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quiver representations

In what
way is a formally smooth algebra a _machine_ producing families of
manifolds? Consider the special case of the path algebra CQ of a
quiver and recall that an n-dimensional representation is an algebra
map CQϕMn(C) or, equivalently, an
n-dimensional left CQ-module Cϕn with the action
determined by the rule a.v=ϕ(a)v vCϕn,aCQ If the ei 1ik are the idempotents
in CQ corresponding to the vertices (see this [post][1]) then we get
a direct sum decomposition Cϕn=ϕ(e1)Cϕnϕ(ek)Cϕn and so every n-dimensional
representation does determine a _dimension vector_ α=(a1,,ak) with ai=dimCVi=dimCϕ(ei)Cϕn with |α|=iai=n. Further,
for every arrow Misplaced & we have (because ej.a.ei=a that ϕ(a)
defines a linear map ϕ(a) : ViVj (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_ Vi 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 repα Q=a Maj×ai(C) and
base-change in the vertex-spaces does determine the action of the
_base-change group_ GL(α)=GLa1××GLak on this space. Finally, as all this started out with fixing
a bases in Cϕn we get the affine variety of all
n-dimensional representations by bringing in the base-change
GLn-action (by conjugation) and have repn CQ=|α|=nGLn×GL(α)repα 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 GLn-orbits in the fiber-bundle GLn×GL(α)repα Q and GL(α)-orbits in the
affine space repα Q. In our main example
Misplaced & an n-dimensional representation
determines vertex-spaces V=ϕ(e)Cϕn and W=ϕ(f)Cϕn 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×n matrices a[00A0] b[0B00]
x[000X] y[000Y] The basechange group
GL(α)=GLp×GLq is the diagonal subgroup of GLn
GL(α)=[GLp00GLq] and
acts on the representation space repα Q=Mq×p(C)Mp×q(C)Mq(C)Mq(C)
(embedded as block-matrices in Mn(C)4 as above) by
simultaneous conjugation. More generally, if A is a formally smooth
algebra, then all its representation varieties repn A are
affine smooth varieties equipped with a GLn-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 AB/I with I a nilpotent ideal of B there exists a lift
AB. The path algebra of a quiver is formally smooth
because for every map ϕ : CQB/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(ej)π1(ϕ(a))ψ(ei). 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
repn 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

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representation spaces

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 repn A, the varieties of finite dimensional
representations. On repn 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=MXss(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
π1(X)=x1,,xg,y1,,yg([x1,y1][xg,yg])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=[C0C[Y]C[Y]] As A has two
orthogonal idempotents, its representation varieties decompose into
connected components according to dimension vectors repm A=p+q=mrep(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//GLp+q=Sq(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 GLp+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 MXss(0,n) bir M(n,gn)(A,θ) Rather than studying the moduli spaces of semi-stable vectorbundles MXss(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 MXss(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 Misplaced & and hence the representation theory of this quiver plays an important role in studying the geometric properties of the moduli spaces MXss(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

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diy psychoanalysis

After several years of total inactivity in the DIY-sector I managed to do the following over the week-end :

  • Replace two bedroom reading lamps (at least two years overdue). We had a two-lamps-in-one IKEA thing but ThePartner’s side broke off two years ago and mine followed a year later. Since then one lamp hung in very unstable equilibrium and in the end the only way to prevent nightly accidents was to position it vertically

  • Fix the halogen spots in the kitchen (at least one year overdue). Two years ago one half of the spots went dead because the transfo overheated. I spend a week trying to find the (totally unreachable) place where it was hidden by the installers, so I wasn’t looking forward to a repeat when the second half went dead last year. Yesterday, after replacing the transfo it became clear that this time it was just a matter of faulty spots

  • Rewire the dinner room (at least three years overdue). For years an outlet was lying around on the floor. It is now replaced by a state of the art cable tray

You do not need to have a master in psychoanalysis to figure out that I am subconsciously trying to regain some control over OurHouseSystem which PseudonymousDaughterOne gave a couple of blows this week. And, running electricity errands has a much higher effect-rate than parental interference.

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