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