QUIVERS WITH A REGULAR RING OF INVARIANTS

RAF BOCKLANDT

Abstract

Many problems in representation theory can be reduced to representations of quivers. Suppose A is a finitely generated algebra and Rep_{n}A is the space of ndimensional complex representations of A. On this space is an action of GL_{n} and one can divide out this action by taking the affine quotient to obtain a new space iss_{n}A := Rep_{n}A/GL_{n} classifying the equivalence classes of n-dimensional semisimple representations of A. (see [2])

If W ∈ Rep_{n}A is a semisimple representation, one can wonder what the structure of iss_{n}A around the point p corresponding to the equivalence class of W looks like. If W is a smooth point in Rep_{n}A there is a neighborhood of p that is étale (or analytically) isomorphic to a neighborhood of the zero representation in the quotient space, iss_{α_{p}}, Q_{p} of a quiver setting (Q_{p}, α_{p}) which is called the local quiver setting of p. This local quiver setting depends on the structure of W as a direct sum of simple representations

W := S_{1}^{⊕a_{1}} ⊕ ⋯ ⊕ S_{k}^{⊕a_{k}}.

(For the exact construction see [3])

So if one for example asks whether iss_{n}A is smooth in the point p one can as well ask whether its local quiver setting has a quotient space that is smooth in zero. Such quiver settings will be called coregular.

In this talk we present a method to determine if a random given quiver setting (Q, α) is indeed coregular. This method consists of some specific reduction steps which can reduce a quiver setting to 3 basic settings if and only if the setting is coregular.

References

[1] Raf Bocklandt. General Quiver Settings with a Regular Ring of Invariants. To appear in Journal of Algebra.

[2] Hanspeter Kraft. Geometrische Methoden in der Invariantentheorie, volume D1 of Aspekte der Mathematik. Vieweg-Verlag, Braunschweig, 1984.

[3] Lieven Le Bruyn and Claudio Procesi. Semisimple representations of quivers. Trans.Amer.Math.Soc, 317:585–598, 1990.

Universiteit Antwerpen (UIA), B-2610 Antwerp (Belgium)

   E-mail address: rbockl@uia.ac.be