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quiver representations

Published September 6, 2004 by lieven

In what
way is a formally smooth algebra a _machine_ producing families of
manifolds? Consider the special case of the path algebra $C Q$ of a
quiver and recall that an $n$ -dimensional representation is an algebra
map $C Q \to^{ϕ} M_{n} (C)$ or, equivalently, an
$n$ -dimensional left $C Q$ -module $C_{ϕ}^{n}$ with the action
determined by the rule $a . v = ϕ (a) v \forall v \in C_{ϕ}^{n}, \forall a \in C Q$ If the $e_{i} 1 \leq i \leq k$ are the idempotents
in $C Q$ corresponding to the vertices (see this [post][1]) then we get
a direct sum decomposition $C_{ϕ}^{n} = ϕ (e_{1}) C_{ϕ}^{n} \oplus \dots \oplus ϕ (e_{k}) C_{ϕ}^{n}$ and so every $n$ -dimensional
representation does determine a _dimension vector_ $α = (a_{1}, \dots, a_{k}) with a_{i} = d i m_{C} V_{i} = d i m_{C} ϕ (e_{i}) C_{ϕ}^{n}$ with $| α | = \sum_{i} a_{i} = n$ . Further,
for every arrow $Misplaced &$ we have (because $e_{j} . a . e_{i} = a$ that $ϕ (a)$
defines a linear map $ϕ (a) : V_{i} \to V_{j}$ (that was the
whole point of writing paths in the quiver from right to left so that a
representation is determined by its _vertex spaces_ $V_{i}$ and as many
linear maps between them as there are arrows). Fixing vectorspace bases
in the vertex-spaces one observes that the space of all
$α$ -dimensional representations of the quiver is just an affine
space ${rep}_{α} Q = \oplus_{a} M_{a_{j} \times a_{i}} (C)$ and
base-change in the vertex-spaces does determine the action of the
_base-change group_ $G L (α) = G L_{a_{1}} \times \dots \times G L_{a_{k}}$ on this space. Finally, as all this started out with fixing
a bases in $C_{ϕ}^{n}$ we get the affine variety of all
$n$ -dimensional representations by bringing in the base-change
$G L_{n}$ -action (by conjugation) and have ${rep}_{n} C Q = ⨆_{| α | = n} G L_{n} \times^{G L (α)} {rep}_{α} Q$ and in this decomposition the connected
components are no longer just affine spaces with a groupaction but
essentially equal to them as there is a natural one-to-one
correspondence between $G L_{n}$ -orbits in the fiber-bundle $G L_{n} \times^{G L (α)} {rep}_{α} Q$ and $G L (α)$ -orbits in the
affine space ${rep}_{α} Q$ . In our main example
$Misplaced &$ an $n$ -dimensional representation
determines vertex-spaces $V = ϕ (e) C_{ϕ}^{n}$ and $W = ϕ (f) C_{ϕ}^{n}$ of dimensions $p, q with p + q = n$ . The arrows
determine linear maps between these spaces $Misplaced &$ and if we fix a set of bases in these two
vertex-spaces, we can represent these maps by matrices
$Misplaced &$ which can be considered as block
$n \times n$ matrices $a \mapsto [\begin{matrix} 0 & 0 \\ A & 0 \end{matrix}] b \mapsto [\begin{matrix} 0 & B \\ 0 & 0 \end{matrix}]$
$x \mapsto [\begin{matrix} 0 & 0 \\ 0 & X \end{matrix}] y \mapsto [\begin{matrix} 0 & 0 \\ 0 & Y \end{matrix}]$ The basechange group
$G L (α) = G L_{p} \times G L_{q}$ is the diagonal subgroup of $G L_{n}$
$G L (α) = [\begin{matrix} G L_{p} & 0 \\ 0 & G L_{q} \end{matrix}]$ and
acts on the representation space ${rep}_{α} Q = M_{q \times p} (C) \oplus M_{p \times q} (C) \oplus M_{q} (C) \oplus M_{q} (C)$
(embedded as block-matrices in $M_{n} (C)^{\oplus 4}$ as above) by
simultaneous conjugation. More generally, if $A$ is a formally smooth
algebra, then all its representation varieties ${rep}_{n} A$ are
affine smooth varieties equipped with a $G L_{n}$ -action. This follows more
or less immediately from the definition and [Grothendieck][2]\’s
characterization of commutative regular algebras. For the record, an
algebra $A$ is said to be _formally smooth_ if for every algebra map $A \to B / I$ with $I$ a nilpotent ideal of $B$ there exists a lift
$A \to B$ . The path algebra of a quiver is formally smooth
because for every map $ϕ : C Q \to B / I$ the images of the
vertex-idempotents form an orthogonal set of idempotents which is known
to lift modulo nilpotent ideals and call this lift $ψ$ . But then also
every arrow lifts as we can send it to an arbitrary element of
$ψ (e_{j}) π^{- 1} (ϕ (a)) ψ (e_{i})$ . In case $A$ is commutative and
$B$ is allowed to run over all commutative algebras, then by
Grothendieck\’s criterium $A$ is a commutative regular algebra. This
also clarifies why so few commutative regular algebras are formally
smooth : being formally smooth is a vastly more restrictive property as
the lifting property extends to all algebras $B$ and whenever the
dimension of the commutative variety is at least two one can think of
maps from its coordinate ring to the commutative quotient of a
non-commutative ring by a nilpotent ideal which do not lift (for an
example, see for example [this preprint][3]). The aim of
non-commutative algebraic geometry is to study _families_ of manifolds
${rep}_{n} A$ associated to the formally-smooth algebra $A$ . [1]:
https://lievenlb.local/wp-trackback.php/10 [2]:
http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Grothendieck.
html [3]: http://www.arxiv.org/abs/math.AG/9904171

representation spaces

Published September 6, 2004 by lieven

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 ${rep}_{n} A$ , the varieties of finite dimensional
representations. On ${rep}_{n} A$ 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 $M = M_{X}^{s s} (0, n)$ 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
$π_{1} (X) = \frac{⟨ x_{1}, \dots, x_{g}, y_{1}, \dots, y_{g} ⟩}{([x_{1}, y_{1}] \dots [x_{g}, y_{g}])} \to U_{n} (C)$ 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 $A = [\begin{matrix} C & 0 \\ C [Y] & C [Y] \end{matrix}]$ As $A$ has two
orthogonal idempotents, its representation varieties decompose into
connected components according to dimension vectors ${rep}_{m} A = ⨆_{p + q = m} {rep}_{(p, q)} A$ 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
${rep}_{(p, q)} A / / G L_{p + q} = S^{q} (Y)$ 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 $- u q + v p < 0$ (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 $M_{(p, q)} (A, θ)$ . 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 $M_{X}^{s s} (0, n) \sim^{b i r} M_{(n, g n)} (A, θ)$ Rather than studying the moduli spaces of semi-stable vectorbundles $M_{X}^{s s} (0, n)$ 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 $⨆_{n} M_{X}^{s s} (0, n)$ 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 $Misplaced &$ and hence the representation theory of this quiver plays an important role in studying the geometric properties of the moduli spaces $M_{X}^{s s} (0, n)$ , 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

path algebras

Published September 3, 2004 by lieven

The previous post can be found [here][1].
Pierre Gabriel invented a lot of new notation (see his book [Representations of finite dimensional algebras][2] for a rather extreme case) and is responsible for calling a directed graph a quiver. For example,

$Misplaced &$

is a quiver. Note than it is allowed to have multiple arrows between vertices, as well as loops in vertices. For us it will be important that a quiver $Q$ depicts how to compute in a certain non-commutative algebra : the path algebra $C Q$ . If the quiver has $k$ vertices and $l$ arrows (including loops) then the path algebra $C Q$ is a subalgebra of the full $k \times k$ matrix-algebra over the free algebra in $l$ non-commuting variables

$C Q \subset M_{k} (C ⟨ x_{1}, \dots, x_{l} ⟩)$

Under this map, a vertex $v_{i}$ is mapped to the basis $i$ -th diagonal matrix-idempotent and an arrow

$Misplaced &$

is mapped to the matrix having all its entries zero except the $(j, i)$ -entry which is equal to $x_{a}$ . That is, in our main example

$Misplaced &$

the corresponding path algebra is the subalgebra of $M_{2} (C ⟨ a, b, x, y ⟩)$ generated by the matrices

$e \mapsto [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]$ $f \mapsto [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$

$a \mapsto [\begin{matrix} 0 & 0 \\ a & 0 \end{matrix}]$ $b \mapsto [\begin{matrix} 0 & b \\ 0 & 0 \end{matrix}]$

$x \mapsto [\begin{matrix} 0 & 0 \\ 0 & x \end{matrix}]$ $y \mapsto [\begin{matrix} 0 & 0 \\ 0 & y \end{matrix}]$

The name \’path algebra\’ comes from the fact that the subspace of $C Q$ at the $(j, i)$ -place is the vectorspace spanned by all paths in the quiver starting at vertex $v_{i}$ and ending in vertex $v_{j}$ . For an easier and concrete example of a path algebra. consider the quiver

$Misplaced &$

and verify that in this case, the path algebra is just

$C Q = [\begin{matrix} C & 0 \\ C [x] a & C [x] \end{matrix}]$

Observe that we write and read paths in the quiver from right to left. The reason for this strange convention is that later we will be interested in left-modules rather than right-modules. Right-minder people can go for the more natural left to right convention for writing paths.
Why are path algebras of quivers of interest in non-commutative geometry? Well, to begin they are examples of _formally smooth algebras_ (some say _quasi-free algebras_, I just call them _qurves_). These algebras were introduced and studied by Joachim Cuntz and Daniel Quillen and they are precisely the algebras allowing a good theory of non-commutative differential forms.
So you should think of formally smooth algebras as being non-commutative manifolds and under this analogy path algebras of quivers correspond to _affine spaces_. That is, one expects path algebras of quivers to turn up in two instances : (1) given a non-commutative manifold (aka formally smooth algebra) it must be \’embedded\’ in some non-commutative affine space (aka path algebra of a quiver) and (2) given a non-commutative manifold, the \’tangent spaces\’ should be determined by path algebras of quivers.
The first fact is easy enough to prove, every affine $C$ -algebra is an epimorphic image of a free algebra in say $l$ generators, which is just the path algebra of the _bouquet quiver_ having $l$ loops

$\xymatrix \vtx \ar @ (d l, l)^{x_{1}} \ar @ (l, u l)^{x_{2}} \ar @ (u r, r)^{x_{i}} \ar @ (r, d r)^{x_{l}}$

The second statement requires more work. For a first attempt to clarify this you can consult my preprint [Qurves and quivers][3] but I\’ll come back to this in another post. For now, just take my word for it : if formally smooth algebras are the non-commutative analogon of manifolds then path algebras of quivers are the non-commutative version of affine spaces!

[1]: https://lievenlb.local/index.php?p=71
[2]: http://www.booxtra.de/verteiler.asp?site=artikel.asp&wea=1070000&sh=homehome&artikelnummer=000000689724
[3]: http://www.arxiv.org/abs/math.RA/0406618

Tag: representations

quiver representations

representation spaces

path algebras