Quivers with relations for symmetrizable Cartan matrices II: Change of symmetrizers

Abstract

For $k \ge 1$ we consider the $K$-algebra $H(k) := H(C,kD,Ω)$ associated to a symmetrizable Cartan matrix $C$, a symmetrizer $D$, and an orientation $Ω$ of $C$, which was defined in Part 1. We construct and analyse a reduction functor from rep$(H(k))$ to rep$(H(k-1))$. As a consequence we show that the canonical decomposition of rank vectors for $H(k)$ does not depend on $k$, and that the rigid locally free $H(k)$-modules are up to isomorphism in bijection with the rigid locally free $H(k-1)$-modules. Finally, we show that for a rigid locally free $H(k)$-module of a given rank vector the Euler characteristic of the variety of flags of locally free submodules with fixed ranks of the subfactors does not depend on the choice of $k$.