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,\ldots,x_d ]] $ where $d $ 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,\ldots,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,\ldots,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](http://wwwmath.uni-muenster.de/u/cuntz/cuntz.html) and

[Quillen](http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Quillen.html). In more details : If A is a qurve, then a simple

$n $-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 $S $-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,\ldots,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,\ldots,x_d \rangle

\rangle) $, but what is the corresponding dual coalgebra. Well,

$\mathbb{C} \langle \langle x_1,\ldots,x_d \rangle \rangle $ is

the algebra dual to the _cofree coalgebra_ on $V = \mathbb{C} x_1 +

\ldots + \mathbb{C}x_d $. As a vectorspace this is the

tensor-algebra $T(V) = \mathbb{C} \langle x_1,\ldots,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} + \ldots +

\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,\ldots,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](http://www.arxiv.org/abs/math.AG/9907136)). 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 \ldots \oplus

M_{n_k}(\mathb{C}) $ and taking again

$\mathfrak{m}=Ker(\pi) $ and

$M=\mathfrak{m}/\mathfrak{m}^2 $, then $M $ 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](http://condor.depaul.edu/~wchin/crt.pdf#search=%22%22A%20brief%20introduction%20to%20coalgebra%20representation%20theory%22%22). 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.

# Tag: Quillen

In this

series of posts I’ll try to make at least part of the recent

[Kontsevich-Soibelman paper](http://www.arxiv.org/abs/math.RA/0606241) a

bit more accessible to algebraists. In non-geometry, the algebras

corresponding to *smooth affine varieties* I’ll call **qurves** (note

that they are called **quasi-free algebras** by Cuntz & Quillen and

**formally smooth** by Kontsevich). By definition, a qurve in an affine

$\mathbb{C} $-algebra A having the lifting property for algebra

maps through nilpotent ideals (extending Grothendieck’s characterization

of smooth affine algebras in the commutative case). Examples of qurves

are : finite dimensional semi-simple algebras (for example, group

algebras $\mathbb{C} G $ of finite groups), coordinate rings of

smooth affine curves or a noncommutative mixture of both, skew-group

algebras $\mathbb{C}[X] \ast G $ whenever G is a finite group of

automorphisms of the affine curve X. These are Noetherian examples but

in general a qurve is quite far from being Noetherian. More typical

examples of qurves are : free algebras $\mathbb{C} \langle

x_1,\ldots,x_k \rangle $ and path algebras of finite quivers

$~\mathbb{C} Q $. Recall that a finite quiver Q s just a

directed graph and its path algebra is the vectorspace spanned by all

directed paths in Q with multiplication induced by concatenation of

paths. Out of these building blocks one readily constructs more

involved qurves via universal algebra operations such as (amalgamated)

free products, universal localizations etc. In this way, the

groupalgebra of the modular group $SL_2(\mathbb{Z}) $ (as well

as that of a congruence subgroup) is a qurve and one can mix groups with

finite groupactions on curves to get qurves like $ (\mathbb{C}[X]

\ast G) \ast_{\mathbb{C} H} \mathbb{C} M $ whenever H is a common

subgroup of the finite groups G and M. So we have a huge class of

qurve-examples obtained from mixing finite and arithmetic groups with

curves and quivers. Qurves can we used as *machines* generating

interesting $A_{\infty} $-categories. Let us start by recalling

some facts about finite closed subschemes of an affine smooth variety Y

in the commutative case. Let **fdcom** be the category of all finite

dimensional commutative $\mathbb{C} $-algebras with morphisms

being onto algebra morphisms, then the study of finite closed subschemes

of Y is essentially the study of the covariant functor **fdcom** –>

**sets** assigning to a f.d. commutative algebra S the set of all onto

algebra maps from $\mathbb{C}[Y] $ to S. S being a f.d.

commutative semilocal algebra is the direct sum of local factors $S

\simeq S_1 \oplus \ldots \oplus S_k $ where each factor has a

unique maximal ideal (a unique point in Y). Hence, our study reduces to

f.d. commutative images with support in a fixed point p of Y. But all

such quotients are also quotients of the completion of the local ring of

Y at p which (because Y is a smooth variety, say of dimension n) is

isomorphic to formal power series

$~\mathbb{C}[[x_1,\ldots,x_n]] $. So the local question, at any

point p of Y, reduces to finding all settings

$\mathbb{C}[[x_1,\ldots,x_n]] \twoheadrightarrow S

\twoheadrightarrow \mathbb{C} $ Now, we are going to do something

strange (at least to an algebraist), we’re going to take duals and

translate the above sequence into a coalgebra statement. Clearly, the

dual $S^{\ast} $ of any finite dimensional commutative algebra

is a finite dimensional cocommutative coalgebra. In particular

$\mathbb{C}^{\ast} \simeq \mathbb{C} $ where the

comultiplication makes 1 into a grouplike element, that is

$\Delta(1) = 1 \otimes 1 $. As long as the (co)algebra is

finite dimensional this duality works as expected : onto maps correspond

to inclusions, an ideal corresponds to a sub-coalgebra a sub-algebra

corresponds to a co-ideal, so in particular a local commutative algebra

corresponds to an pointed irreducible cocommutative coalgebra (a

coalgebra is said to be irreducible if any two non-zero subcoalgebras

have non-zero intersection, it is called simple if it has no non-zero

proper subcoalgebras and is called pointed if all its simple

subcoalgebras are one-dimensional. But what about infinite dimensional

algebras such as formal power series? Well, here the trick is not to

take all dual functions but only those linear functions whose kernel

contains a cofinite ideal (which brings us back to the good finite

dimensional setting). If one takes only those good linear functionals,

the ‘fancy’-dual $A^o $of an algebra A is indeed a coalgebra. On

the other hand, the full-dual of a coalgebra is always an algebra. So,

between commutative algebras and cocommutative coalgebras we have a

duality by associating to an algebra its fancy-dual and to a coalgebra

its full-dual (all this is explained in full detail in chapter VI of

Moss Sweedler’s book ‘Hopf algebras’). So, we can dualize the above pair

of onto maps to get coalgebra inclusions $\mathbb{C} \subset

S^{\ast} \subset U(\mathfrak{a}) $ where the rightmost coalgebra is

the coalgebra structure on the enveloping algebra of the Abelian Lie

algebra of dimension n (in which all Lie-elements are primitive, that is

$\Delta(x) = x \otimes 1 + 1 \otimes x $ and indeed we have that

$U(\mathfrak{a})^{\ast} \simeq \mathbb{C}[[x_1,\ldots,x_n]] $.

We have translated our local problem to finding all f.d. subcoalgebras

(containing the unique simple) of the enveloping algebra. But what is

the point of this translation? Well, we are not interested in the local

problem, but in the global problem, so we somehow have to **sum over all

points**. Now, on the algebra level that is a problem because the sum of

all local power series rings over all points is no longer an algebra,

whereas the direct sum of all pointed irreducible coalgebras $~B_Y

= \oplus_{p \in Y} U(\mathfrak{a}_p) $ is again a coalgebra! That

is, we have found a huge coalgebra (which we call the coalgebra of

‘distributions’ on Y) such that for every f.d. commutative algebra S we

have $Hom_{comm alg}(\mathbb{C}[Y],S) \simeq Hom_{cocomm

coalg}(S^{\ast},B_Y) $ Can we get Y back from this coalgebra of

districutions? Well, in a way, the points of Y correspond to the

group-like elements, and if g is the group-like corresponding to a point

p, we can recover the tangent-space at p back as the g-primitive

elements of the coalgebra of distributions, that is the elements such

that $\Delta(x) = x \otimes g + g \otimes x $. Observe that in

this commutative case, there are no **skew-primitives**, that is

elements such that $\Delta(x) = x \otimes g + h \otimes x $ for

different group-likes g and h. This is the coalgebra translation of the

fact that a f.d. semilocal commutative algebra is the direct sum of

local components. This is something that will definitely change if we

try to extend the above to the case of qurves (to be continued).

A couple of days ago Ars Mathematica had a post Cuntz on noncommutative topology pointing to a (new, for me) paper by Joachim Cuntz

A couple of years ago, the Notices of the AMS featured an article on noncommutative geometry a la Connes: Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz. The hallmark of this approach is the heavy reliance on K theory. The first few pages of the article are fairly elementary (and full of intriguing pictures), before the K theory takes over.

A few comments are in order. To begin, the paper is **not** really about noncommutative geometry a la Connes, but rather about noncommutative geometry a la Cuntz&Quillen (based on quasi-free algebras) or, equivalently, a la Kontsevich (formally smooth algebras) or if I may be so bold a la moi (qurves).

About the **intruiging pictures** : it seems to be a recent trend in noncommutative geometry research papers to include meaningless pictures to lure the attention of the reader. But, unlike aberrations such as the recent pastiche by Alain Connes and Mathilde Marcolli A Walk in the Noncommutative Garden, Cuntz is honest about their true meaning

I am indebted to my sons, Nicolas and Michael,

for the illustrations to the examples above. Since

these pictures have no technical meaning, they

are only meant to provide a kind of suggestive

visualization of the corresponding quantum spaces.

As one of these pictures made it to the cover of the **Notices** an explanation was included by the cover-editor

About the Cover :

The image on this month’s cover arose from

Joachim Cuntz’s effort to render into visible art

his own internal vision of a noncommutative

torus, an object otherwise quite abstract. His

original idea was then implemented by his son

Michael in a program written in Pascal. More

explicitly, he says that the construction started

out with a triangle in a square, then translated

the triangle by integers times a unit along a line

with irrational slope; plotted the images thus

obtained in a periodic manner; and stopped

just before the figure started to seem cluttered.

Many mathematicians carry around inside

their heads mental images of the abstractions

they work with, and manipulate these objects

somehow in conformity with their mental imagery. They probably also make aesthetic judgements of the value of their work according to

the visual qualities of the images. These presumably common phenomena remain a rarely

explored domain in either art or psychology.â€”Bill Casselman(covers@ams.org)

There can be no technical meaning to the pictures as in the Connes and Cuntz&Quillen approach there is only a noncommutative algebra and _not_ an underlying geometric space, so there is no topology, let alone a noncommutative topology. Of course, I do understand why Cuntz&others name it as such. They view the noncommutative algebra as the ring of functions on some virtual noncommutative space and they compute topological invariants (such as K-groups) of the algebras and interprete them as information about the noncommutative topology of these virtual and unspecified spaces.

Still, it is perfectly possible to associate to a qurve (aka quasi-free algebra or formally smooth algebra) a genuine noncommutative topological space. In this series of posts I’ll explain the little I know of the history of this topic, the thing I posted about it a couple of years ago, why I abandoned the project and the changes I made to it since and the applications I have in mind, both to new problems (such as the birational_classification of qurves) as well as classical problems (such as rationality problems for $PGL_n $ quotient spaces).

Although others have tried to define noncommutative topologies before, I learned about them from Fred Van Oystaeyen. Fred spend the better part of his career constructing structure sheaves associated to noncommutative algebras, mainly to prime Noetherian algebras (the algebras of preference for the majority of non-commutative algebraists). So, suppose you have an ordinary (meaning, the usual commutative definition) topological space X associated to this algebra R, he wants to define an algebra of sections on every open subset $X(\sigma) $ by taking a suitable localization of the algebra $Q_{\sigma}(R) $. This localization is taken with respect to a suitable filter of left ideals $\mathcal{L}(\sigma) $ of R and is defined to be the subalgebra of the classiocal quotient ring $Q(R) $ (which exists because $R$ is prime Noetherian in which case it is a simple Artinian algebra)

$Q_{\sigma}(R) = { q \in Q(R)~|~\exists L \in \mathcal{L}(\sigma)~:~L q \subset R } $

(so these localizations are generalizations of the usual Ore-type rings of fractions). But now we come to an essential point : if we want to glue this rings of sections together on an intersection $X(\sigma) \cap X(\tau) $ we want to do this by ‘localizing further’. However, there are two ways to do this, either considering $~Q_{\sigma}(Q_{\tau}(R)) $ or considering $Q_{\tau}(Q_{\sigma}(R)) $ and these two algebras are only the same if we impose fairly heavy restrictions on the filters (or on the algebra) such as being compatible.

As this gluing property is essential to get a sheaf of noncommutative algebras we seem to get stuck in the general (non compatible) case. Fred’s way out was to make a distinction between the intersection $X_{\sigma} \cap X_{\tau} $ (on which he put the former ring as its ring of sections) and the intersection $X_{\tau} \cap X_{\sigma} $ (on which he puts the latter one). So, the crucial new ingredient in a noncommutative topology is that the order of intersections of opens matter !!!

Of course, this is just the germ of an idea. He then went on to properly define what a noncommutative topology (and even more generally a noncommutative Grothendieck topology) should be by using this localization-example as guidance. I will not state the precise definition here (as I will have to change it slightly later on) but early version of it can be found in the Antwerp Ph.D. thesis by Luc Willaert (1995) and in Fred’s book Algebraic geometry for associative algebras.

Although _qurves_ are decidedly non-Noetherian (apart from trivial cases), one can use Fred’s idea to associate a noncommutative topological space to a qurve as I will explain next time. The quick and impatient may already sneak at my old note a non-commutative topology on rep A but please bear in mind that I changed my mind since on several issues…

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