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: geometry

OK! I asked to get side-tracked by comments so now that there is one I’d better deal with it at once. So, is there any relation between the non-commutative (algebraic) geometry based on formally smooth algebras and the non-commutative _differential_ geometry advocated by Alain Connes?

Short answers to this question might be (a) None whatsoever! (b) Morally they are the same! and (c) Their objectives are quite different!

As this only adds to the confusion, let me try to explain each point separately after issuing a _disclaimer_ that I am by no means an expert in Connes’ NOG neither in $C^* $-algebras. All I know is based on sitting in some lectures by Alain Connes, trying at several times to make sense of his terribly written book and indeed by reading the Landi notes in utter desperation.

(a) _None whatsoever!_ : Connes’ approach via spectral triples is modelled such that one gets (suitable) ordinary (that is, commutative) manifolds into this framework. The obvious algebraic counterpart for this would be a statement to the effect that the affine coordinate ring $\mathbb{C}[X] $ of a (suitable) smooth affine variety X would be formally smooth. Now you’re in for a first shock : the only affine smooth varieties for which this holds are either _points_ or _curves_! Not much of a geometry huh? In fact, that is the reason why I prefer to call formally smooth algebras just _qurves_ …

(b) _Morally they are the same_ : If you ever want to get some differential geometry done, you’d better have a connection on the tangent bundle! Now, Alain Connes extended the notion of a connection to the non-commutative world (see for example _the_ book) and if you take the algebraic equivalent of it and ask for which algebras possess such a connection, you get _precisely_ the formally smooth algebras (see section 8 of the Cuntz-Quillen paper “Algebra extensions and nonsingularity” Journal AMS Vol 8 (1991). Besides there is a class of $C^* $-algebras which are formally smooth algebras : the AF-algebras which also feature prominently in the Landi notes (although they are virtually never affine, that is, finitely generated as an algebra).

(c) _Their objectives are quite different!_ : Connes’ formalism aims to define a length function on a non-commutative manifold associated to a $C^* $-algebra. Non-commutative geometry based on formally smooth algebras has no interest in defining some sort of space associated to the algebra. The major importance of formally smooth algebras (as advocated by Maxim Kontsevich is that such an algebra A can be seen as a _machine_ producing an infinite family of ordinary commutative manifolds via its _representation varieties_ $\mathbf{rep}_n~A $ which are manifolds equipped with a $GL_n $-action. Non-commutative functions and diifferential forms defined at the level of the formally smooth algebra A do determine similar $GL_n $-invariant features on _all_ of these representation varieties at once.

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

Now that the preparation for my undergraduate courses in the first semester is more or less finished, I can begin to think about the courses I’ll give this year in the master class

non-commutative geometry. For a change I’d like to introduce the main ideas and concepts by a very concrete example : Ginzburg’s coadjoint-orbit result for the Calogero-Moser space and its

relation to the classification of one-sided ideals in the first Weyl algebra. Not only will this example give me the opportunity to say things about formally smooth algebras, non-commutative

differential forms and even non-commutative symplectic geometry, but it also involves what some people prefer to call _non-commutative algebraic geometry_ (that is the study of graded Noetherian

rings having excellent homological properties) via the projective space associated to the homogenized Weyl algebra. Besides, I have some affinity with this example.

A long time ago I introduced

the moduli spaces for one-sided ideals in the Weyl algebra in Moduli spaces for right ideals of the Weyl algebra and when I was printing a _very_ preliminary version of Ginzburg’s paper

Non-commutative Symplectic Geometry, Quiver varieties, and Operads (probably because he send a preview to Yuri Berest and I was in contact with him at the time about the moduli spaces) the

idea hit me at the printer that the right way to look at the propblem was to consider the quiver

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

which eventually led to my paper together with Raf Bocklandt Necklace Lie algebras and noncommutative symplectic geometry.

Apart from this papers I would like to explain the following

papers by illustrating them on the above example : Michail Kapranov Noncommutative geometry based on commutator expansions Maxim Kontsevich and Alex Rosenberg Noncommutative smooth

spaces Yuri Berest and George Wilson Automorphisms and Ideals of the Weyl Algebra Yuri Berest and George Wilson Ideal Classes of the Weyl Algebra and Noncommutative Projective

Geometry Travis Schedler A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver and of course the seminal paper by Joachim Cuntz and Daniel Quillen on

quasi-free algebras and their non-commutative differential forms which, unfortunately, in not available online.

I plan to write a series of posts here on all this material but I will be very

happy to get side-tracked by any comments you might have. So please, if you are interested in any of this and want to have more information or explanation do not hesitate to post a comment (only

your name and email is required to do so, you do not have to register and you can even put some latex-code in your post but such a posting will first have to viewed by me to avoid cluttering of

nonsense GIFs in my directories).

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.

After yesterday’s post I had to explain today what

*point-modules* and *line-modules* are and that one can really

describe them as points in a (commutative) variety. Seemingly, the

present focus on categorical methods scares possibly interested students

away and none of them seems to know that this non-commutative projective

algebraic geometry once dealt with very concrete examples.

Let

us fix the setting : A will be a *quadratic algebra*, that is, A is

a positively graded algebra, part of degree zero the basefield k,

generated by its homogeneous part A_1 of degree one (which we take to be

of k-dimension n 1) and with all defining relations quadratic in these

generators. Take m k-independent linear terms (that is, elements of A_1)

: l1,…,lm and consider the graded *left A-module*

L = A/(Al1 + ... + Alm)

Clearly, the Hilbert series of this

module (that is, the formal power series in t with coefficient of t^a

the k-dimension of the homogeneous part of L of degree a) starts off

with

Hilb(L,t) = 1 + (n+1-m) t + ...

and

we call L a *linear d-dimensional module* if the Hilbert series is

the power series expansion of

1/(1-t)^{d +1} = 1 + (d+1)t +(d +1)(d +2)/2 t^2 ...

In particular, if d=0 (that is, m=n) then L

is said to be a **point-module** and if d=1 (that is, m=n-1) then L

is said to be a **line-module**. To a d-dimensional linear module L

one can associate a d-dimensional linear subspace of ordinary (that is,

commutative) projective n-space **P^n**. To do this, identify

P^n = P(A 1^*)

the projective space of the n 1 dimensional space of

linear functions on the homogeneous part of degree one. Then each of the

linear elements li determines a hyperplane V(li) in **P^n** and the

intersection of the m hyperplanes V(l1),…,V(lm) is the wanted

subspace. In particular, to a point-module corresponds a *point* in

**P^n** and to a line-module a *line* in **P^n**. So, where

is the non-commutativity of A hidden? Well, if P is a point-module

P = P0 + P1 + P2 +...

(with all components P_a one dimensional)

then the *twisted* module

P' = P1 + P2 + P3 + ...

is

again a point-module and the map P–>P’ defines an automorphism on the

*point variety*. In low dimensions, it is often possible to

reconstruct A from the point-variety and automorphism. In higher

dimensions, one has to consider also the higher dimensional linear

modules.

When I explained all this (far clumsier as it was a

long time since I worked with this) I was asked for an elementary text

on all this. ‘Why hasn’t anybody written a book on all this?’ Well,

Paul Smith wrote such a book so have a look at his

homepage. But then, it turned out that the version one can download from

one of his course pages is a more recent and a lot more

categorical version. There is no more sight of a useful book on

non-commutative projective spaces and their linear modules which might

give starting students an interesting way to learn some non-commutative

algebra and the beginnings of algebraic geometry (commutative and

non-commutative). So, hopefully Paul still has the old version around

and will make it available… The only webpage on this I could find in

short time are the slides of a talk by Michaela Vancliff.

One

of the best collections of links to homepages of people working in

non-commutative algebra and/or geometry is maintained by Paul Smith. At regular intervals I use it to check

up on some people, usually in vain as nobody seems to update their

homepage… So, today I wrote a simple spider to check for updates in

this list. The idea is simple : it tries to get the link (and when this

fails it reports that the link seems to be broken), it saves a text-copy

of the page (using *lynx*) on disc which it will check on a future

check-up for changes with *diff*. Btw. for OS X-people I got

*lynx* from the Fink Project. It then collects all data (broken

links, time of last visit and time of last change and recent updates) in

RSS-feeds for which an HTML-version is maintained at the geoMetry-site, again

using server side includes. If you see a 1970-date this means that I

have never detected a change since I let this spider loose (today).

Also, the list of pages is not alphabetic, even to me it is a surprise

how the next list will look. As I check for changes with *diff* the

claimed number of changed lines is by far accurate (the total of lines

from the first change made to the end of the file might be a better

approximation of reality… I will change this soon).

Clearly,

all of this is still experimental so please give me feedback if you

notice something wrong with these lists. Also I plan to extend this list

substantially over the next weeks (for example, Paul Smith himself is

not present in his own list…). So, if you want your pages to be

included, let me know at lieven.lebruyn@ua.ac.be.

For those on Paul\’s list, if you looked at your log-files today

you may have noticed a lot of traffic from *www.matrix.ua.ac.be* as

I was testing the script. I\’ll keep my further visits down to once a

day, at most…