representation spaces

The previous part of this sequence was quiver representations. 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 $GLn$ 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 $M = MX^{ss}(0,n)$ 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 $\pi1(X) = \frac{\langle x1,\hdots,xg,y1,\hdots,yg \rangle}{([x1,y1] \hdots [xg,yg])} \rightarrow Un(\C)$ 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 $A = \begin{bmatrix} \C & 0 \ \C[Y] & \C[Y] \end{bmatrix}$ 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
$MX^{ss}(0,n)~\sim^{bir}~M{(n,gn)}(A,\theta)$ 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 $\bigsqcupn~M^{ss}X(0,n)$ 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 $QA$ (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.

Previous in series

Next in series