Spring, 2006

(an offshoot of the UMass
Representation Theory seminar)

Time: Wednesdays at 4:15

Place: LGRT 1634 (for now, at least)

Resources:

Some
references compiled by Jim Humphreys

Notes from a seminar on
quiver varieties held at Berkeley in 2000

Web page of summer school "Geometry of quiver-representations and preprojective algebras", 2000

Web page of summer school "Geometry of quiver-representations and preprojective algebras", 2000

The talks:

- Thursday, March 9. Jim Humphreys "Quivers and their representations" abstract
- Wednesday, March 16. Ralf Schiffler "Varieties of representations of quivers" abstract
- Wednesday, March 30. Tom Braden "GIT quotients and moduli of quivers" abstract
- Wednesday, April 5, LGRT 1114 (new location!) Tom Braden
"Nakajima's quiver varieties" abstract

- Wednesday, April 19. Eric Sommers "Quiver varieties in type A" abstract
- Tuesday, May 2 and 9. Eyal Markman "Quiver Varieties and Kac-Moody Algebras, after Nakajima"

Jim Humphreys "Quivers and their representations"

This talk is intended to provide
background material for further seminar talks on quiver varieties. It
should be accessible to graduate students and nonspecialists. We look
at quivers (directed graphs) and their "representations". Quivers of
finite representation type turn out to be parametrized by the
ubiquitous Coxeter-Dynkin diagrams of types A-D-E (work of Gabriel,
Tits, and others).

Ralf Schiffler "Varieties of representations of quivers"

In this talk, I will present certain
varieties of representations for quivers of type A,D,E and their role
in the definition of the canonical basis of quantized enveloping
algebras of type A,D,E.

The group GL(d_{1}) x ... x GL(d_{n}) acts on the
space of representation of a quiver with dimension vector (d_{1},
... ,d_{n}) by conjugation. The isoclass of a representation is
precisely the orbit of
the representation under this action. The varieties we are interested
in are the Zariski closures of these orbits.

Tom
Braden "GIT quotients and moduli spaces of quivers"The group GL(d

When a quiver is not of finite type,
the group which changes the bases at the vertices will not act with
finitely many orbits. To study the way representations of the quiver
can vary in families, we use geometric invariant theory. This gives a
way to throw out certain "unstable" orbits so that the remaining orbits
have a good quotient.

Tom Braden "Nakajima's quiver varieties"

In a series of celebrated papers,
Nakajima used certain moduli spaces of
quiver representations to study the representation theory of Kac-Moody
Lie
algebras. Their construction involves a modification of the GIT
quotients
discussed in last week's talk to a hyperkaehler or holomorphic
symplectic
setting, where we replace complex affine space by its cotangent
bundle. We
explain the idea of hyperkaehler quotients, introduce Nakajima's
varieties, and
then show how special cases give familiar varieties such as cotangent
bundles
to flag varieties.

Eric Sommers "Quiver varieties in type A"

I'll talk about a result of Maffei in
type A that relates nilpotent orbits (via something called the Slodowy
slice) to quiver varieties. This proves a conjecture of Nakajima
and provides a link between his work and that of Ginzburg. I'll
also mention a conjectural extension to other simply-laced types.