moduli spaces

In the previous part we saw that moduli spaces of suitable representations of the quiver locally determine the moduli spaces of vectorbundles over smooth projective curves. There is yet another classical problem related to this quiver (which also illustrates the idea of looking at families of moduli spaces rather than individual ones) : linear control systems. Such a system with an $n$ dimensional state space and $m$ controls (or inputs) is determined by the following system of linear differential equations $ \frac{d x}{d t} = A.x + B.u$ where $x(t) \in \C^n$ is the state of the system at time $t$, $u(t) \in \C^m$ is the control-vector at time $t$ and $A \in Mn(\C), B \in M{n \times m}(\C)$ are the matrices describing the evolution of the system $\Sigma$ (after fixing bases in the state- and control-space). That is, $\Sigma$ determines a representation of the above quiver of dimension-vector $\alpha = (m,n)$

Whereas in control theory (see for example Allen Tannenbaum\’s Lecture Notes in Mathematics 845 for a mathematical introduction) it is natural to call two systems equivalent when they only differ up to base change in the state-space, one usually fixes the control knobs so it is not natural to allow for base change in the control-space. So, at first sight the control theoretic problem of classifying equivalent systems is not the same problem as classifying representations of the quiver up to isomorphism. Fortunately, there is an elegant way round this which is called deframing. That is, for a fixed number $m$ of controls one considers the quiver $Qf$ having precisely $m$ arrows from the first to the second vertex and the system $\Sigma$ does determine a representation of this new quiver of dimension vector $\beta=(1,n)$ by assigning to the arrows the different columns of the matrix $B$. Isomorphism classes of these quiver-representations do correspond precisely to equivalence classes of linear control systems. In part 4 we introduced stable and semi-stable representations. In this framed-quiver setting call a representation $(A,B1,\hdots,Bm)$ stable if there is no proper subrepresentation of dimension vector $(1,p)$ for some $p \lneq n$. Perhaps remarkable this algebraic notion has a counterpart in system-theory : the systems corresponding to stable quiver-representations are precisely the completely controllable systems. That is, those which can be brought to any wanted state by varying the controls. Hence, the moduli space
$M^s{(1,n)}(Qf,\theta)$ classifying stable representations is exactly the moduli space of completely controllable linear systems studied in control theory. For an excellent account of this moduli space one can read the paper [Introduction to moduli spaces associated to quivers by Christof Geiss. Fixing the number $m$ of controls but varying the dimensions of teh state-spaces one would like to take all the moduli spaces $ \bigsqcupn~M^s{(1,n)}(Qf,\theta)$
together as they are all determined by the same formally smooth algebra $\C Qf$. This was done in a joint paper with Markus Reineke called Canonical systems and non-commutative geometry in which we prove that this disjoint union can be identified with the infinite Grassmannian $ \bigsqcupn~M^s{(1,n)}(Qf,\theta) = \mathbf{Gras}m(\infty)$ of $m$-dimensional subspaces of an infinite dimensional space. This result can be seen as a baby-version of George Wilson\’s result relating the disjoint union of Calogero-Moser spaces to the adelic Grassmannian. But why do we stress this particular quiver so much? This will be partly explained next time.