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Tag: quivers

coalgebras and non-geometry 3

Published September 11, 2006 by lievenlb

Last
time we saw that the _coalgebra of distributions_ of a
noncommutative manifold can be described as a coalgebra
Takeuchi-equivalent to the path coalgebra of a huge quiver. This
infinite quiver has as its vertices the isomorphism classes of finite
dimensional simple representations of the qurve A (the coordinate ring
of the noncommutative manifold) and there are as many directed arrows
between the vertices corresponding to the simples S and T as is the
dimension of $Ext^1_A(S,T) $.

The fact that this
coalgebra of distributions is equivalent to the path coalgebra of
_some_ quiver is in the Kontsevich-Soibelman
paper though it would have been nice if they had given reference for
this fact to the paper Wedge Products and
Cotensor Coalgebras in Monoidal Categories by Ardizzoni or to
previous work by P. Jara, D. Llena, L. Merino and D. Stefan,
“Hereditary and formally smooth coalgebras”, Algebr.
Represent. Theory 8 (2005), 363-374. In those papers it is shown that a
coalgebra with coseparable coradical is hereditary if and only if it
is formally smooth if and only if it is a cotensor coalgebra of some
bicomodule.

At first this looks just like the dual version of
the classical result that a finite dimensional hereditary algebra is
Morita equivalent to the path algebra of a quiver (which is indeed what
the proof does) but again the condition that the coradical is
coseparable does not require the coradical to be finite dimensional…
In our case, the coradical is indeed coseparable being the direct sum
over all matrix coalgebras corresponding to the simple representations.
Hence, we can again recover the _points_ of our noncommutative manifold
from the direct summands of the coradical. Fortunately, one can
compute this huge coalgebra of distributions from a small quiver, the
_one quiver to rule them all_, but as I’ve been babbling about all of
this here [numerous
times](http://www.neverendingbooks.org/?s=one+quiver) I’ll let the
interested find out for themselves how you use it (a) to get at the
isoclasses of all simples (hint : morally they are the smooth points of
the quotient varieties of n-dimensional representations and enough tools
have been developed recently to spot some fake simples, that is smooth
proper semi-simple points) and (b) to compute the _ragball_, that is the
huge quiver with vertex set the simples and arows as described
above. Over the years I’ve calculated several one-quivers for a
variety of qurves (such as amalgamated free products of finite groups
and smooth curves). If you are in for a puzzle, try to determine it for
the qurve $~(\mathbb{C}[x] \ast C_2) \ast_{\mathbb{C}
C_2} \mathbb{C} PSL_2(\mathbb{Z}) \ast_{\mathbb{C} C_3}
(\mathbb{C}[x] \ast C_3) $ The answer is a mysterious
hexagon

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coalgebras and non-geometry

Published September 6, 2006 by lievenlb

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).

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non-(commutative) geometry

Published June 21, 2006 by lievenlb

Now
that my non-geometry
post is linked via the comments in this
string-coffee-table post which in turn is available through a
trackback from the Kontsevich-Soibelman
paper it is perhaps useful to add a few links.

The little
I’ve learned from reading about Connes-style non-commutative geometry is
this : if you have a situation where a discrete group is acting with a
bad orbit-space (for example, $GL_2(\mathbb{Z})$ acting on the whole
complex-plane, rather than just the upper half plane) you can associate
to this a $C^*$-algebra and study invariants of it and interprete them
as topological information about this bad orbit space. An intruiging
example is the one I mentioned and where the additional noncommutative
points (coming from the orbits on the real axis) seem to contain a lot
of modular information as clarified by work of Manin&Marcolli and
Zagier. Probably the best introduction into Connes-style
non-commutative geometry from this perspective are the Lecture on
Arithmetic Noncommutative Geometry by Matilde Marcolli. To
algebraists : this trick is very similar to looking at the
skew-group algebra $\mathbb{C}[x_1,\ldots,x_n] * G$ if
you want to study the _orbifold_ for a finite group action on affine
space. But as algebraist we have to stick to affine varieties and
polynomials so we can only deal with the case of a finite group,
analysts can be sloppier in their functions, so they can also do
something when the group is infinite.

By the way, the
skew-group algebra idea is also why non-commutative algebraic
geometry enters string-theory via the link with orbifolds. The
easiest (and best understood) example is that of Kleinian singularities.
The best introduction to this idea is via the Representations
of quivers, preprojective algebras and deformations of quotient
singularities notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka
non-commutative projective geometry originated from the
work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in
the east) to understand Sklyanin algebras associated to elliptic curves
and automorphisms via ‘geometric’ objects such as point- (and
fat-point-) modules, line-modules and the like. An excellent survey
paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby
Stafford and
Michel Van den Bergh. The best introduction is the (also
neverending…) book-project Non-
commutative algebraic geometry by Paul Smith who
maintains a
noncommutative geometry and algebra resource page page (which is
also available from the header).

Non-geometry
started with the seminal paper ‘Algebra extensions and
nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim
Cuntz and Daniel Quillen but which is not available online. An
online introduction is Noncommutative smooth
spaces by Kontsevich and Rosenberg. Surely, different people have
different motivations to study non-geometry. I assume Cuntz got
interested because inductive limits of separable algebras are quasi-free
(aka formally smooth aka qurves). Kontsevich and Soibelman want to study
morphisms and deformations of $A_{\infty}$-categories as they explain in
their recent
paper. My own motivation to be interested in non-geometry is the
hope that in the next decades one will discover new exciting connections
between finite groups, algebraic curves and arithmetic groups (monstrous
moonshine being the first, and still not entirely understood,
instance of this). Part of the problem is that these three topics seem
to be quite different, yet by taking group-algebras of finite or
arithmetic groups and coordinate rings of affine smooth curves they all
turn out to be quasi-free algebras, so perhaps non-geometry is the
unifying theory behind these seemingly unrelated topics.

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non-geometry

Published June 16, 2006 by lievenlb

Here’s
an appeal to the few people working in Cuntz-Quillen-Kontsevich-whoever
noncommutative geometry (the one where smooth affine varieties
correspond to quasi-free or formally smooth algebras) : let’s rename our
topic and call it non-geometry. I didn’t come up with
this term, I heard in from Maxim Kontsevich in a talk he gave a couple
of years ago in Antwerp. There are some good reasons for this name
change.

The term _non-commutative geometry_ is already taken by
much more popular subjects such as _Connes-style noncommutative
differential geometry_ and _Artin-style noncommutative algebraic
geometry_. Renaming our topic we no longer have to include footnotes
(such as the one in the recent Kontsevich-Soibelman
paper) :

We use “formal” non-commutative geometry
in tensor categories, which is different from the non-commutative
geometry in the sense of Alain Connes.

or to make a
distinction between _noncommutative geometry in the small_ (which is
Artin-style) and _noncommutative geometry in the large_ (which in
non-geometry) as in the Ginzburg notes.

Besides, the stress in _non-commutative geometry_ (both in Connes-
and Artin-style) in on _commutative_. Connes-style might also be called
‘K-theory of $C^*$-algebras’ and they use the topological
information of K-theoretic terms in the commutative case as guidance to
speak about geometrical terms in the nocommutative case. Similarly,
Artin-style might be called ‘graded homological algebra’ and they
use Serre’s homological interpretation of commutative geometry to define
similar concepts for noncommutative algebras. Hence, non-commutative
geometry is that sort of non-geometry which is almost
commutative…

But the main point of naming our subject
non-geometry is to remind us not to rely too heavily on our
(commutative) geometric intuition. For example, we would expect a
manifold to have a fixed dimension. One way to define the dimension is
as the trancendence degree of the functionfield. However, from the work
of Paul Cohn (I learned about it through Aidan Schofield) we know that
quasi-free algebras usually do’nt have a specific function ring of
fractions, rather they have infinitely many good candidates for it and
these candidates may look pretty unrelated. So, at best we can define a
_local dimension_ of a noncommutative manifold at a point, say given by
a simple representation. It follows from the Cunz-Quillen tubular
neighborhood result that the local ring in such a point is of the
form

$M_n(\mathbb{C} \langle \langle z_1,\ldots,z_m \rangle
\rangle) $

(this s a noncommutative version of the classical fact
than the local ring in a point of a d-dimensional manifold is formal
power series $\mathbb{C} [[ z_1,\ldots,z_d ]] $) but in non-geometry both
m (the _local_ dimension) and n (the dimension of the simple
representation) vary from point to point. Still, one can attach to the
quasi-free algebra A a finite amount of data (in fact, a _finite_ quiver
and dimension vector) containing enough information to compute the (n,m)
couples for _all_ simple points (follows from the one quiver to rule them
all paper or see this for more
details).

In fact, one can even extend this to points
corresponding to semi-simple representations in which case one has to
replace the matrix-ring above by a ring Morita equivalent to the
completion of the path algebra of a finite quiver, the _local quiver_ at
the point (which can also be computer from the one-quiver of A. The
local coalgebras of distributions at such points of
Kontsevich&Soibelman are just the dual coalgebras of these local
algebras (in math.RA/0606241 they
merely deal with the n=1 case but no doubt the general case will appear
in the second part of their paper).

The case of the semi-simple
point illustrates another major difference between commutative geometry
and non-geometry, whereas commutative simples only have self-extensions
(so the distribution coalgebra is just the direct sum of all the local
distributions) noncommutative simples usually have plenty of
non-isomorphic simples with which they have extensions, so to get at the
global distribution coalgebra of A one cannot simply add the locals but
have to embed them in more involved coalgebras.

The way to do it
is somewhat concealed in the
third version of my neverending book (the version that most people
found incomprehensible). Here is the idea : construct a huge uncountable
quiver by taking as its vertices the isomorphism classes of all simple
A-representations and with as many arrows between the simple vertices S
and T as the dimension of the ext-group between these simples (and
again, these dimensions follow from the knowledge of the one-quiver of
A). Then, the global coalgebra of distributions of A is the limit over
all cotensor coalgebras corresponding to finite subquivers). Maybe I’ll
revamp this old material in connection with the Kontsevich&Soibelman
paper(s) for the mini-course I’m supposed to give in september.

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why nag? (1)

Published March 26, 2006 by lievenlb

Let us
take a hopeless problem, motivate why something like non-commutative
algebraic geometry might help to solve it, and verify whether this
promise is kept.

Suppose we want to know all solutions in invertible
matrices to the braid relation (or Yang-Baxter equation)

X Y X = Y X Y

All such solutions (for varying size of matrices)
form an additive Abelian category \mathbf{rep}~B_3, so a big step forward would be to know all its
simple solutions (that is, those whose matrices cannot be brought in
upper triangular block form). A literature check shows that even this
task is far too ambitious. The best result to date is the classification
due to Imre Tuba and
Hans Wenzl of simple solutions of which the matrix size is at most
5.

For fixed matrix size n, finding solutions in \mathbf{rep}~B_3 is the same as solving a system of n^2 cubic
polynomial relations in 2n^2
unknowns, which quickly becomes a daunting task. Algebraic geometry
tells us that all solutions, say \mathbf{rep}_n~B_3 form an affine closed subvariety of n^2-dimensional affine space. If we assume that \mathbf{rep}_n~B_3 is a smooth variety (that is, a manifold) and
if we know one solution explicitly, then we can use the tangent space in
this point to linearize the problem and to get at all solutions in a
neighborhood.

So, here is an idea : assume that \mathbf{rep}~B_3 itself would be a non-commutative manifold, then
we might linearize our problem by considering tangent spaces and obtain
new solutions out of already known ones. But, what is a non-commutative
manifold? Well, by the above we at least require that for all integers n
the commutative variety \mathbf{rep}_n~B_3 is a commutative manifold.

But, there
is still some redundancy in our problem : if (X,Y) is a
solution, then so is any conjugated pair (g^{-1}Xg,g^{-1}Yg) where g \in GL_n is a basechange matrix. In categorical terms, we are only
interested in isomorphism classes of solutions. Again, if we fix the
size n of matrix-solutions, we consider the affine variety \mathbf{rep}_n~B_3 as a variety with a GL_n-action
and we like to classify the orbits of simple solutions. If \mathbf{rep}_n~B_3 is a manifold then the theory of Luna slices
provides a method, both to linearize the problem as well as to reduce
its complexity. Instead of the tangent space we consider the normal
space N to the GL_n-orbit
(in a suitable solution). On this affine space, the stabilizer subgroup
GL(\alpha) acts and there is a natural one-to-one
correspondence between GL_n-orbits
in \mathbf{rep}_n~B_3 and GL(\alpha)-orbits in the normal space N (at least in a
neighborhood of the solution).

So, here is a refinement of the
idea : we would like to view \mathbf{rep}~B_3 as a non-commutative manifold with a group action
given by the notion of isomorphism. Then, in order to get new isoclasses
of solutions from a constructed one we want to reduce the size of our
problem by considering a linearization (the normal space to the orbit)
and on it an easier isomorphism problem.

However, we immediately
encounter a problem : calculating ranks of Jacobians we discover that
already \mathbf{rep}_2~B_3 is not a smooth variety so there is not a
chance in the world that \mathbf{rep}~B_3 might be a useful non-commutative manifold.
Still, if (X,Y) is a
solution to the braid relation, then the matrix (XYX)^2
commutes with both X and Y.

If (X,Y) is a
simple solution, this means that after performing a basechange, C=(XYX)^2 becomes a scalar matrix, say \lambda^6 1_n. But then, (X_1,Y_1) = (\lambda^{-1}X,\lambda^{-1}Y) is a solution to

XYX = YXY , (XYX)^2 = 1

and all such solutions form a
non-commutative closed subvariety, say \mathbf{rep}~\Gamma of \mathbf{rep}~B_3 and if we know all (isomorphism classes of)
simple solutions in \mathbf{rep}~\Gamma we have solved our problem as we just have to
bring in the additional scalar \lambda \in \mathbb{C}^*.

Here we strike gold : \mathbf{rep}~\Gamma is indeed a non-commutative manifold. This can
be seen by identifying \Gamma
with one of the most famous discrete infinite groups in mathematics :
the modular group PSL_2(\mathbb{Z}). The modular group acts by Mobius
transformations on the upper half plane and this action can be used to
write PSL_2(\mathbb{Z}) as the free group product \mathbb{Z}_2 \ast \mathbb{Z}_3. Finally, using
classical representation theory of finite groups it follows that indeed
all \mathbf{rep}_n~\Gamma are commutative manifolds (possibly having
many connected components)! So, let us try to linearize this problem by
looking at its non-commutative tangent space, if we can figure out what
this might be.

Here is another idea (or rather a dogma) : in the
world of non-commutative manifolds, the role of affine spaces is played
by \mathbf{rep}~Q the representations of finite quivers Q. A quiver
is just on oriented graph and a representation of it assigns to each
vertex a finite dimensional vector space and to each arrow a linear map
between the vertex-vector spaces. The notion of isomorphism in \mathbf{rep}~Q is of course induced by base change actions in all
of these vertex-vector spaces. (to be continued)

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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.

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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…

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sexing up curves

Published June 10, 2005 by lievenlb

Here the
story of an idea to construct new examples of non-commutative compact
manifolds, the computational difficulties one runs into and, when they
are solved, the white noise one gets. But, perhaps, someone else can
spot a gem among all gibberish…
[Qurves](http://www.neverendingbooks.org/toolkit/pdffile.php?pdf=/TheLibrary/papers/qaq.pdf) (aka quasi-free algebras, aka formally smooth
algebras) are the \’affine\’ pieces of non-commutative manifolds. Basic
examples of qurves are : semi-simple algebras (e.g. group algebras of
finite groups), [path algebras of
quivers](http://www.lns.cornell.edu/spr/2001-06/msg0033251.html) and
coordinate rings of affine smooth curves. So, let us start with an
affine smooth curve $X$ and spice it up to get a very non-commutative
qurve. First, we bring in finite groups. Let $G$ be a finite group
acting on $X$, then we can form the skew-group algebra $A = \mathbfk[X]
\bigstar G$. These are examples of prime Noetherian qurves (aka
hereditary orders). A more pompous way to phrase this is that these are
precisely the [one-dimensional smooth Deligne-Mumford
stacks](http://www.math.lsa.umich.edu/~danielch/paper/stacks.pdf).
As the 21-st century will turn out to be the time we discovered the
importance of non-Noetherian algebras, let us make a jump into the
wilderness and consider the amalgamated free algebra product $A =
(\mathbf k[X] \bigstar G) \ast_{\mathbf k G} \mathbfk H$ where $G
\subset H$ is an interesting extension of finite groups. Then, $A$ is
again a qurve on which $H$ acts in a way compatible with the $G$-action
on $X$ and $A$ is hugely non-commutative… A very basic example :
let $\mathbb{Z}/2\mathbb{Z}$ act on the affine line $\mathbfk[x]$ by
sending $x \mapsto -x$ and consider a finite [simple
group](http://mathworld.wolfram.com/SimpleGroup.html) $M$. As every
simple group has an involution, we have an embedding
$\mathbb{Z}/2\mathbb{Z} \subset M$ and can construct the qurve
$A=(\mathbfk[x] \bigstar \mathbb{Z}/2\mathbb{Z}) \ast_{\mathbfk
\mathbb{Z}/2\mathbb{Z}} \mathbfk M$ on which the simple group $M$ acts
compatible with the involution on the affine line. To study the
corresponding non-commutative manifold, that is the Abelian category
$\mathbf{rep}~A$ of all finite dimensional representations of $A$ we have
to compute the [one quiver to rule them
all](http://www.matrix.ua.ac.be/master/coursenotes/onequiver.pdf) for
$A$. Because $A$ is a qurve, all its representation varieties
$\mathbf{rep}_n~A$ are smooth affine varieties, but they may have several
connected components. The direct sum of representations turns the set of
all these components into an Abelian semigroup and the vertices of the
\’one quiver\’ correspond to the generators of this semigroup whereas
the number of arrows between two such generators is given by the
dimension of $Ext^1_A(S_i,S_j)$ where $S_i,S_j$ are simple
$A$-representations lying in the respective components. All this
may seem hard to compute but it can be reduced to the study of another
quiver, the Zariski quiver associated to $A$ which is a bipartite quiver
with on the left the \’one quiver\’ for $\mathbfk[x] \bigstar
\mathbb{Z}/2\mathbb{Z}$ which is just $\xymatrix{\vtx{}
\ar@/^/[rr] & & \vtx{} \ar@/^/[ll]} $ (where the two vertices
correspond to the two simples of $\mathbb{Z}/2\mathbb{Z}$) and on the
right the \’one quiver\’ for $\mathbf k M$ (which just consists of as
many verticers as there are simple representations for $M$) and where
the number of arrows from a left- to a right-vertex is the number of
$\mathbb{Z}/2\mathbb{Z}$-morphisms between the respective simples. To
make matters even more concrete, let us consider the easiest example
when $M = A_5$ the alternating group on $5$ letters. The corresponding
Zariski quiver then turns out to be $\xymatrix{& & \vtx{1} \\\
\vtx{}\ar[urr] \ar@{=>}[rr] \ar@3[drr] \ar[ddrr] \ar[dddrr] \ar@/^/[dd]
& & \vtx{4} \\\ & & \vtx{5} \\\ \vtx{} \ar@{=>}[uurr] \ar@{=>}[urr]
\ar@{=>}[rr] \ar@{=>}[drr] \ar@/^/[uu] & & \vtx{3} \\\ & &
\vtx{3}} $ The Euler-form of this quiver can then be used to
calculate the dimensions of the EXt-spaces giving the number of arrows
in the \’one quiver\’ for $A$. To find the vertices, that is, the
generators of the component semigroup we have to find the minimal
integral solutions to the pair of equations saying that the number of
simple $\mathbb{Z}/2\mathbb{Z}$ components based on the left-vertices is
equal to that one the right-vertices. In this case it is easy to see
that there are as many generators as simple $M$ representations. For
$A_5$ they correspond to the dimension vectors (for the Zariski quiver
having the first two components on the left) $\begin{cases}
(1,2,0,0,0,0,1) \\ (1,2,0,0,0,1,0) \\ (3,2,0,0,1,0,0) \\
(2,2,0,1,0,0,0) \\ (1,0,1,0,0,0,0) \end{cases}$ We now have all
info to determine the \’one quiver\’ for $A$ and one would expect a nice
result. Instead one obtains a complete graph on all vertices with plenty
of arrows. More precisely one obtains as the one quiver for $A_5$
$\xymatrix{& & \vtx{} \ar@{=}[dll] \ar@{=}[dddl] \ar@{=}[dddr]
\ar@{=}[drr] & & \\\ \vtx{} \ar@(ul,dl)|{4} \ar@{=}[rrrr]|{6}
\ar@{=}[ddrrr]|{8} \ar@{=}[ddr]|{4} & & & & \vtx{} \ar@(ur,dr)|{8}
\ar@{=}[ddlll]|{6} \ar@{=}[ddl]|{10} \\\ & & & & & \\\ & \vtx{}
\ar@(dr,dl)|{4} \ar@{=}[rr]|{8} & & \vtx{} \ar@(dr,dl)|{11} & } $
with the number of arrows (in each direction) indicated. Not very
illuminating, I find. Still, as the one quiver is symmetric it follows
that all quotient varieties $\mathbf{iss}_n~A$ have a local Poisson
structure. Clearly, the above method can be generalized easily and all
examples I did compute so far have this \’nearly complete graph\’
feature. One might hope that if one would start with very special
curves and groups, one might obtain something more interesting. Another
time I\’ll tell what I got starting from Klein\’s quartic (on which the
simple group $PSL_2(\mathbb{F}_7)$ acts) when the situation was sexed-up
to the sporadic simple Mathieu group $M_{24}$ (of which
$PSL_2(\mathbb{F}_7)$ is a maximal subgroup).

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TheLibrary

Published April 25, 2005 by lievenlb

Some
people objected to the set-up of TheLibrary because it was serving only one-page at a
time. They’d rather have a longer download-time if they can then
browse through the paper/book, download it and print if they decide to
do so.

Fine! Today I spend some hours refilling TheLibrary with
texts. As before you are able to search any document for specific words
(as explained in elsewhere )
and click on any section or page to view the wanted material in your
browser (assuming you have the proper plugin installed). This time I
used pdfscreen to make the notes more readable on your
screen.

If you prefer, you can download the text, safe it on
your hard-disk and browse at leasure. Whereas these versions are
intended to be read from screen, you can also print them if you have to
at 150dpi. In the next couple of weeks I hope to add some material :
older and undergraduate courses and I’ll add a _papers_ section
where I’ll put all my papers of which I can recover a TeX-file. For
starters, I included the revision of the Qurves and quivers
paper in the courses-folder.

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pdfsync

Published April 13, 2005 by lievenlb

I
expect to be writing a lot in the coming months. To start, after having
given the course once I noticed that I included a lot of new material
during the talks (mainly concerning the component coalgebra and some
extras on non-commutative differential forms and symplectic forms) so
I\’d better update the Granada notes
soon as they will also be the basis of the master course I\’ll start
next week. Besides, I have to revise the Qurves and
Quivers-paper and to start drafting the new bachelor courses for
next academic year (a course on representation theory of finite groups,
another on Riemann surfaces and an upgrade of the geometry-101 course).

So, I\’d better try to optimize my LaTeX-workflow and learn
something about the pdfsync package.
Here is what it is supposed to do :

pdfsync is
an acronym for synchronization between a pdf file and the TeX or so
source file used in the production process. As TeX system is not a
WYSIWYG editor, you cannot modify the output directly, instead, you must
edit a source file then run the production process. The pdfsync helps
you finding what part of the output corresponds to what line of the
source file, and conversely what line of the source file corresponds to
a location of a given page in the ouput. This feature is achieved with
the help of an auxiliary file: foo.pdfsync corresponding to a foo.pdf.

All you have to do is to put the pdfsync.sty file
in the directory _~/Library/texmf/tex/latex/pdfsync.sty_ and to
include the pdfsync-package in the preamble of the LaTeX-document. Under
my default iTex-front-end TeXShop it
works well to go from a spot in the PDF-file to the corresponding place
in the source-code, but in the other direction it only shows the
appropriate page rather than indicate the precise place with a red dot
as it does in the alternative front-end iTeXMac.

A major
drawback for me is that pdfsync doesn\’t live in harmony with my
favorite package for drawing commutative diagrams diagrams.sty. For example, the 75 pages of the current
version of the Granada notes become blown-up to 96 pages because each
commutative diagram explodes to nearly page size! So I will also have to
translate everything to xymatrix&#
8230;

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