Representations of valued quivers and finite group actions
by
Zongzhu Lin
Kansas State Univerity

Given a quiver \Gamma with a finite group G acting on it. An representation X of \Gamma is said to be G invariant if for each g in G , the twisted representation X^g is isomorphic to X . Tanisaki studied the representations of quivers with a finite group action and determined when the category of all G -invariant representations to be of finite type and the indecomposable representations are exactly those corresponding to the positive roots of the root system of a finite dimensional complex semisimple Lie algebra. The Dynkin diagram of the root system arises naturally as the quotient graph of the quiver under the finite group action. Dlab and Ringel also described the positive roots of finite root systems as indecomposable representations of valued quivers associated to a Dynkin diagram.

In this talk we will relate the representations of valued quivers over finite fields and the G-invariant representations of a cover quiver. Each valued quiver can be realized as a quotient quiver of a quiver with a finite abelian group G action. Under this correspondence, the category of representations of the valued quiver and the category of G -invariant representations of the quiver \Gamma. Then we make a few conjectures, similar to the conjecture of Kac, relating the number of absolutely indecomposable representations valued quiver and the dimensions of root spaces of Kac-Moody Lie algebras corresponding to a symmetrizable generalized Cartan matrix that defines the valued quiver.. We will then show a few examples to indicate how Kac's conjecture in the symmetric Cartan matrix situation can be related to the non-symmetric situation.

Date received: July 31, 2000