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

Abstract

For $k\ge 1$ we consider the $K$-algebra $H(k):=H(C,kD,\Omega )$ 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 $k$.

24 pages. This article, together with "Quiver with relations for symmetrizable Cartan matrices III: Convolution Algebras" (arXiv:1511.06216) replaces arXiv:1502.01565, which has been withdrawn. V3: acknowledgements added. V4: exposition improved and a few typos fixed after referee report, references updated. Final version, to appear in IMRN

References in corpus (3)

• Quivers with relations for symmetrizable Cartan matrices I : Foundations
• Quivers with relations for symmetrizable Cartan matrices III: Convolution algebras
• The Hall algebra and the composition monoid

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