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
$G{L}_n$ 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
${\pi }_1(X)=\frac{⟨x_1,\dots ,x_g,y_1,\dots ,y_g⟩}{([x_1,y_1]\dots [x_g,y_g])}\to U_n(\mathbb{C})$ 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 $A=\left[\begin{array}{cc}\mathbb{C}& 0\\ \mathbb{C}[Y]& \mathbb{C}[Y]\end{array}\right]$ As $A$ has two
orthogonal idempotents, its representation varieties decompose into
connected components according to dimension vectors ${\mathbf{rep}}_{m}A=\underset{p+q=m}{⨆}{\mathbf{rep}}_{(p,q)}A$ 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
${\mathbf{rep}}_{(p,q)}A//G{L}_{p+q}=S^q(Y)$ 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 $G{L}_{p+q}$ 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 $M_X^{ss}(0,n)$ 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 $\underset{n}{⨆}{M}_X^{ss}(0,n)$ 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 and hence the representation theory of this quiver plays an important role in studying the geometric properties of the moduli spaces $M_X^{ss}(0,n)$, for instance it allows to determine the smooth loci of these varieties. Move on the the [next part. [1]: https://lievenlb.local/index.php/quiver-representations.html