This morning,

Michel Van den Bergh posted an interesting paper on the arXiv

entitled Double

Poisson Algebras. His main motivation was the construction of a

natural Poisson structure on quotient varieties of representations of

deformed multiplicative preprojective algebras (introduced by

Crawley-Boevey and Shaw in Multiplicative

preprojective algebras, middle convolution and the Deligne-Simpson

problem) which he achieves by extending his double Poisson structure

on the path algebra of the quiver to the 'obvious' universal

localization, that is the one by inverting all $1+aa^{\star} $ for $a $ an

arrow and $a^{\star} $ its double (the one in the other direction).

For me the more interesting fact of this paper is that his double

bracket on the path algebra of a double quiver gives finer information

than the _necklace Lie algebra_ as defined in my (old) paper with Raf

Bocklandt Necklace

Lie algebras and noncommutative symplectic geometry. I will

certainly come back to this later when I have more energy but just to

wet your appetite let me point out that Michel calls a _double bracket_

on an algebra $A $ a bilinear map

$\{ \{ -,- \} \}~:~A \times A

\rightarrow A \otimes A $

which is a derivation in the _second_

argument (for the outer bimodulke structure on $A $) and satisfies

$\{ \{ a,b \} \} = – \{ \{ b,a \} \}^o $ with $~(u \otimes v)^0 = v

\otimes u $

Given such a double bracket one can define an ordinary

bracket (using standard Hopf-algebra notation)

$\{ a,b \} = \sum

\{ \{ a,b \} \}_{(1)} \{ \{ a,b \} \}_{(2)} $

which makes $A $ into

a Loday

algebra and induces a Lie algebra structure on $A/[A,A] $. He then

goes on to define such a double bracket on the path algebra of a double

quiver in such a way that the associated Lie structure above is the

necklace Lie algebra.

# Tag: geometry

In my geometry 101 course I'm doing the rotation-symmetry groups

of the Platonic solids right now. This goes slightly slower than

expected as it turned out that some secondary schools no longer give a

formal definition of what a group is. So, a lot of time is taken up

explaining permutations and their properties as I want to view the

Platonic groups as subgroups of the permutation groups on the vertices.

To prove that the _tetrahedral group_ is isomorphic to $A_4$ was pretty

straigthforward and I'm half way through proving that the

_octahedral group_ is just $S_4$ (using the duality of the octahedron

with the cube and using the $4$ body diagonals of the cube).

Next

week I have to show that the _icosahedral group_ is isomorphic to $A_5$

which is a lot harder. The usual proof (that is, using the duality

between the icosahedron and the dodecahedron and using the $5$ cubes

contained in the dodecahedron, one for each of the diagonals of a face)

involves too much calculations to do in one hour. An alternative road is

to view the icosahedral group as a subgroup of $S_6$ (using the main

diagonals on the $12$ vertices of the icosahedron) and identifying this

subgroup as $A_5$. A neat exposition of this approach is given by John Baez in his

post Some thoughts on

the number $6$. (He also has another post on the icosahedral group

in his Week 79's

finds in mathematical physics).

But

probably I'll go for an “In Gap we

thrust”-argument. Using the numbers on the left, the rotation by

$72^o$ counter-clockwise in the top face we get the permutation in

$S_{20}$

$(1,2,3,4,5)(6,8,10,12,14)(7,9,11,13,15)(16,17,18,19,20)$

and the

rotation by $72^o$ counterclockwise along the face $(1,2,8,7,8)$ gives

the permutation

$(1,6,7,8,2)(3,5,15,16,9)(4,14,20,17,10)(12,13,19,18,11)$

GAP

calculates that the subgroup $dode$ of $S_{20}$ generated by these two

elements is $60$ (the correct number) and with $IsSimplegroup(dode);$ we

find that this group must be simple. Finally using

$IsomorphismTypeInfoFiniteSimplegroup(dode);$

we get the required

result that the group is indeed isomorphic to $A_5$. The time saved I

can then use to tell something about the classification project of

finite simple groups which might be more inspiring than tedious

calculations…

Are

there hidden relations between mathematical and physical constants such

as

$\frac{e^2}{4 \pi \epsilon_0 h c} \sim \frac{1}{137} $

or are these numerical relations mere accidents? A couple of years

ago, Pierre Cartier proposed in his paper A mad day’s work : from Grothendieck to Connes and

Kontsevich : the evolution of concepts of space and symmetry that

there are many reasons to believe in a cosmic Galois group acting on the

fundamental constants of physical theories and responsible for relations

such as the one above.

The Euler-Zagier numbers are infinite

sums over $n_1 > n_2 > ! > n_r \geq 1 $ of the form

$\zeta(k_1,\dots,k_r) = \sum n_1^{-k_1} \dots n_r^{-k_r} $

and there are polynomial relations with rational coefficients between

these such as the product relation

$\zeta(a)\zeta(b)=\zeta(a+b)+\zeta(a,b)+\zeta(b,a) $

It is

conjectured that all polynomial relations among Euler-Zagier numbers are

consequences of these product relations and similar explicitly known

formulas. A consequence of this conjecture would be that

$\zeta(3),\zeta(5),\dots $ are all trancendental!

Drinfeld

introduced the Grothendieck-Teichmuller group-scheme over $\mathbb{Q} $

whose Lie algebra $\mathfrak{grt}_1 $ is conjectured to be the free Lie

algebra on infinitely many generators which correspond in a natural way

to the numbers $\zeta(3),\zeta(5),\dots $. The Grothendieck-Teichmuller

group itself plays the role of the Galois group for the Euler-Zagier

numbers as it is conjectured to act by automorphisms on the graded

$\mathbb{Q} $-algebra whose degree $d $-term are the linear combinations

of the numbers $\zeta(k_1,\dots,k_r) $ with rational coefficients and

such that $k_1+\dots+k_r=d $.

The Grothendieck-Teichmuller

group also appears mysteriously in non-commutative geometry. For

example, the set of all Kontsevich deformation quantizations has a

symmetry group which Kontsevich conjectures to be isomorphic to the

Grothendieck-Teichmuller group. See section 4 of his paper Operads and motives in

deformation quantzation for more details.

It also appears

in the renormalization results of Alain Connes and Dirk Kreimer. A very

readable introduction to this is given by Alain Connes himself in Symmetries Galoisiennes

et renormalisation. Perhaps the latest news on Cartier’s dream of a

cosmic Galois group is the paper by Alain Connes and Matilde Marcolli posted

last month on the arXiv : Renormalization and

motivic Galois theory. A good web-page on all of this, including

references, can be found here.

The

previous post in this sequence was [moduli spaces][1]. Why did we spend

time explaining the connection of the quiver

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

to moduli spaces of vectorbundles on curves and moduli spaces of linear

control systems? At the start I said we would concentrate on its _double

quiver_ $\tilde{Q}~:~\xymatrix{\vtx{} \ar@/^/[rr]^a && \vtx{}

\ar@(u,ur)^x \ar@(d,dr)_{x^*} \ar@/^/[ll]^{a^*}} $ Clearly,

this already gives away the answer : if the path algebra $C Q$

determines a (non-commutative) manifold $M$, then the path algebra $C

\tilde{Q}$ determines the cotangent bundle of $M$. Recall that for a

commutative manifold $M$, the cotangent bundle is the vectorbundle

having at the point $p \in M$ as fiber the linear dual $(T_p M)^*$ of

the tangent space. So, why do we claim that $C \tilde{Q}$

corresponds to the cotangent bundle of $C Q$? Fix a dimension vector

$\alpha = (m,n)$ then the representation space

$\mathbf{rep}_{\alpha}~Q = M_{n \times m}(C) \oplus M_n(C)$ is just

an affine space so in its point the tangent space is the representation

space itself. To define its linear dual use the non-degeneracy of the

_trace pairings_ $M_{n \times m}(C) \times M_{m \times n}(C)

\rightarrow C~:~(A,B) \mapsto tr(AB)$ $M_n(C) \times M_n(C)

\rightarrow C~:~(C,D) \mapsto tr(CD)$ and therefore the linear dual

$\mathbf{rep}_{\alpha}~Q^* = M_{m \times n}(C) \oplus M_n(C)$ which is

the representation space $\mathbf{rep}_{\alpha}~Q^s$ of the quiver

$Q^s~:~\xymatrix{\vtx{} & & \vtx{} \ar[ll] \ar@(ur,dr)} $

and therefore we have that the cotangent bundle to the representation

space $\mathbf{rep}_{\alpha}~Q$ $T^* \mathbf{rep}_{\alpha}~Q =

\mathbf{rep}_{\alpha}~\tilde{Q}$ Important for us will be that any

cotangent bundle has a natural _symplectic structure_. For a good

introduction to this see the [course notes][2] “Symplectic geometry and

quivers” by [Geert Van de Weyer][3]. As a consequence $C \tilde{Q}$

can be viewed as a non-commutative symplectic manifold with the

symplectic structure determined by the non-commutative 2-form

$\omega = da^* da + dx^* dx$ but before we can define all this we

will have to recall some facts on non-commutative differential forms.

Maybe [next time][4]. For the impatient : have a look at the paper by

Victor Ginzburg [Non-commutative Symplectic Geometry, Quiver varieties,

and Operads][5] or my paper with Raf Bocklandt [Necklace Lie algebras

and noncommutative symplectic geometry][6]. Now that we have a

cotangent bundle of $C Q$ is there also a _tangent bundle_ and does it

again correspond to a new quiver? Well yes, here it is

$\xymatrix{\vtx{} \ar@/^/[rr]^{a+da} \ar@/_/[rr]_{a-da} & & \vtx{}

\ar@(u,ur)^{x+dx} \ar@(d,dr)_{x-dx}} $ and the labeling of the

arrows may help you to work through some sections of the Cuntz-Quillen

paper…

[1]: http://www.neverendingbooks.org/index.php?p=39

[2]: http://www.win.ua.ac.be/~gvdwey/lectures/symplectic_moment.pdf

[3]: http://www.win.ua.ac.be/~gvdwey/

[4]: http://www.neverendingbooks.org/index.php?p=41

[5]: http://www.arxiv.org/abs/math.QA/0005165

[6]: http://www.arxiv.org/abs/math.AG/0010030

In [the previous part][1] we saw that moduli spaces of suitable representations

of the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{}

\ar@(ur,dr)} $ locally determine the moduli spaces of

vectorbundles over smooth projective curves. There is yet another

classical problem related to this quiver (which also illustrates the

idea of looking at families of moduli spaces rather than individual

ones) : _linear control systems_. Such a system with an $n$ dimensional

_state space_ and $m$ _controls_ (or inputs) is determined by the

following system of linear differential equations $ \frac{d x}{d t}

= A.x + B.u$ where $x(t) \in \mathbb{C}^n$ is the state of the system at

time $t$, $u(t) \in \mathbb{C}^m$ is the control-vector at time $t$ and $A \in

M_n(\mathbb{C}), B \in M_{n \times m}(\mathbb{C})$ are the matrices describing the

evolution of the system $\Sigma$ (after fixing bases in the state- and

control-space). That is, $\Sigma$ determines a representation of the

above quiver of dimension-vector $\alpha = (m,n)$

$\xymatrix{\vtx{m} \ar[rr]^B & & \vtx{n} \ar@(ur,dr)^A} $

Whereas in control theory (see for example Allen Tannenbaum\’s Lecture

Notes in Mathematics 845 for a mathematical introduction) it is natural

to call two systems equivalent when they only differ up to base change

in the state-space, one usually fixes the control knobs so it is not

natural to allow for base change in the control-space. So, at first

sight the control theoretic problem of classifying equivalent systems is

not the same problem as classifying representations of the quiver up to

isomorphism. Fortunately, there is an elegant way round this which is

called _deframing_. That is, for a fixed number $m$ of controls one

considers the quiver $Q_f$ having precisely $m$ arrows from the first to

the second vertex $\xymatrix{\vtx{1} \ar@/^4ex/[rr]^{B_1}

\ar@/^/[rr]^{B_2} \ar@/_3ex/[rr]_{B_m} & & \vtx{n} \ar@(ur,dr)^A} $

and the system $\Sigma$ does determine a representation of this new

quiver of dimension vector $\beta=(1,n)$ by assigning to the arrows the

different columns of the matrix $B$. Isomorphism classes of these

quiver-representations do correspond precisely to equivalence classes of

linear control systems. In [part 4][1] we introduced stable and

semi-stable representations. In this framed-quiver setting call a

representation $(A,B_1,\ldots,B_m)$ stable if there is no proper

subrepresentation of dimension vector $(1,p)$ for some $p \lneq n$.

Perhaps remarkable this algebraic notion has a counterpart in

system-theory : the systems corresponding to stable

quiver-representations are precisely the completely controllable

systems. That is, those which can be brought to any wanted state by

varying the controls. Hence, the moduli space

$M^s_{(1,n)}(Q_f,\theta)$ classifying stable representations is

exactly the moduli space of completely controllable linear systems

studied in control theory. For an excellent account of this moduli space

one can read the paper [Introduction to moduli spaces associated to

quivers by [Christof Geiss][2]. Fixing the number $m$ of controls but

varying the dimensions of teh state-spaces one would like to take all

the moduli spaces $ \bigsqcup_n~M^s_{(1,n)}(Q_f,\theta)$

together as they are all determined by the same formally smooth algebra

$\mathbb{C} Q_f$. This was done in a joint paper with [Markus Reineke][3] called

[Canonical systems and non-commutative geometry][4] in which we prove

that this disjoint union can be identified with the _infinite

Grassmannian_ $ \bigsqcup_n~M^s_{(1,n)}(Q_f,\theta) =

\mathbf{Gras}_m(\infty)$ of $m$-dimensional subspaces of an

infinite dimensional space. This result can be seen as a baby-version of

George Wilson\’s result relating the disjoint union of Calogero-Moser

spaces to the _adelic_ Grassmannian. But why do we stress this

particular quiver so much? This will be partly explained [next time][5].

[1]: http://www.neverendingbooks.org/index.php?p=350

[2]: http://www.matem.unam.mx/~christof/

[3]: http://wmaz1.math.uni-wuppertal.de/reineke/

[4]: http://www.arxiv.org/abs/math.AG/0303304

[5]: http://www.neverendingbooks.org/index.php?p=352

In a few

weeks I will give a _geometry 101_ course! It was decided that in

this course I should try to explain what rotations in $\mathbb{R}^3’$

are, so the classification of all finite rotation groups seemed like a

fun topic. Along the way I’ll have to introduce groups so bringing in a

little bit of GAP

may be a good idea. Clearly, the real power of GAP is lost on the

symmetry groups of the Platonic solids so I’ll do the traditional

computation of the transformation group of the Rubik’s cube. But

then I discovered that there is also a version of it on the dodecahedron

which is called megaminx so I couldn’t resist trying to work out the order of its

transformation group. Fortunately Coreyanne Rickwalt did already the

hard work giving a presentation as

a permutation group. So giving the generators to GAP

f1:=(1,3,5,7,9)(2,4,6,8,10)(20,31,42,53,64)(19,30,41,52,63)(18,29,40,51,62);

f2:=(12,14,16,18,20)(13,15,17,19,21)(1,60,73,84,31)(3,62,75,86,23)(2,61,74,85,32);

f3:=(23,25,27,29,31)(24,26,28,30,32)(82,95,42,3,16)(83,96,43,4,17)(84,97,34,5,18);

f4:=(34,36,38,40,42)(35,37,39,41,43)(27,93,106,53,5)(28,94,107,54,6)(29,95,108,45,7);

f5:=(45,47,49,51,53)(46,48,50,52,54)(38,104,117,64,7)(39,105,118,65,8),(40,106,119,56,9);

f6:=(56,58,60,62,64)(57,59,61,63,65)(49,115,75,20,9)(50,116,76,21,10),(51,117,67,12,1);

f7:=(67,69,71,73,75)(68,70,72,74,76)(58,113,126,86,12)(59,114,127,7,13),(60,115,128,78,14);

f8:=(78,80,82,84,86)(79,81,83,85,87)(71,124,97,23,14)(72,125,98,24,15),(73,126,89,25,16);

f9:=(89,91,93,95,97)(90,92,94,96,98)(80,122,108,34,25)(81,123,109,35,26),(82,124,100,36,27);

f10:=(100,102,104,106,108)(101,103,105,107,109)(91,130,119,45,36),(92,131,120,46,37)(93,122,111,47,38);

f11:=(111,113,115,117,119)(112,114,116,118,120)(102,128,67,56,47),(103,129,68,57,48)(104,130,69,58,49);

f12:=(122,124,126,128,130)(123,125,127,129,131)(100,89,78,69,111),(101,90,79,70,112)(102,91,80,71,113);

and defining the

megaminx group by

megaminx:=Group(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12); Size(megaminx);

and asking for its order I was a bit surprised to get

after a couple of minutes the following awkward number

33447514567245635287940590270451862933763731665690149051478356761508167786224814946834370826

35992490654078818946607045276267204294704060929949240557194825002982480260628480000000000000

000000000000000

or if you prefer it is

$2^{115} 3^{58} 5^{28} 7^{19} 11^{10} 13^9 17^7 19^6 23^5 29^4 31^3

37^3 41^2 43^2 47^2 53^2 59^2 61 .67 .71. 73. 79 .83 .89 .97. 101 .103.

107 .109 .113$

In what

way is a formally smooth algebra a _machine_ producing families of

manifolds? Consider the special case of the path algebra $\mathbb{C} Q$ of a

quiver and recall that an $n$-dimensional representation is an algebra

map $\mathbb{C} Q \rightarrow^{\phi} M_n(\mathbb{C})$ or, equivalently, an

$n$-dimensional left $\mathbb{C} Q$-module $\mathbb{C}^n_{\phi}$ with the action

determined by the rule $a.v = \phi(a) v~\forall v \in \mathbb{C}^n_{\phi},

\forall a \in \mathbb{C} Q$ If the $e_i~1 \leq i \leq k$ are the idempotents

in $\mathbb{C} Q$ corresponding to the vertices (see this [post][1]) then we get

a direct sum decomposition $\mathbb{C}^n_{\phi} = \phi(e_1)\mathbb{C}^n_{\phi} \oplus

\ldots \oplus \phi(e_k)\mathbb{C}^n_{\phi}$ and so every $n$-dimensional

representation does determine a _dimension vector_ $\alpha =

(a_1,\ldots,a_k)~\text{with}~a_i = dim_{\mathbb{C}} V_i = dim_{\mathbb{C}}

\phi(e_i)\mathbb{C}^n_{\phi}$ with $ | \alpha | = \sum_i a_i = n$. Further,

for every arrow $\xymatrix{\vtx{e_i} \ar[rr]^a & &

\vtx{e_j}} $ we have (because $e_j.a.e_i = a$ that $\phi(a)$

defines a linear map $\phi(a)~:~V_i \rightarrow V_j$ (that was the

whole point of writing paths in the quiver from right to left so that a

representation is determined by its _vertex spaces_ $V_i$ and as many

linear maps between them as there are arrows). Fixing vectorspace bases

in the vertex-spaces one observes that the space of all

$\alpha$-dimensional representations of the quiver is just an affine

space $\mathbf{rep}_{\alpha}~Q = \oplus_a~M_{a_j \times a_i}(\mathbb{C})$ and

base-change in the vertex-spaces does determine the action of the

_base-change group_ $GL(\alpha) = GL_{a_1} \times \ldots \times

GL_{a_k}$ on this space. Finally, as all this started out with fixing

a bases in $\mathbb{C}^n_{\phi}$ we get the affine variety of all

$n$-dimensional representations by bringing in the base-change

$GL_n$-action (by conjugation) and have $\mathbf{rep}_n~\mathbb{C} Q =

\bigsqcup_{| \alpha | = n} GL_n \times^{GL(\alpha)}

\mathbf{rep}_{\alpha}~Q$ and in this decomposition the connected

components are no longer just affine spaces with a groupaction but

essentially equal to them as there is a natural one-to-one

correspondence between $GL_n$-orbits in the fiber-bundle $GL_n

\times^{GL(\alpha)} \mathbf{rep}_{\alpha}~Q$ and $GL(\alpha)$-orbits in the

affine space $\mathbf{rep}_{\alpha}~Q$. In our main example

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

\ar@(d,dr)_y \ar@/^/[ll]^b} $ an $n$-dimensional representation

determines vertex-spaces $V = \phi(e) \mathbb{C}^n_{\phi}$ and $W = \phi(f)

\mathbb{C}^n_{\phi}$ of dimensions $p,q~\text{with}~p+q = n$. The arrows

determine linear maps between these spaces $\xymatrix{V

\ar@/^/[rr]^{\phi(a)} & & W \ar@(u,ur)^{\phi(x)} \ar@(d,dr)_{\phi(y)}

\ar@/^/[ll]^{\phi(b)}} $ and if we fix a set of bases in these two

vertex-spaces, we can represent these maps by matrices

$\xymatrix{\mathbb{C}^p \ar@/^/[rr]^{A} & & \mathbb{C}^q \ar@(u,ur)^{X}

\ar@(d,dr)_{Y} \ar@/^/[ll]^{B}} $ which can be considered as block

$n \times n$ matrices $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 basechange group

$GL(\alpha) = GL_p \times GL_q$ is the diagonal subgroup of $GL_n$

$GL(\alpha) = \begin{bmatrix} GL_p & 0 \\ 0 & GL_q \end{bmatrix}$ and

acts on the representation space $\mathbf{rep}_{\alpha}~Q = M_{q \times

p}(\mathbb{C}) \oplus M_{p \times q}(\mathbb{C}) \oplus M_q(\mathbb{C}) \oplus M_q(\mathbb{C})$

(embedded as block-matrices in $M_n(\mathbb{C})^{\oplus 4}$ as above) by

simultaneous conjugation. More generally, if $A$ is a formally smooth

algebra, then all its representation varieties $\mathbf{rep}_n~A$ are

affine smooth varieties equipped with a $GL_n$-action. This follows more

or less immediately from the definition and [Grothendieck][2]\’s

characterization of commutative regular algebras. For the record, an

algebra $A$ is said to be _formally smooth_ if for every algebra map $A

\rightarrow B/I$ with $I$ a nilpotent ideal of $B$ there exists a lift

$A \rightarrow B$. The path algebra of a quiver is formally smooth

because for every map $\phi~:~\mathbb{C} Q \rightarrow B/I$ the images of the

vertex-idempotents form an orthogonal set of idempotents which is known

to lift modulo nilpotent ideals and call this lift $\psi$. But then also

every arrow lifts as we can send it to an arbitrary element of

$\psi(e_j)\pi^{-1}(\phi(a))\psi(e_i)$. In case $A$ is commutative and

$B$ is allowed to run over all commutative algebras, then by

Grothendieck\’s criterium $A$ is a commutative regular algebra. This

also clarifies why so few commutative regular algebras are formally

smooth : being formally smooth is a vastly more restrictive property as

the lifting property extends to all algebras $B$ and whenever the

dimension of the commutative variety is at least two one can think of

maps from its coordinate ring to the commutative quotient of a

non-commutative ring by a nilpotent ideal which do not lift (for an

example, see for example [this preprint][3]). The aim of

non-commutative algebraic geometry is to study _families_ of manifolds

$\mathbf{rep}_n~A$ associated to the formally-smooth algebra $A$. [1]:

http://www.matrix.ua.ac.be/wp-trackback.php/10 [2]:

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.

html [3]: http://www.arxiv.org/abs/math.AG/9904171

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

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.