Linear quivers, generic extensions and Kashiwara operators

In the present paper, we introduce the generic extension graph G of a Dynkin or cyclic quiver Q and then compare this graph with the crystal graph C for the quantized enveloping algebra associated to Q via two maps ℘Q, Q : Ω → ΛQ induced by generic extensions and Kashiwara operators, respectively, where ΛQ is the set of isoclasses of nilpotent representations of Q, and Ω is the set of all words on the alphabet I, the vertex set of Q. We prove that, if Q is a (finite or infinite) linear quiver, then the intersection of the fibres ℘Q−1 (λ) and KQ−1 (λ) is non-empty for every λ ∈ Λ Q. We will also show that this non-emptyness property fails for cyclic quivers.