The

previous part of this sequence was [quiver representations][1]. When $A$

is a formally smooth algebra, we have an infinite family of smooth

affine varieties $\mathbf{rep}_n~A$, the varieties of finite dimensional

representations. On $\mathbf{rep}_n~A$ there is a basechange action of

$GL_n$ and we are really interested in _isomorphism classes_ of

representations, that is, orbits under this action. Mind you, an orbit

space does not always exist due to the erxistence of non-closed orbits

so one often has to restrict to suitable representations of $A$ for

which it _is_ possible to construct an orbit-space. But first, let us

give a motivating example to illustrate the fact that many interesting

classification problems can be translated into the setting of this

non-commutative algebraic geometry. Let $X$ be a smooth projective

curve of genus $g$ (that is, a Riemann surface with $g$ holes). A

classical object of study is $M = M_X^{ss}(0,n)$ the _moduli space

of semi-stable vectorbundles on $X$ of rank $n$ and degree $0$_. This

space has an open subset (corresponding to the _stable_ vectorbundles)

which classify the isomorphism classes of unitary simple representations

$\pi_1(X) = \frac{\langle x_1,\ldots,x_g,y_1,\ldots,y_g

\rangle}{([x_1,y_1] \ldots [x_g,y_g])} \rightarrow U_n(\mathbb{C})$ of the

fundamental group of $X$. Let $Y$ be an affine open subset of the

projective curve $X$, then we have the formally smooth algebra $A =

\begin{bmatrix} \mathbb{C} & 0 \\ \mathbb{C}[Y] & \mathbb{C}[Y] \end{bmatrix}$ As $A$ has two

orthogonal idempotents, its representation varieties decompose into

connected components according to dimension vectors $\mathbf{rep}_m~A

= \bigsqcup_{p+q=m} \mathbf{rep}_{(p,q)}~A$ all of which are smooth

varieties. As mentioned before it is not possible to construct a

variety classifying the orbits in one of these components, but there are

two methods to approximate the orbit space. The first one is the

_algebraic quotient variety_ of which the coordinate ring is the ring of

invariant functions. In this case one merely recovers for this quotient

$\mathbf{rep}_{(p,q)}~A // GL_{p+q} = S^q(Y)$ the symmetric product

of $Y$. A better approximation is the _moduli space of semi-stable

representations_ which is an algebraic quotient of the open subset of

all representations having no subrepresentation of dimension vector

$(u,v)$ such that $-uq+vp < 0$ (that is, cover this open set by
$GL_{p+q}$ stable affine opens and construct for each the algebraic
quotient and glue them together). Denote this moduli space by
$M_{(p,q)}(A,\theta)$. It is an unpublished result of Aidan Schofield
that the moduli spaces of semi-stable vectorbundles are birational
equivalent to specific ones of these moduli spaces
$M_X^{ss}(0,n)~\sim^{bir}~M_{(n,gn)}(A,\theta)$ Rather than studying
the moduli spaces of semi-stable vectorbundles $M^{ss}_X(0,n)$ on the
curve $X$ one at a time for each rank $n$, non-commutative algebraic
geometry allows us (via the translation to the formally smooth algebra
$A$) to obtain common features on all these moduli spaces and hence to
study $\bigsqcup_n~M^{ss}_X(0,n)$ the moduli space of all
semi-stable bundles on $X$ of degree zero (but of varying ranks).
There exists a procedure to associate to any formally smooth algebra $A$
a quiver $Q_A$ (playing roughly the role of the tangent space to the
manifold determined by $A$). If we do this for the algebra described
above we find the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{}
\ar@(ur,dr)}$ and hence the representation theory of this quiver plays
an important role in studying the geometric properties of the moduli
spaces $M^{ss}_X(0,n)$, for instance it allows to determine the smooth
loci of these varieties. Move on the the [next part.
[1]: http://www.neverendingbooks.org/index.php/quiver-representations.html

# Tag: representations

The previous post can be found [here][1].

Pierre Gabriel invented a lot of new notation (see his book [Representations of finite dimensional algebras][2] for a rather extreme case) and is responsible for calling a directed graph a quiver. For example,

$\xymatrix{\vtx{} \ar@/^/[rr] & & \vtx{} \ar@(u,ur) \ar@(d,dr) \ar@/^/[ll]} $

is a quiver. Note than it is allowed to have multiple arrows between vertices, as well as loops in vertices. For us it will be important that a quiver $Q $ depicts how to compute in a certain non-commutative algebra : the path algebra $\mathbb{C} Q $. If the quiver has $k $ vertices and $l $ arrows (including loops) then the path algebra $\mathbb{C} Q $ is a subalgebra of the full $k \times k $ matrix-algebra over the free algebra in $l $ non-commuting variables

$\mathbb{C} Q \subset M_k(\mathbb{C} \langle x_1,\ldots,x_l \rangle) $

Under this map, a vertex $v_i $ is mapped to the basis $i $-th diagonal matrix-idempotent and an arrow

$\xymatrix{\vtx{v_i} \ar[rr]^{x_a} & & \vtx{v_j}} $

is mapped to the matrix having all its entries zero except the $(j,i) $-entry which is equal to $x_a $. That is, in our main example

$\xymatrix{\vtx{e} \ar@/^/[rr]^a & & \vtx{f} \ar@(u,ur)^x \ar@(d,dr)_y \ar@/^/[ll]^b} $

the corresponding path algebra is the subalgebra of $M_2(\mathbb{C} \langle a,b,x,y \rangle) $ generated by the matrices

$e \mapsto \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} $ $ f \mapsto \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} $

$a \mapsto \begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} $ $b \mapsto \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix} $

$x \mapsto \begin{bmatrix} 0 & 0 \\ 0 & x \end{bmatrix} $ $y \mapsto \begin{bmatrix} 0 & 0 \\ 0 & y \end{bmatrix} $

The name \’path algebra\’ comes from the fact that the subspace of $\mathbb{C} Q $ at the $(j,i) $-place is the vectorspace spanned by all paths in the quiver starting at vertex $v_i $ and ending in vertex $v_j $. For an easier and concrete example of a path algebra. consider the quiver

$\xymatrix{\vtx{e} \ar[rr]^a & & \vtx{f} \ar@(ur,dr)^x} $

and verify that in this case, the path algebra is just

$\mathbb{C} Q = \begin{bmatrix} \mathbb{C} & 0 \\ \mathbb{C}[x]a & \mathbb{C}[x] \end{bmatrix} $

Observe that we write and read paths in the quiver from right to left. The reason for this strange convention is that later we will be interested in left-modules rather than right-modules. Right-minder people can go for the more natural left to right convention for writing paths.

Why are path algebras of quivers of interest in non-commutative geometry? Well, to begin they are examples of _formally smooth algebras_ (some say _quasi-free algebras_, I just call them _qurves_). These algebras were introduced and studied by Joachim Cuntz and Daniel Quillen and they are precisely the algebras allowing a good theory of non-commutative differential forms.

So you should think of formally smooth algebras as being non-commutative manifolds and under this analogy path algebras of quivers correspond to _affine spaces_. That is, one expects path algebras of quivers to turn up in two instances : (1) given a non-commutative manifold (aka formally smooth algebra) it must be \’embedded\’ in some non-commutative affine space (aka path algebra of a quiver) and (2) given a non-commutative manifold, the \’tangent spaces\’ should be determined by path algebras of quivers.

The first fact is easy enough to prove, every affine $\mathbb{C} $-algebra is an epimorphic image of a free algebra in say $l $ generators, which is just the path algebra of the _bouquet quiver_ having $l $ loops

$\xymatrix{\vtx{} \ar@(dl,l)^{x_1} \ar@(l,ul)^{x_2} \ar@(ur,r)^{x_i} \ar@(r,dr)^{x_l}} $

The second statement requires more work. For a first attempt to clarify this you can consult my preprint [Qurves and quivers][3] but I\’ll come back to this in another post. For now, just take my word for it : if formally smooth algebras are the non-commutative analogon of manifolds then path algebras of quivers are the non-commutative version of affine spaces!

[1]: http://www.neverendingbooks.org/index.php?p=71

[2]: http://www.booxtra.de/verteiler.asp?site=artikel.asp&wea=1070000&sh=homehome&artikelnummer=000000689724

[3]: http://www.arxiv.org/abs/math.RA/0406618

Before the vacation I finished a rewrite of the One quiver to rule them

all note. The main point of that note was to associate to any qurve

$A$ (formerly known as a quasi-free algebra in the terminology of

Cuntz-Quillen or a formally smooth algebra in the terminology of

Kontsevich-Rosenberg) a quiver $Q(A)$ and a dimension vector $\alpha_A$

such that $A$ is etale isomorphic (in a yet to be defined

non-commutative etale toplogy) to a ring Morita equivalent to the path

algebra $lQ(A)$ where the Morita setting is determined by the dimension

vector $\alpha_A$. These “one-quiver settings” are easy to

work out for a group algebra $lG$ if $G$ is the amalgamated free product

of finite groups $G = H_1 \bigstar_H H_2$.

Here is how to do

this : construct a bipartite quiver with the left vertices corresponding

to the irreducible representations of $H_1$, say ${ S_1, .. ,S_k }$ of

dimensions $(d_1, .. ,d_k)$ and the right vertices corresponding to the

irreducible representations of $H_2$, ${ T_1, .. ,T_l }$ of dimensions

$(e_1, .. ,e_l)$. The number of arrows from the $i$-th left vertex to

the $j$-th right vertex is given by the dimension of $Hom_H(S_i,T_j)$

This is the quiver I call the Zariski quiver for $G$ as the finite

dimensional $G$-representations correspond to $\theta$-semistable

representations of this quiver for the stability structure $\theta=(d_1,

.. ,d_k ; -e_1, .. ,-e_l)$. The one-quiver $Q(G)$ has vertices

corresponding to the minimal $\theta$-stable dimension vectors (say

$\alpha,\beta, .. $of the Zariski quiver and with the number of arrows

between two such vertices determined by $\delta_{\alpha

\beta}-\chi(\alpha,\beta)$ where $\chi$ is the Euler form of the Zariski

quiver. In the old note I've included the example of the projective

modular group $PSL_2(Z) = Z_2 \bigstar Z_3$ (which can easily be

generalized to the modular group $SL_2(Z) = Z_4 \bigstar_{Z_2} Z_6$)

which turns out to be the double of the extended Dynkin quiver

$\tilde{A_5}$. In the rewrite I've also included an example of a

congruence subgroup $\Gamma_0(2) = Z_4 \bigstar_{Z_2}^{HNN}$ which is an

HNN-extension. These are somehow the classical examples of interesting

amalgamated (HNN) groups and one would like to have plenty of other

interesting examples. Yesterday I read a paper by Karen Vogtmann called

Automorphisms of free groups and outer space in which I encountered

an amalgamated product decomposition for $GL_2(Z) = D_8 \bigstar_{Z_2

\times Z_2} (S_3 \times Z_2)$where $D_8$ is the diheder group of 8

elements. When I got back from vacation I found a reference to this

result in my mail-box from Warren Dicks. Theorem 23.1, p. 82, in Heiner

Zieschang, Finite Groups of Mapping Classes of Surfaces, LNM 875,

Springer, Berlin, 1981.

I worked out the one-quiver and it has

the somewhat strange form depicted above. It is perfectly possible that

I made mistakes so if you find another result, please let me know.

**added material (febr 2007)** : mistakes were made and

the correct one quiver can be found elsewhere on this blog.

Today Travis Schedler posted a nice paper on the arXiv

“A Hopf algebra quantizing a necklace Lie algebra

canonically associated to a quiver”. I heard the first time about

necklace Lie algebras from Jacques Alev who had heard a talk by Kirillov

who constructed an infinite dimensional Lie algebra on the monomials in

two non-commuting variables X and Y (upto cyclic permutation of the

word, whence ‘necklace’). Later I learned that this Lie algebra was

defined by Maxim Kontsevich for the free algebra in an even number of

variables in his “Formal (non)commutative symplectic geometry” paper

(published in the Gelfand seminar proceedings 1993). Later I extended

this construction together with Raf Bocklandt in “Necklace Lie algebras and non-commutative symplectic

geometry” (see also Victor Ginzburg’s paper “Non-commutative symplectic geometry, quiver

varieties and operads”. Here, the necklace Lie algebra appears from

(relative) non-commutative differential forms on a symmetric quiver and

its main purpose is to define invariant symplectic flows on quotient

varieties of representations of the quiver.

Travis Schedler

extends this construction in two important ways. First, he shows that

the Lie-algebra is really a Lie-bialgebra hence there is some sort of

group-like object acting on all the representation varieties. Even more

impoprtant, he is able to define a quantization of this structure

defining a Hopf algebra. In this quantization, necklaces play a role

similar to that of (projected) flat links in the plane whereas their

quantization (necklaces with a height) are similar to genuine links in

3-space.

Sadly, at the moment there is no known natural

representations for this Hopf algebra playing a similar role to the

quotient varieties of quiver-varieties in the case of the necklace Lie

bialgebra.

Can

it be that one forgets an entire proof because the result doesn’t seem

important or relevant at the time? It seems the only logical explanation

for what happened last week. Raf Bocklandt asked me whether a

classification was known of all group algebras **l G** which are

noncommutative manifolds (that is, which are formally smooth a la Kontsevich-Rosenberg or, equivalently, quasi-free

a la Cuntz-Quillen). I said I didn’t know the answer and that it looked

like a difficult problem but at the same time it was entirely clear to

me how to attack this problem, even which book I needed to have a look

at to get started. And, indeed, after a visit to the library borrowing

Warren Dicks

lecture notes in mathematics 790 “Groups, trees and projective

modules” and browsing through it for a few minutes I had the rough

outline of the classification. As the proof is basicly a two-liner I

might as well sketch it here.

If **l G** is quasi-free it

must be hereditary so the augmentation ideal must be a projective

module. But Martin Dunwoody proved that this is equivalent to

**G** being a group acting on a (usually infinite) tree with finite

group vertex-stabilizers all of its orders being invertible in the

basefield **l**. Hence, by Bass-Serre theory **G** is the

fundamental group of a graph of finite groups (all orders being units in

**l**) and using this structural result it is then not difficult to

show that the group algebra **l G** does indeed have the lifting

property for morphisms modulo nilpotent ideals and hence is

quasi-free.

If **l** has characteristic zero (hence the

extra order conditions are void) one can invoke a result of Karrass

saying that quasi-freeness of **l G** is equivalent to **G** being

*virtually free* (that is, **G** has a free subgroup of finite

index). There are many interesting examples of virtually free groups.

One source are the discrete subgroups commensurable with **SL(2,Z)**

(among which all groups appearing in monstrous moonshine), another

source comes from the classification of rank two vectorbundles over

projective smooth curves over finite fields (see the later chapters of

Serre’s Trees). So

one can use non-commutative geometry to study the finite dimensional

representations of virtually free groups generalizing the approach with

Jan Adriaenssens in Non-commutative covers and the modular group (btw.

Jan claims that a revision of this paper will be available soon).

In order to avoid that I forget all of this once again, I’ve

written over the last couple of days a short note explaining what I know

of representations of virtually free groups (or more generally of

*fundamental algebras* of finite graphs of separable

**l**-algebras). I may (or may not) post this note on the arXiv in

the coming weeks. But, if you have a reason to be interested in this,

send me an email and I’ll send you a sneak preview.

Tomorrow

I’ll start with the course *Projects in non-commutative geometry*

in our masterclass. The idea of this course (and its companion

*Projects in non-commutative algebra* run by Fred Van Oystaeyen) is

that students should make a small (original if possible) work, that may

eventually lead to a publication.

At this moment the students

have seen the following : definition and examples of quasi-free algebras

(aka formally smooth algebras, non-commutative curves), their

representation varieties, their connected component semigroup and the

Euler-form on it. Last week, Markus Reineke used all this in his mini-course

Rational points of varieties associated to quasi-free

algebras. In it, Markus gave a method to compute (at least in

principle) the number of points of the *non-commutative Hilbert
scheme* and the varieties of

*simple representations*over a

finite field. Here, in principle means that Markus demands a lot of

knowledge in advance : the number of points of all connected components

of all representation schemes of the algebra as well as of its scalar

extensions over finite field extensions, together with the action of the

Galois group on them … Sadly, I do not know too many examples were all

this information is known (apart from path algebras of quivers).

Therefore, it seems like a good idea to run through Markus’

calculations in some specific examples were I think one can get all this

:

*free products of semi-simple algebras*. The motivating examples

being the groupalgebra of the (projective)

*modular group*

**PSL(2,Z) = Z(2) * Z(3)**and the free matrix-products $M(n,F_q) *

M(m,F_q)$. I will explain how one begins to compute things in these

examples and will also explain how to get the One

quiver to rule them all in these cases. It would be interesting to

compare the calculations we will find with those corresponding to the

path algebra of this

*one quiver*.

As Markus set the good

example of writing out his notes and posting them, I will try to do the

same for my previous two sessions on quasi-free algebras over the next

couple of weeks.

Again I

spend the whole morning preparing my talks for tomorrow in the master

class. Here is an outline of what I will cover :

– examples of

noncommutative points and curves. Grothendieck’s characterization of

commutative regular algebras by the lifting property and a proof that

this lifting property in the category **alg** of all l-algebras is

equivalent to being a noncommutative curve (using the construction of a

generic square-zero extension).

– definition of the affine

scheme **rep(n,A)** of all n-dimensional representations (as always,

**l** is still arbitrary) and a proof that these schemes are smooth

using the universal property of **k(rep(n,A))** (via generic

matrices).

– whereas **rep(n,A)** is smooth it is in general

a disjoint union of its irreducible components and one can use the

sum-map to define a semigroup structure on these components when

**l** is algebraically closed. I’ll give some examples of this

semigroup and outline how the construction can be extended over

arbitrary basefields (via a cocommutative coalgebra).

–

definition of the Euler-form on **rep A**, all finite dimensional

representations. Outline of the main steps involved in showing that the

Euler-form defines a bilinear form on the connected component semigroup

when **l** is algebraically closed (using Jordan-Holder sequences and

upper-semicontinuity results).

After tomorrow’s

lectures I hope you are prepared for the mini-course by Markus Reineke on non-commutative Hilbert schemes

next week.

Never thought that I would ever consider Galois descent of *semigroup
coalgebras* but in preparing for my talks for the master-class it

came about naturally. Let

**A**be a formally smooth algebra

(sometimes called a quasi-free algebra, I prefer the terminology

noncommutative curve) over an arbitrary base-field

**k**. What, if

anything, can be said about the connected components of the affine

**k**-schemes

**rep(n,A)**of

**n**-dimensional representations

of

**A**? If

**k**is algebraically closed, then one can put a

commutative semigroup structure on the connected components induced by

the

*sum map*

rep(n,A) x rep(m,A) -> rep(n + m,A) : (M,N) -> M + N

as introduced and studied by Kent

Morrison a long while ago. So what would be a natural substitute for

this if **k** is arbitrary? Well, define **pi(n)** to be the

*maximal* unramified sub **k**-algebra of **k(rep(n,A))**,

the coordinate ring of **rep(n,A)**, then corresponding to the

sum-map above is a map

pi(n + m) -> pi(n) \\otimes pi(m)

and these maps define on the *graded
space*

Pi(A) = pi(0) + pi(1) + pi(2) + ...

the

structure of a graded commutative **k**-coalgebra with

comultiplication

pi(n) -> sum(a + b=n) pi(a) \\otimes pi(b)

The relevance of **Pi(A)** is that if we consider it

over the algebraic closure **K** of **k** we get the *semigroup
coalgebra*

K G with g -> sum(h.h\' = g) h \\otimes h\'

where **G** is Morrison\’s connected component

semigroup. That is, **Pi(A)** is a **k**-form of this semigroup

coalgebra. Perhaps it is a good project for one of the students to work

this out in detail (and correct possible mistakes I made) and give some

concrete examples for formally smooth algebras **A**. If you know of

a reference on this, please let me know.

Yesterday I made reservations for lecture rooms to run the

master class on non-commutative geometry sponsored by the ESF-NOG project. We have a lecture room on

monday- and wednesday afternoon and friday the whole day which should be

enough. I will run two courses in the program : *non-commutative
geometry* and

*projects in non-commutative geometry*both 30

hours. I hope that Raf Bocklandt will do most of the work on the

*Geometric invariant theory*course so that my contribution to it

can be minimal. Here are the first ideas of topics I want to cover in my

courses. As always, all suggestions are wellcome (just add a

*comment*).

**non-commutative geometry** : As

I am running this course jointly with Markus Reineke and as Markus will give a

mini-course on his work on non-commutative Hilbert schemes, I will explain

the theory of *formally smooth algebras*. I will cover most of the

paper by Joachim Cuntz and Daniel Quillen “Algebra extensions and

nonsingularity”, Journal of AMS, v.8, no. 2, 1995, 251?289. Further,

I’ll do the first section of the paper by Alexander Rosenberg and Maxim Kontsevich,

“Noncommutative smooth spaces“. Then, I will

explain some of my own work including the “One

quiver to rule them all” paper and my recent attempts to classify

all formally smooth algebras up to non-commutative birational

equivalence. When dealing with the last topic I will explain some of Aidan Schofield‘s paper

“Birational classification of moduli spaces of representations of quivers“.

**projects in
non-commutative geometry** : This is one of the two courses (the other

being “projects in non-commutative algebra” run by Fred Van Oystaeyen)

for which the students have to write a paper so I will take as the topic

of my talks the application of non-commutative geometry (in particular

the theory of orders in central simple algebras) to the resolution of

commutative singularities and ask the students to carry out the detailed

analysis for one of the following important classes of examples :

quantum groups at roots of unity, deformed preprojective algebras or

symplectic reflexion algebras. I will explain in much more detail three talks I gave on the subject last fall in

Luminy. But I will begin with more background material on central simple

algebras and orders.