Spring, 2006

(an offshoot of the UMass Representation Theory seminar)

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

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).

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(d₁) x ... x GL(d_n) acts on the space of representation of a quiver with dimension vector (d₁, ... ,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.

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.

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.

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.