Skip to content →

Tag: Riemann

representation spaces

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]:

Leave a Comment