Tag: topology

noncommutative topology (3)

Published February 3, 2006 by lievenlb

For
finite dimensional hereditary algebras, one can describe its
noncommutative topology (as developed in part 2)
explicitly, using results of Markus
Reineke in The monoid
of families of quiver representations. Consider a concrete example,
say

$A = \begin{bmatrix} \mathbb{C} & V \\ 0 & \mathbb{C}
\end{bmatrix}$ where $V$ is an n-dimensional complex vectorspace, or
equivalently, A is the path algebra of the two point, n arrow quiver
$\xymatrix{\vtx{} \ar@/^/[r] \ar[r] \ar@/_/[r] & \vtx{}} $
Then, A has just 2 simple representations S and T (the vertex reps) of
dimension vectors s=(1,0) and t=(0,1). If w is a word in S and T we can
consider the set $\mathbf{r}_w$ of all A-representations having a
Jordan-Holder series with factors the terms in w (read from left to
right) so $\mathbf{r}_w \subset \mathbf{rep}_{(a,b)}~A$ when there are a
S-terms and b T-terms in w. Clearly all these subsets can be given the
structure of a monoid induced by concatenation of words, that is
$\mathbf{r}_w \star \mathbf{r}_{w’} = \mathbf{r}_{ww’}$ which is
Reineke’s *composition monoid*. In this case it is generated by
$\mathbf{r}_s$ and $\mathbf{r}_t$ and in the composition monoid the
following relations hold among these two generators
$\mathbf{r}_t^{\star n+1} \star \mathbf{r}_s = \mathbf{r}_t^{\star n}
\star \mathbf{r}_s \star \mathbf{r}_t \quad \text{and} \quad
\mathbf{r}_t \star \mathbf{r}_s^{\star n+1} = \mathbf{r}_s \star
\mathbf{r}_t \star \mathbf{r}_s^{\star n}$ With these notations we can
now see that the left basic open set in the noncommutative topology
(associated to a noncommutative word w in S and T) is of the form
$\mathcal{O}^l_w = \bigcup_{w’} \mathbf{r}_{w’}$ where the union is
taken over all words w’ in S and T such that in the composition monoid
the relation holds $\mathbf{r}_{w’} = \mathbf{r}_w \star \mathbf{r}_{u}$
for another word u. Hence, each op these basic opens hits a large number
of $~\mathbf{rep}_{\alpha}$, in fact far too many for our purposes….
So, what do we want? We want to define a noncommutative notion of
birationality and clearly we want that if two algebras A and B are
birational that this is the same as saying that some open subsets of
their resp. $\mathbf{rep}$’s are homeomorphic. But, what do we
understand by *noncommutative birationality*? Clearly, if A and B are
prime Noethrian, this is clear. Both have a ring of fractions and we
demand them to be isomorphic (as in the commutative case). For this
special subclass the above noncommutative topology based on the Zariski
topology on the simples may be fine.

However, most qurves don’t have
a canonical ‘ring of fractions’. Usually they will have infinitely
many simple Artinian algebras which should be thought of as being
_a_ ring of fractions. For example, in the finite dimensional
example A above, if follows from Aidan Schofield‘s work Representations of rings over skew fields that
there is one such for every (a,b) with gcd(a,b)=1 and (a,b) satisfying
$a^2+b^2-n a b < 1$ (an indivisible Shur root for A).

And
what is the _noncommutative birationality result_ we are aiming
for in each of these cases? Well, the inspiration for this comes from
another result by Aidan (although it is not stated as such in the
paper…) Birational
classification of moduli spaces of representations of quivers. In
this paper Aidan proves that if you take one of these indivisible Schur
roots (a,b) above, and if you look at $\alpha_n = n(a,b)$ that then the
moduli space of semi-stable quiver representations for this multiplied
dimension vector is birational to the quotient variety of
$1-(a^2+b^2-nab)$-tuples of $ n \times n $-matrices under simultaneous
conjugation.

So, *morally speaking* this should be stated as the
fact that A is (along the ray determined by (a,b)) noncommutative
birational to the free algebra in $1-(a^2+b^2-nab)$ variables. And we
want a noncommutative topology on $\mathbf{rep}~A$ to encode all these
facts… As mentioned before, this can be done by replacing simples with
bricks (or if you want Schur representations) but that will have to wait
until next week.

a noncommutative topology 2

Published February 1, 2006 by lievenlb

A *qurve*
is an affine algebra such that $~\Omega^1~A$ is a projective
$~A~$-bimodule. Alternatively, it is an affine algebra allowing lifts of
algebra morphisms through nilpotent ideals and as such it is the ‘right’
noncommutative generalization of Grothendieck’s smoothness criterium.
Examples of qurves include : semi-simple algebras, coordinate rings of
affine smooth curves, hereditary orders over curves, group algebras of
virtually free groups, path algebras of quivers etc.

Hence, qurves
behave a lot like curves and as such one might hope to obtain one day a
‘birational’ classification of them, if we only knew what we mean
by this. Whereas the etale classification of them is understood (see for
example One quiver to
rule them all or Qurves and quivers )
we don’t know what the Zariski topology of a qurve might be.

Usually, one assigns to a qurve $~A~$ the Abelian category of all its
finite dimensional representations $\mathbf{rep}~A$ and we would like to
equip this set with a topology of sorts. Because $~A~$ is a qurve, its
scheme of n-dimensional representations $\mathbf{rep}_n~A$ is a smooth
affine variety for each n, so clearly $\mathbf{rep}~A$ being the disjoint
union of these acquires a trivial but nice commutative topology.
However, we would like open sets to hit several of the components
$\mathbf{rep}_n~A$ thereby ‘connecting’ them to form a noncommutative
topological space associated to $~A~$.

In a noncommutative topology on
rep A I proposed a way to do this and though the main idea remains a
good one, I’ll ammend the construction next time. Whereas we don’t know
of a topology on the whole of $\mathbf{rep}~A$, there is an obvious
ordinary topology on the subset $\mathbf{simp}~A$ of all simple finite
dimensional representations, namely the induced topology of the Zariski
topology on $~\mathbf{spec}~A$, the prime spectrum of twosided prime ideals
of $~A~$. As in commutative algebraic geometry the closed subsets of the
prime spectrum consist of all prime ideals containing a given twosided
ideal. A typical open subset of the induced topology on $\mathbf{simp}~A$
hits many of the components $\mathbf{rep}_n~A$, but how can we extend it to
a topology on the whole of the category $\mathbf{rep}~A$ ?

Every
finite dimensional representation has (usually several) Jordan-Holder
filtrations with simple successive quotients, so a natural idea is to
use these filtrations to extend the topology on the simples to all
representations by restricting the top (or bottom) of the Jordan-Holder
sequence. Let W be the set of all words w such as $U_1U_2 \ldots U_k$
where each $U_i$ is an open subset of $\mathbf{simp}~A$. We can now define
the *left basic open set* $\mathcal{O}_w^l$ consisting of all finite
dimensional representations M having a Jordan-Holder sequence such that
the i-th simple factor (counted from the bottom) belongs to $U_i$.
(Similarly, we can define a *right basic open set* by counting from the
top or a *symmetric basic open set* by merely requiring that the simples
appear in order in the sequence). One final technical (but important)
detail is that we should really consider equivalence classes of left
basic opens. If w and w’ are two words we will denote by $\mathbf{rep}(w
\cup w’)$ the set of all finite dimensional representations having a
Jordan-Holder filtration with enough simple factors to have one for each
letter in w and w’. We then define $\mathcal{O}^l_w \equiv
\mathcal{O}^l_{w’}$ iff $\mathcal{O}^l_w \cap \mathbf{rep}(w \cup w’) =
\mathcal{O}^l_{w’} \cap \mathbf{rep}(w \cup w’)$. Equivalence classes of
these left basic opens form a partially ordered set (induced by
set-theoretic inclusion) with a unique minimal element 0 (the empty set
corresponding to the empty word) and a uunique maximal element 1 (the
set $\mathbf{rep}~A$ corresponding to the letter $w=\mathbf{simp}~A$).
Set-theoretic union induces an operation $\vee$ and the operation
$~\wedge$ is induced by concatenation of words, that is,
$\mathcal{O}^l_w \wedge \mathcal{O}^l_{w’} \equiv \mathcal{O}^l_{ww’}$.
This then defines a **left noncommutative topology** on $\mathbf{rep}~A$ in
the sense of Van Oystaeyen (see [part
1](http://www.neverendingbooks.org/index.php/noncommutative-topology-1 $
). To be precise, it satisfies the axioms in the left and middle column
of the following picture and
similarly, the right basic opens give a right noncommutative topology
(satisfying the axioms of the middle and right columns) whereas the
symmetric opens satisfy all axioms giving the basis of a noncommutative
topology. Even for very simple finite dimensional qurves such as
$\begin{bmatrix} \mathbb{C} & \mathbb{C} \\ 0 & \mathbb{C}
\end{bmatrix}$ this defines a properly noncommutative topology on the
Abelian category of all finite dimensional representations which
obviously respect isomorphisms so is really a noncommutative topology on
the orbits. Still, while this may give a satisfactory local definition,
in gluing qurves together one would like to relax simple representations
to *Schurian* representations. This can be done but one has to replace
the topology coming from the Zariski topology on the prime spectrum by
the partial ordering on the *bricks* of the qurve, but that will have to
wait until next time…

One Comment

noncommutative topology (1)

Published January 29, 2006 by lievenlb

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.

—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…