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Tag: representations

representation spaces

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

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path algebras

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

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driven by ambition and sloth

Here’s a part of yesterday’s post by bitch ph.d. :

But first of all I have to figure out what the hell I’m going to teach my graduate students this semester, and really more to the point, what I am not going to bother to try to cram into this class just because it’s my first graduate class and I’m feeling like teaching everything I know in one semester is a realistic and desireable possibility. Yes! Here it all is! Everything I have ever learned! Thank you, and goodnight!

Ah, the perpetual motion machine of last-minute course planning, driven by ambition and sloth!.

I’ve had similar experiences, even with undergraduate courses (in Belgium there is no fixed curriculum so the person teaching the course is responsible for its contents). If you compare the stuff I hoped to teach when I started out with the courses I’ll be giving in a few weeks, you would be more than disappointed.
The first time I taught _differential geometry 1_ (a third year course) I did include in the syllabus everything needed to culminate in an outline of Donaldson’s result on exotic structures on $\mathbb{R}^4 $ and Connes’ non-commutative GUT-model (If you want to have a good laugh, here is the set of notes). As far as I remember I got as far as classifying compact surfaces!
A similar story for the _Lie theory_ course. Until last year this was sort of an introduction to geometric invariant theory : quotient variety of conjugacy classes of matrices, moduli space of linear dynamical systems, Hilbert schemes and the classification of $GL_n $-representations (again, smile! here is the set of notes).
Compared to these (over)ambitious courses, next year’s courses are lazy sunday-afternoon walks! What made me change my mind? I learned the hard way something already known to the ancient Greeks : mathematics does not allow short-cuts, you cannot expect students to run before they can walk. Giving an over-ambitious course doesn’t offer the students a quicker road to research, but it may result in a burn-out before they get even started!

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the one quiver for GL(2,Z)

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.

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the necklace Lie bialgebra

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.

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