of (projective) commensurable 2-dimensional lattices.

Start with a fixed two dimensional lattice $L_1 = \Z e_1 + \Z e_2 = \langle e_1,e_2 \rangle$ and we want to name all lattices of the form $L = \langle v_1= a e_1+ b e_2, v_2 = c e_1 + d e_2 \rangle$ that are commensurable to L_1 . Again this means that the intersection $L \cap L_1$ is of finite index in both lattices. From this it follows immediately that all coefficients a,b,c,d are rational numbers.

It simplifies matters enormously if we do not look at lattices individually but rather at projective equivalence classes, that is $~L=\langle v_1, v_2 \rangle \sim L' = \langle v'_1,v'_2 \rangle$ if there is a rational number $\lambda \in \mathbb{Q}$ such that $~\lambda v_1 = v'_1, \lambda v_2=v'_2$ . Further, we are of course allowed to choose a different ‘basis’ for our lattices, that is, $~L = \langle v_1,v_2 \rangle = \langle w_1,w_2 \rangle$ whenever $~(w_1,w_2) = (v_1,v_2).\gamma$ for some $\gamma \in PSL_2(\Z)$ . Using both operations we can get any lattice in a specific form. For example,

$\langle \frac{1}{2}e_1+3e_2,e_1-\frac{1}{3}e_2 \overset{(1)}{=} \langle 3 e_1+18e_2,6e_1-2e_2 \rangle \overset{(2)}{=} \langle 3 e_1+18 e_2,38 e_2 \rangle \overset{(3)}{=} \langle \frac{3}{38}e_1+\frac{9}{19}e_2,e_2 \rangle$

Here, identities (1) and (3) follow from projective equivalence and identity (2) from a base-change. In general, any lattice commensurable to the standard lattice L_1 can be rewritten uniquely as $L = \langle Me_1 + \frac{g}{h} e_2,e_2 \rangle$ where a positive rational number and with $0 \leq \frac{g}{h} < 1$ .

Another major feature is that one can define a symmetric hyper-distance between (equivalence classes of) such lattices. Take $L=\langle Me_1 + \frac{g}{h} e_2,e_2 \rangle$ and $L'=\langle N e_1 + \frac{i}{j} e_2,e_2 \rangle$ and consider the matrix

$D_{LL'} = \begin{bmatrix} M & \frac{g}{h} \\ 0 & 1 \end{bmatrix} \begin{bmatrix} N & \frac{i}{j} \\ 0 & 1 \end{bmatrix}^{-1}$ and let $\alpha$ be the smallest positive rational number such that all entries of the matrix $\alpha.D_{LL'}$ are integers, then

$\delta(L,L') = det(\alpha.D_{LL'}) \in \mathbb{N}$ defines a symmetric hyperdistance which depends only of the equivalence classes of lattices (hyperdistance because the log of it behaves like an ordinary distance).

Conway’s big picture is the graph obtained by taking as its vertices the equivalence classes of lattices commensurable with L_1 and with edges connecting any two lattices separated by a prime number hyperdistance. Here’s part of the 2-picture, that is, only depicting the edges of hyperdistance 2.

The 2-picture is an infinite 3-valent tree as there are precisely 3 classes of lattices at hyperdistance 2 from any lattice $L = \langle v_1,v_2 \rangle$ namely (the equivalence classes of) $\langle \frac{1}{2}v_1,v_2 \rangle~,~\langle v_1, \frac{1}{2} v_2 \rangle$ and $\langle \frac{1}{2}(v_1+v_2),v_2 \rangle$ .

Similarly, for any prime hyperdistance p, the p-picture is an infinite p+1-valent tree and the big picture is the product over all these prime trees. That is, two lattices at square-free hyperdistance $N=p_1p_2\hdots p_k$ are two corners of a k-cell in the big picture! (Astute readers of this blog (if such people exist…) may observe that Conway’s big picture did already appear here prominently, though in disguise. More on this another time).

The big picture presents a simple way to look at arithmetic groups and makes many facts about them visually immediate. For example, the point-stabilizer subgroup of L_1 clearly is the modular group $PSL_2(\Z)$ . The point-stabilizer of any other lattice is a certain conjugate of the modular group inside $PSL_2(\mathbb{R})$ . For example, the stabilizer subgroup of the lattice $L_N = \langle Ne_1,e_2 \rangle$ (at hyperdistance N from L_1 ) is the subgroup

$\{ \begin{bmatrix} a & \frac{b}{N} \\ Nc & d \end{bmatrix}~|~\begin{bmatrix} a & b \\ c & d \end{bmatrix} \in PSL_2(\Z)~\}$

Now the intersection of these two groups is the modular subgroup $\Gamma_0(N)$ (consisting of those modular group element whose lower left-hand entry is divisible by N). That is, the proper way to look at this arithmetic group is as the joint stabilizer of the two lattices L_1,L_N . The picture makes it trivial to compute the index of this subgroup.

Consider the ball B(L_1,N) with center L_1 and hyper-radius N (on the left, the ball with hyper-radius 4). Then, it is easy to show that the modular group acts transitively on the boundary lattices (including the lattice L_N ), whence the index $[ \Gamma : \Gamma_0(N)]$ is just the number of these boundary lattices. For N=4 the picture shows that there are exactly 6 of them. In general, it follows from our knowledge of all the p-trees the number of all lattices at hyperdistance N from L_1 is equal to $N \prod_{p | N}(1+ \frac{1}{p})$ , in accordance with the well-known index formula for these modular subgroups!

But, there are many other applications of the big picture giving a simple interpretation for the Hecke operators, an elegant proof of the Atkin-Lehner theorem on the normalizer of $\Gamma_0(N)$ (the whimsical source of appearances of the number 24) and of Helling’s theorem characterizing maximal arithmetical groups inside $PSL_2(\mathb{C})$ as conjugates of the normalizers of $\Gamma_0(N)$ for square-free N. J.H. Conway’s paper “Understanding groups like $\Gamma_0(N)$ ” containing all this material is a must-read! Unfortunately, I do not know of an online version.

Previous in series

Next in series

Print Friendly

Conway, groups, modular, monster, moonshine, representations

the monster graph and McKay’s observation

Apr 22nd

Posted by lievenlb in groups

7 comments

Moonshine and E(8)

the monster graph and McKay’s observation
Conway’s big picture
looking for the moonshine picture
E(8) from moonshine groups

While the verdict on a neolithic Scottish icosahedron is still open, let us recall Kostant’s group-theoretic construction of the icosahedron from its rotation-symmetry group A_5 .

The alternating group A_5 has two conjugacy classes of order 5 elements, both consisting of exactly 12 elements. Fix one of these conjugacy classes, say and construct a graph with vertices the 12 elements of and an edge between two $u,v \in C$ if and only if the group-product $u.v \in C$ still belongs to the same conjugacy class.

Observe that this relation is symmetric as from $u.v = w \in C$ it follows that $v.u=u^{-1}.u.v.u = u^{-1}.w.u \in C$ . The graph obtained is the icosahedron, depicted on the right with vertices written as words in two adjacent elements u and v from , as indicated.

Kostant writes : “Normally it is not a common practice in group theory to consider whether or not the product of two elements in a conjugacy class is again an element in that conjugacy class. However such a consideration here turns out to be quite productive.”

Still, similar constructions have been used in other groups as well, in particular in the study of the largest sporadic group, the monster group $\mathbb{M}$ .

There is one important catch. Whereas it is quite trivial to multiply two permutations and verify whether the result is among 12 given ones, for most of us mortals it is impossible to do actual calculations in the monster. So, we’d better have an alternative way to get at the icosahedral graph using only A_5 -data that is also available for the monster group, such as its character table.

Let be any finite group and consider three of its conjugacy classes C(i),C(j) and C(k) . For any element $w \in C(k)$ we can compute from the character table of the number of different products u.v = w such that $u \in C(i)$ and $v \in C(j)$ . This number is given by the formula

$\frac{|G|}{|C_G(g_i)||C_G(g_j)|} \sum_{\chi} \frac{\chi(g_i) \chi(g_j) \overline{\chi(g_k)}}{\chi(1)}$

where the sum is taken over all irreducible characters $\chi$ and where $g_i \in C(i),g_j \in C(j)$ and $g_k \in C(k)$ . Note also that |C_G(g)| is the number of -elements commuting with and that this number is the order of divided by the number of elements in the conjugacy class of .

The character table of A_5 is given on the left : the five columns correspond to the different conjugacy classes of elements of order resp. 1,2,3,5 and 5 and the rows are the character functions of the 5 irreducible representations of dimensions 1,3,3,4 and 5.

Let us fix the 4th conjugacy class, that is 5a, as our class . By the general formula, for a fixed $w \in C$ the number of different products u.v=w with $u,v \in C$ is equal to

$\frac{60}{25}(\frac{1}{1} + \frac{(\frac{1+\sqrt{5}}{2})^3}{3} + \frac{(\frac{1-\sqrt{5}}{2})^3}{3} – \frac{1}{4} + \frac{0}{5}) = \frac{60}{25}(1 + \frac{4}{3} – \frac{1}{4}) = 5$

Because for each $x \in C$ also its inverse $x^{-1} \in C$ , this can be rephrased by saying that there are exactly 5 different products $w^{-1}.u \in C$ , or equivalently, that the valency of every vertex $w^{-1} \in C$ in the graph is exactly 5.

That is, our graph has 12 vertices, each with exactly 5 neighbors, and with a bit of extra work one can show it to be the icosahedral graph.

For the monster group, the Atlas tells us that it has exactly 194 irreducible representations (and hence also 194 conjugacy classes). Of these conjugacy classes, the involutions (that is the elements of order 2) are of particular importance.

There are exactly 2 conjugacy classes of involutions, usually denoted 2A and 2B. Involutions in class 2A are called “Fischer-involutions”, after Bernd Fischer, because their centralizer subgroup is an extension of Fischer’s baby Monster sporadic group.

Likewise, involutions in class 2B are usually called “Conway-involutions” because their centralizer subgroup is an extension of the largest Conway sporadic group.

Let us define the monster graph to be the graph having as its vertices the Fischer-involutions and with an edge between two of them $u,v \in 2A$ if and only if their product u.v is again a Fischer-involution.

Because the centralizer subgroup is $2.\mathbb{B}$ , the number of vertices is equal to 97239461142009186000 = 2^4 * 3^7 * 5^3 * 7^4 * 11 * 13^2 * 29 * 41 * 59 * 71 .

From the general result recalled before we have that the valency in all vertices is equal and to determine it we have to use the character table of the monster and the formula. Fortunately GAP provides the function ClassMultiplicationCoefficient to do this without making errors.

gap> table:=CharacterTable("M"); CharacterTable( "M" ) gap> ClassMultiplicationCoefficient(table,2,2,2); 27143910000

Perhaps noticeable is the fact that the prime decomposition of the valency 27143910000 = 2^4 * 3^4 * 5^4 * 23 * 31 * 47 is symmetric in the three smallest and three largest prime factors of the baby monster order.

Robert Griess proved that one can recover the monster group $\mathbb{M}$ from the monster graph as its automorphism group!

As in the case of the icosahedral graph, the number of vertices and their common valency does not determine the monster graph uniquely. To gain more insight, we would like to know more about the sizes of minimal circuits in the graph, the number of such minimal circuits going through a fixed vertex, and so on.

Such an investigation quickly leads to a careful analysis which other elements can be obtained from products u.v of two Fischer involutions $u,v \in 2A$ . We are in for a major surprise, first observed by John McKay:

Printing out the number of products of two Fischer-involutions giving an element in the i-th conjugacy class of the monster, where i runs over all 194 possible classes, we get the following string of numbers :

97239461142009186000, 27143910000, 196560, 920808, 0, 3, 1104, 4, 0, 0, 5, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

That is, the elements of only 9 conjugacy classes can be written as products of two Fischer-involutions! These classes are :

1A = { 1 } written in 97239461142009186000 different ways (after all involutions have order two)
2A, each element of which can be written in exactly 27143910000 different ways (the valency)
2B, each element of which can be written in exactly 196560 different ways. Observe that this is the kissing number of the Leech lattice leading to a permutation representation of .
3A, each element of which can be written in exactly 920808 ways. Note that this number gives a permutation representation of the maximal monster subgroup $3.Fi_{24}'$ .
3C, each element of which can be written in exactly 3 ways.
4A, each element of which can be written in exactly 1104 ways.
4B, each element of which can be written in exactly 4 ways.
5A, each element of which can be written in exactly 5 ways.
6A, each element of which can be written in exactly 6 ways.

Let us forget about the actual numbers for the moment and concentrate on the orders of these 9 conjugacy classes : 1,2,2,3,3,4,4,5,6. These are precisely the components of the fundamental root of the extended Dynkin diagram $\tilde{E_8}$ !

This is the content of John McKay’s E(8)-observation : there should be a precise relation between the nodes of the extended Dynkin diagram and these 9 conjugacy classes in such a way that the order of the class corresponds to the component of the fundamental root. More precisely, one conjectures the following correspondence

This is similar to the classical McKay correspondence between finite subgroups of SU(2) and extended Dynkin diagrams (the binary icosahedral group corresponding to extended E(8)). In that correspondence, the nodes of the Dynkin diagram correspond to irreducible representations of the group and the edges are determined by the decompositions of tensor-products with the fundamental 2-dimensional representation.

Here, however, the nodes have to correspond to conjugacy classes (rather than representations) and we have to look for another procedure to arrive at the required edges! An exciting proposal has been put forward recently by John Duncan in his paper Arithmetic groups and the affine E8 Dynkin diagram.

It will take us a couple of posts to get there, but for now, let’s give the gist of it : monstrous moonshine gives a correspondence between conjugacy classes of the monster and certain arithmetic subgroups of $PSL_2(\mathbb{R})$ commensurable with the modular group $\Gamma = PSL_2(\mathbb{Z})$ . The edges of the extended Dynkin E(8) diagram are then given by the configuration of the arithmetic groups corresponding to the indicated 9 conjugacy classes! (to be continued…)

Next in series

Print Friendly

arxiv, Brauer, Conway, groups, modular, monster, moonshine, permutation representation, representations, symmetry

noncommutative F_un geometry (2)

Oct 17th

Posted by lievenlb in geometry

No comments

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

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

which over $\C$ resp. $\Z$ 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

$D~:~\wis{groups} \rightarrow \wis{sets} \qquad G \mapsto G_2 \times G_3$ (here G_n 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)

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

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

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

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

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

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

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

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

Kontsevich and Soibelman proved that every noncommutative thin scheme $\mathbb{X}$ is representable by a -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 -algebra we have

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

In the case of interest to us, that is for the functor $\wis{rep}(A)$ the coalgebra of distributions is Kostant’s dual coalgebra A^o . 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 A^o , but where’s the geometry in all this? Well, let’s look at the commutative case. Suppose $A= \C[X]$ is the coordinate ring of a smooth affine variety , then its dual coalgebra looks like

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

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

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

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 . 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 $\Z$ (or more general, of Dedekind domains) one can repeat Kostant’s proof word for word provided one takes as the definition of the dual $\Z$ -coalgebra of an algebra (which is $\Z$ -torsion free)

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

(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 $\Z$ 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 $\C \Gamma$ and $\Z \Gamma$ where $\Gamma = PSL_2(\Z)$ 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 )

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).

Print Friendly

arxiv, coalgebras, Connes, Dedekind, geometry, google, Grothendieck, groups, Kontsevich, modular, noncommutative, permutation representation, representations

noncommutative F_un geometry (1)

Oct 13th

Posted by lievenlb in geometry

No comments

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

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

from finite abelian groups to sets, together with additional data to the effect that there is a unique minimal integral scheme associated to . 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

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

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 $\C$ -algebra together with a natural transformation (the ‘evaluation’)

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

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

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

any affine $\Z$ -algebra having a gadget-morphism $~(N,\wis{maxi}(R)) \rightarrow (\wis{mini}(T),\wis{maxi}(T \otimes_{\Z} \C))$ comes from a $\Z$ -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 $\wis{groups} \rightarrow \wis{sets}$ still has as its integral form the integral torus $\Z[x,x^{-1}]$ . However, both theories quickly diverge beyond this example.

For example, consider the functor

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

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

But then, perhaps there isn’t a minimal integral $\Z$ -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 $\Z \mathcal{F}_2$ does the trick.

Again, the proof-strategy is the same. Given a gadget-morphism we have an algebra map $f~:~T \mapsto \C \mathcal{F}_2$ and we have to show, using the universal property that the image of is contained in the integral group-algebra $\Z \mathcal{F}_2$ . Take a generator of then the degree of the image f(z) is bounded say by 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 $\C \mathcal{F}_2 \rightarrow \C \mathcal{F}_2/H$ 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 $\Z$ 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 has a Belyi-map $X \rightarrow \mathbb{P}^1_{\overline{\mathbb{Q}}}$ ramified only in three points $\{ 0,1,\infty \}$ . The “drawing” corresponding to is a bipartite graph, drawn on the Riemann surface $X_{\C}$ obtained by lifting the unit interval [0,1] to . 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 est une extension de type fini du corps premier) sur des groupes fondamentaux géométriques (profinis) de variétés algébriques (définies sur ), 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 $Gal(\overline{\Q}/\Q)$ 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 $Gal(\overline{\Q},\Q)$ ), on trouve essentiellement les groupes fondamentaux de toutes les courbes algébriques définis sur des corps de nombres , 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 $\Z \Gamma$ (with $\Gamma = PSL_2(\Z)$ ? Yes, here it is

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

where G_n is the set of all elements of order in . 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…

Print Friendly

Connes, Galois, geometry, Grothendieck, groups, modular, noncommutative, profinite, representations, Riemann

Posts tagged representations

Pollock your own noncommutative space

Conway’s big picture

Moonshine and E(8)

the monster graph and McKay’s observation

Moonshine and E(8)

noncommutative F_un geometry (2)

noncommutative F_un geometry (1)

My latest tweets

Blogroll

Series of posts

Who's Online

About Me

Posts tagged representations

Pollock your own noncommutative space

Conway’s big picture

Moonshine and E(8)

the monster graph and McKay’s observation

Moonshine and E(8)

noncommutative F_un geometry (2)

noncommutative F_un geometry (1)

My latest tweets

User Login

Blogroll

Series of posts

Who's Online

About Me