Posts tagged Quillen

what does the monster see?

Jul 16th

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

g^2 = h^3 = (gh)^7 = 1

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 $C_2 \ast C_3 = PSL_2(\mathbb{Z}) \rightarrow \mathbb{M}$ from the modular group $PSL_2(\mathbb{Z})$ onto the Monster-group.

In noncommutative geometry, the group-algebra of the modular group $\mathbb{C} PSL_2$ 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 $\mathbb{C} \mathbb{M}$ 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 $\mathbb{C}[X] \rightarrow \mathbb{C}$ from the coordinate ring of the variety $\mathbb{C}[X]$ onto $\mathbb{C}$ and the kernel of this map is the maximal ideal $\mathfrak{m}_P$ of $\mathbb{C}[X]$ consisting of all functions vanishing in P.

Equivalently, we can view the point $P= \wis{spec}~\mathbb{C}[X]/\mathfrak{m}_P$ as the scheme corresponding to the quotient $\mathbb{C}[X]/\mathfrak{m}_P$ . 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 $\wis{spec}~\C[X]/\mathfrak{m}_P^{n+1}$ the n-th forml neighborhood of P, then the first neighborhood, that is with coordinate ring $\mathbb{C}[X]/\mathfrak{m}_P^2$ gives us tangent-information. Alternatively, it gives the best linear approximation of functions near P. The second neighborhood $\mathbb{C}[X]/\mathfrak{m}_P^3$ gives us the best quadratic approximation of function near P, etc. etc.

These successive quotients by powers of the maximal ideal $\mathfrak{m}_P$ form a system of algebra epimorphisms

$\hdots \frac{\mathbb{C}[X]}{\mathfrak{m}_P^{n+1}} \rightarrow \frac{\mathbb{C}[X]}{\mathfrak{m}_P^{n}} \rightarrow \hdots \hdots \rightarrow \frac{\mathbb{C}[X]}{\mathfrak{m}_P^{2}} \rightarrow \frac{\mathbb{C}[X]}{\mathfrak{m}_P} = \mathbb{C}$

and its inverse limit $\underset{\leftarrow}{lim}~\frac{\mathbb{C}[X]}{\mathfrak{m}_P^{n}} = \hat{\mathcal{O}}_{X,P}$ 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 $\hat{\mathcal{O}}_{X,P}$ 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 $\hat{\mathcal{O}}_{X,P}$ is isomorphic to the algebra of formal power series $\mathbb{C}[[ x_1,x_2,\hdots,x_d ]]$ where the x_i 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

$\pi~:~\C PSL_2(\mathbb{Z}) \rightarrow \C \mathbb{M}$

and in analogy with the commutative case, all information the Monster can gain from the modular group is contained in the $\mathfrak{m}$ -adic completion

$\widehat{\C PSL_2(\mathbb{Z})}_{\mathfrak{m}} = \underset{\leftarrow}{lim}~\frac{\mathbb{C} PSL_2(\mathbb{Z})}{\mathfrak{m}^n}$

where $\mathfrak{m}$ is the kernel of the epimorphism $\pi$ sending the two free generators of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$ 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 $\mathfrak{m}$ let alone the completed algebra… But, (surprise) we can compute $\widehat{\C PSL_2(\mathbb{Z})}_{\mathfrak{m}}$ as explicitly as in the commutative case we have $\hat{\mathcal{O}}_{X,P} \simeq \mathbb{C}[[ x_1,x_2,\hdots,x_d ]]$ for a point P on a manifold X.

Here the details : the quotient $\mathfrak{m}/\mathfrak{m}^2$ has a natural structure of $\mathbb{C} \mathbb{M}$ -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,

$\mathbb{C} \mathbb{M} \simeq \mathbb{C} \oplus M_{196883}(\mathbb{C}) \oplus M_{21296876}(\mathbb{C}) \oplus \hdots \hdots \oplus M_{258823477531055064045234375}(\mathbb{C})$

with exactly 194 components (the number of irreducible Monster-representations). For any $\mathbb{C} \mathbb{M}$ -bimodule one can form the tensor-algebra

$T_{\mathbb{C} \mathbb{M}}(M) = \mathbb{C} \mathbb{M} \oplus M \oplus (M \otimes_{\mathbb{C} \mathbb{M}} M) \oplus (M \otimes_{\mathbb{C} \mathbb{M}} M \otimes_{\mathbb{C} \mathbb{M}} M) \oplus \hdots \hdots$

and applying the formal neighborhood theorem for formally smooth algebras (such as $\mathbb{C} PSL_2(\mathbb{Z})$ ) due to Joachim Cuntz (left) and Daniel Quillen (right) we have an isomorphism of algebras

$\widehat{\C PSL_2(\mathbb{Z})}_{\mathfrak{m}} \simeq \widehat{T_{\mathbb{C} \mathbb{M}}(\mathfrak{m}/\mathfrac{m}^2)}$

where the right-hand side is the completion of the tensor-algebra (at the unique graded maximal ideal) of the $\mathbb{C} \mathbb{M}$ -bimodule $\mathfrak{m}/\mathfrak{m}^2$ , so we’d better describe this bimodule explicitly.

Okay, so what’s a bimodule over a semisimple algebra of the form $S=M_{n_1}(\mathbb{C}) \oplus \hdots \oplus M_{n_k}(\mathbb{C})$ ? Well, a simple S-bimodule must be either (1) a factor $M_{n_i}(\mathbb{C})$ with all other factors acting trivially or (2) the full space of rectangular matrices $M_{n_i \times n_j}(\mathbb{C})$ with the factor $M_{n_i}(\mathbb{C})$ acting on the left, $M_{n_j}(\mathbb{C})$ 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 $\mathfrak{m}/\mathfrak{m}^2$ 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 S_i and S_j is equal to

$dim_{\mathbb{C}}~Ext^1_{\mathbb{C} PSL_2(\mathbb{Z})}(S_i,S_j)-\delta_{ij}$

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 S_i we have to determine the corresponding dimension-vector ~(a_1,a_2;b_1,b_2,b_3) for the quiver

$\xymatrix{ & & & & \vtx{b_1} \\ \vtx{a_1} \ar[rrrru]^(.3){B_{11}} \ar[rrrrd]^(.3){B_{21}} \ar[rrrrddd]_(.2){B_{31}} & & & & \\ & & & & \vtx{b_2} \\ \vtx{a_2} \ar[rrrruuu]_(.7){B_{12}} \ar[rrrru]_(.7){B_{22}} \ar[rrrrd]_(.7){B_{23}} & & & & \\ & & & & \vtx{b_3}}$

Now the dimensions a_i are the dimensions of the +/-1 eigenspaces for the order 2 element g in the representation and the b_i 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 ~(a_1,a_2;b_1,b_2,b_3) .

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 a_1+a_2=196883, a_1-275=a_2 and b_1+b_2+b_3=196883, b_1-53=b_2=b_3 giving us for that representation the dimension vector of the quiver above ~(98579,98304,65663,65610,65610) .

Okay, so for each of the 194 irreducibles S_i we have determined a dimension vector ~(a_1(i),a_2(i);b_1(i),b_2(i),b_3(i)) , then standard quiver-representation theory asserts that the number of loops in the vertex corresponding to S_i is equal to

dim(S_i)^2 + 1 – a_1(i)^2-a_2(i)^2-b_1(i)^2-b_2(i)^2-b_3(i)^2

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

dim(S_i)dim(S_j) – a_1(i)a_1(j)-a_2(i)a_2(j)-b_1(i)b_1(j)-b_2(i)b_2(j)-b_3(i)b_3(j)

This data then determines completely the $\mathbb{C} \mathbb{M}$ -bimodule $\mathfrak{m}/\mathfrak{m}^2$ and hence the structure of the completion $\widehat{\mathbb{C} PSL_2}_{\mathfrak{m}}$ 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 $2.\mathbb{B}$ of the Baby-Monster. The corresponding permutation representation is of dimension 97239461142009186000 and decomposes into Monster-irreducibles

$S_1 \oplus S_2 \oplus S_4 \oplus S_5 \oplus S_9 \oplus S_{14} \oplus S_{21} \oplus S_{34} \oplus S_{35}$

(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…

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arxiv, Brauer, Cuntz, geometry, groups, hyperbolic, Klein, Kontsevich, modular, monster, noncommutative, permutation representation, Quillen, quivers, representations, Riemann

“God given time”

Feb 20th

Posted by lievenlb in geometry

2 comments

Noncommutative geometry and the Riemann zeta function

If you ever sat through a lecture by Alain Connes you will know about his insistence on the ‘canonical dynamic nature of noncommutative manifolds’. If you haven’t, he did write a blog post Heart bit 1 about it.

I’ll try to explain here that there is a definite “supplément d’âme” obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that “noncommutative spaces generate their own time” and moreover can undergo thermodynamical operations such as cooling, distillation etc…

Here a section from his paper A view of mathematics :

Indeed even at the coarsest level of understanding of a space provided by measure theory, which in essence only cares about the “quantity of points” in a space, one ﬁnds unexpected completely new features in the noncommutative case. While it had been long known by operator algebraists that the theory of von-Neumann algebras represents a far reaching extension of measure theory, the main surprise which occurred at the beginning of the seventies is that such an algebra M inherits from its noncommutativity a god-given time evolution: $\delta~:~\mathbb{R} \rightarrow Out(M)$ where is the quotient of the group of automorphisms of M by the normal subgroup of inner automorphisms. This led in my thesis to the reduction from type III to type II and their automorphisms and eventually to the classiﬁcation of injective factors.

Even a commutative manifold has a kind of dynamics associated to it. Take a suitable vectorfield, consider the flow determined by it and there’s your ‘dynamics’, or a one-parameter group of automorphisms on the functions. Further, other classes of noncommutative algebras have similar features. For example, Cuntz and Quillen showed that also formally smooth algebras (the noncommutative manifolds in the algebraic world) have natural Yang-Mills flows associated to them, giving a one-parameter subgroup of automorphisms.

Let us try to keep far from mysticism and let us agree that by ‘time’ (let alone ‘god given time’) we mean a one-parameter subgroup of algebra automorphisms of the noncommutative algebra. In nice cases, such as some von-Neumann algebras this canonical subgroup is canonical in the sense that it is unique upto inner automorphisms.

In the special case of the Bost-Connes algebra these automorphisms $\sigma_t$ are given by $\sigma_t(X_n) = n^{it} X_n$ and $\sigma_t(Y_{\lambda}) = Y_{\lambda}$ .

This one-parameter subgroup is crucial in the definition of the so called KMS-states (for Kubo-Martin and Schwinger) which is our next goal.

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Connes, Cuntz, geometry, noncommutative, Quillen

Segal’s formal neighbourhood result

Dec 8th

Posted by lievenlb in geometry

1 comment

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

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

Let be a $\mathbb{C}$ -algebra and let $M = S_1 \oplus \hdots \oplus S_k$ be a finite dimensional semi-simple representation with distinct simple components. Let $\mathfrak{m}$ be the kernel of the algebra epimorphism $A \rightarrow S$ to the semi-simple algebra S=End(M) . Then, the $\mathfrak{m}$ -adic completion of is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the $A_{\infty}$ -structure on the Ext-algebra $Ext^{\bullet}_A(M,M)$ . More precisely, there is an isomorphism

$\hat{A}_{\mathfrak{m}} \simeq \frac{\hat{T}_S(Ext^1_A(M,M)^{\ast})}{(Im(HMC)^{\ast})}$

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

$HMC = \oplus_i m_i~:~T_S(Ext_A^1(M,M)) \rightarrow Ext^2_A(M,M)$

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

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

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

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arxiv, Calabi-Yau, Cuntz, necklace, noncommutative, Quillen, simples, superpotential

time for selfcriticism

Feb 22nd

Posted by lievenlb in rants

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The problem with criticizing others is that you have to apply the same standards to your own work. So, as of this afternoon, I do agree with all those who said so before : my book is completely unreadable and should either be dumped or entirely rewritten!

Here’s what happened : Last week I did receive the contract to publish noncommutative geometry@n in a reputable series. One tiny point though, the editors felt that the title was somewhat awkward and would stand out with respect to the other books in the series, so they proposed as an alternative title Noncommutative Geometry. A tall order, I thought, but then, if others are publishing books with such a title why shouldn’t I do the same?

The later chapters are quite general, anyway, and if I would just spice them up a little adding recent material it might even improve the book. So, rewriting two chapters and perhaps adding another “motivational chapter” aimed at physicists… should be doable in a month, or two at the latest which would fit in nicely with the date the final manuscript is due.

This week, I got myself once again in writing mode : painfully drafting new sections at a pace of 5 to 6 pages a day. Everything was going well. Today I wanted to finish the section on the “one quiver to rule them all”-trick and was already mentally planning the next section in which I would give details for groups like $PSL_2(\mathbb{Z})$ and $GL_2(\mathbb{Z})$ , all I needed was to type in a version of the proof of the last proposition.

The proof uses a standard argument, which clearly should be in the book so I had to give the correct reference and started browsing through the print-out of the latest version (about 600 pages long..) but… I could not find it!??? And, it was not just some minor technical lemma, but a result which is crucial to the book’s message (for the few who want to know, the result is the construction and properties of the local quiver at a semi-simple representation of a Quillen-smooth algebra). Of course, there is a much more general result contained in the book, but you have to be me (or have to be drilled by me) to see the connection… Not good at all! I’d better sleep on this before taking further steps

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geometry, groups, noncommutative, Quillen

coalgebras and non-geometry 2

Sep 7th

Posted by lievenlb in geometry

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Last time we have seen that the coalgebra of distributions of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form $\mathbb{C}[[ x_1,\hdots,x_d ]]$ where is the dimension of the variety. Generalizing this to non-commutative manifolds, the first questions are : “What is the analogon of the power-series algebra?” and do all ‘points’ of our non-commutative manifold do have such local algebras? Surely, we no longer expect the variables to commute, so a non-commutative version of the power series algebra should be $\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle$ , the ring of formal power series in non-commuting variables. However, there is still another way to add non-commutativity and that is to go from an algebra to matrices over the algebra. So, in all we would expect to be our local algebras at points of our non-commutative manifold to be isomorphic to $M_n(\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle)$ As to the second question : _qurves (that is, the coordinate rings of non-commutative manifolds) do have such algebras as local rings provided we take as the ‘points’ of the non-commutative variety the set of all simple finite dimensional representations of the qurve. This is a consequence of the tubular neighborhood theorem due to Cuntz and Quillen. In more details : If A is a qurve, then a simple -dimensional representation corresponds to an epimorphism $\pi~:~A \rightarrow S = M_n(\mathbb{C})$ and if we take $\mathfrak{m}=Ker(\pi)$ , then $M=\mathfrak{m}/\mathfrak{m}^2$ is an -bimodule and the $\mathfrak{m}$ -adic completion of A is isomorphic to the completed tensor-algebra $\hat{T}_S(M) \simeq M_n(\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle)$ In contrast with the commutative case however where the dimension remains constant over all points, here the numbers n and d can change from simple to simple. For n this is clear as it gives the dimension of the simple representation, but also d changes (it is the local dimension of the variety classifying simple representations of the same dimension). Here an easy example : Consider the skew group algebra $A = \mathbb{C}[x] \star C_2$ with the action given by sending $x \mapsto -x$ . Then A is a qurve and its center is $\mathbb{C}[y]$ with y=x^2 . Over any point $y \not= 0$ there is a unique simple 2-dimensional representation of A giving the local algebra $M_2(\mathbb{C}[[y]])$ . If y=0 the situation is more complicated as the local structure of A is given by the algebra $\begin{bmatrix} \mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix}$ So, over this point there are precisely 2 one-dimensional simple representations corresponding to the maximal ideals $\mathfrak{m}_1 = \begin{bmatrix} (y) & \mathbb{C}[[y]] \\ (y) & \mathbb{C}[[y]] \end{bmatrix}~\qquad \text{and}~\qquad \mathfrak{m}_2 = \begin{bmatrix} \mathbb{C}[[y]] & \mathbb{C}[[y]] \\ (y) & (y) \end{bmatrix}$ and both ideals are idempotent, that is $\mathfrak{m}_i^2 = \mathfrak{m}_i$ whence the corresponding bimodule M_i =
0 so the local algebra in either of these two points is just $\mathbb{C}$ . Ok, so the comleted local algebra at each point is of the form $M_n(\mathbb{C}\langle \langle x_1,\hdots,x_d \rangle \rangle)$ , but what is the corresponding dual coalgebra. Well, $\mathbb{C} \langle \langle x_1,\hdots,x_d \rangle \rangle$ is the algebra dual to the _cofree coalgebra on $V = \mathbb{C} x_1 + \hdots + \mathbb{C}x_d$ . As a vectorspace this is the tensor-algebra $T(V) = \mathbb{C} \langle x_1,\hdots,x_d \rangle$ with the coalgebra structure induced by the bialgebra structure defined by taking all varaibales to be primitives, that is $\Delta(x_i) = x_i \otimes 1 + 1 \otimes x_i$ . That is, the coproduct on a monomial gives all different expressions $m_1 \otimes m_2$ such that m_1m_2 = m . For example, $\Delta(x_1x_2) = x_1x_2 \otimes 1 + x_1 \otimes x_2 + 1 \otimes x_1x_2$ . On the other hand, the dual coalgebra of $M_n(\mathbb{C})$ is the matrix coalgebra which is the n^2 -dimensional vectorspace $\mathbb{C}e_{11} + \hdots + \mathbb{C}e_{nn}$ with comultiplication $\Delta(e_{ij}) = \sum_k e_{ik} \otimes e_{kj}$ The coalgebra corresponding to the local algebra $M_n(\mathbb{C}\langle \langle x_1,\hdots,x_d \rangle \rangle)$ is then the tensor-coalgebra of the matrix coalgebra and the cofree coalgebra. Having obtained the coalgebra at each point (=simple representation) of our noncommutative manifold one might think that the _coalgebra of non-commutative distributions should be the direct sum of all this coalgebras, summed over all points, as in the commutative case. But then we would forget about a major difference between the commutative and the non-commutative world : distinct simples can have non-trivial extensions! The mental picture one might have about simples having non-trivial extensions is that these points lie ‘infinitesimally close’ together. In the $\mathbb{C}[x] \star C_2$ example above, the two one-dimensional simples have non-trivial extensions so they should be thought of as a cluster of two infinitesimally close points corresponding to the point y=0 (that is, this commutative points splits into two non-commutative points). Btw. this is the reason why non-commutative algebras can be used to resolve commutative singularities (excessive tangents can be split over several non-commutative points). While this is still pretty harmless when the algebra is finite over its center (as in the above example where only the two one-dimensionals have extensions), the situation becomes weird over general qurves as ‘usually’ distinct simples have non-trivial extensions. For example, for the free algebra $\mathbb{C}\langle x,y \rangle$ this is true for all simples… So, if we want to continue using this image of points lying closely together this immediately means that non-commutative ‘affine’ manifolds behave like compact ones (in fact, it turns out to be pretty difficult to ‘glue’ together qurves into ‘bigger’ non-commutative manifolds, apart from the quiver examples of this old paper). So, how to bring this new information into our coalgebra of distributions? Well, let’s repeat the previous argument not with just one point but with a set of finitely many points. Then we have a _semi-simple algebra quotient $\pi~:~A \rightarrow S = M_{n_1}(\mathbb{C}) \oplus \hdots \oplus M_{n_k}(\mathb{C})$ and taking again $\mathfrak{m}=Ker(\pi)$ and $M=\mathfrak{m}/\mathfrak{m}^2$ , then is again an S-bimodule. Now, any S-bimodule can be encoded into a quiver Q on k points, the number of arrows from vertex i to vertex j being the number of components in M of the form $M_{n_i \times n_j}(\mathbb{C})$ . Again, it follows from the tubular neighborhood theorem that the $\mathfrak{m}$ -adic completion of A is isomorphic to the completion of an algebra Morita equivalent to the _path algebra $\mathbb{C} Q$ (being the tensor algebra T_S(M) ). As all the local algebras of the points are quotients of this quiver-like completion, on the coalgebra level our local coalgebras will be sub coalgebras of the coalgebra which is co-Morita equivalent (and believe it or not but coalgebraists have a name for this : _Takeuchi equivalence) to the quiver coalgebra which is the vectorspace of the path algebra $\mathbb{C} Q$ with multiplication induced by making all arrows from i to j skew-primitives, that is, $\Delta(a) = e_i \otimes a + a \otimes e_j$ where the e_i are group-likes corresponding to the vertices. If all of ths is a bit too much co to take in at once, I suggest the paper by Bill Chin A brief introduction to coalgebra representation theory. The _coalgebra of noncommutative distributions we are after at is now the union of all these Takeuchi-equivalent quiver coalgebras. In easy examples such as the $\mathbb{C}[x] \star C_2$ -example this coalgebra is still pretty small (the sum of the local coalgebras corresponding to the local algebras $M_2(\mathbb{C}[[x]])$ summed over all points $y \not= 0$ summed with the quiver coalgebra of the quiver $\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@/^/[ll]}$ In general though this is a huge object and we would like to have a recipe to construct it from a manageable blue-print and that is what we will do next time.