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

TL;DR

This paper studies a family of algebras associated with symmetrizable Cartan matrices, showing that key properties like canonical decompositions and rigid modules are invariant under changes in a parameter, and analyzing related geometric structures.

Contribution

It introduces a reduction functor between module categories of these algebras, proving invariance of canonical decompositions, rigid modules, and flag variety Euler characteristics across different parameter values.

Findings

Canonical decomposition of rank vectors is independent of the parameter k.

Rigid locally free modules correspond bijectively across different k values.

Euler characteristics of flag varieties are invariant under change of k.

Abstract

For $k \ge 1$ we consider the $K$-algebra $H(k) := H(C,kD,\Omega)$ associated to a symmetrizable Cartan matrix $C$, a symmetrizer $D$, and an orientation $\Omega$ 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 $k$.