In [the previous part][1] we saw that moduli spaces of suitable representations

of the quiver $\xymatrix{\vtx{} \ar[rr] & & \vtx{}

\ar@(ur,dr)} $ 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 $x(t) \in \mathbb{C}^n$ is the state of the system at

time $t$, $u(t) \in \mathbb{C}^m$ is the control-vector at time $t$ and $A \in

M_n(\mathbb{C}), B \in M_{n \times m}(\mathbb{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 $\alpha = (m,n)$

$\xymatrix{\vtx{m} \ar[rr]^B & & \vtx{n} \ar@(ur,dr)^A} $

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 $Q_f$ having precisely $m$ arrows from the first to

the second vertex $\xymatrix{\vtx{1} \ar@/^4ex/[rr]^{B_1}

\ar@/^/[rr]^{B_2} \ar@/_3ex/[rr]_{B_m} & & \vtx{n} \ar@(ur,dr)^A} $

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][1] we introduced stable and

semi-stable representations. In this framed-quiver setting call a

representation $(A,B_1,\ldots,B_m)$ 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

$M^s_{(1,n)}(Q_f,\theta)$ 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][2]. Fixing the number $m$ of controls but

varying the dimensions of teh state-spaces one would like to take all

the moduli spaces $ \bigsqcup_n~M^s_{(1,n)}(Q_f,\theta)$

together as they are all determined by the same formally smooth algebra

$\mathbb{C} Q_f$. This was done in a joint paper with [Markus Reineke][3] called

[Canonical systems and non-commutative geometry][4] in which we prove

that this disjoint union can be identified with the _infinite

Grassmannian_ $ \bigsqcup_n~M^s_{(1,n)}(Q_f,\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][5].

[1]: http://www.neverendingbooks.org/index.php?p=350

[2]: http://www.matem.unam.mx/~christof/

[3]: http://wmaz1.math.uni-wuppertal.de/reineke/

[4]: http://www.arxiv.org/abs/math.AG/0303304

[5]: http://www.neverendingbooks.org/index.php?p=352

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