Geometric characterizations of the representation type of hereditary algebras and of canonical algebras

TL;DR

This paper characterizes tame hereditary and canonical algebras using geometric properties of moduli spaces and fields of rational invariants, linking algebraic and geometric perspectives.

Contribution

It provides a new geometric characterization of tame hereditary and canonical algebras through moduli space structures and invariant fields.

Findings

Tame quivers have moduli spaces that are projective spaces for all dimension vectors and weights.

The field of rational invariants for Schur roots is either trivial or rational function field for tame cases.

A canonical algebra is tame if and only if its generic roots have invariant fields isomorphic to k or k(t).

Abstract

We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector $\mathbf{d}$ and each integral weight $\theta$ of Q, the moduli space $\mathcal{M}(Q,\mathbf{d})^{ss}_{\theta}$ of $\theta$-semi-stable $\mathbf{d}$-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight spaces of semi-invariants satisfy a certain log-concavity property. Furthermore, we characterize the tame quivers as being those quivers Q with the property that for each Schur root $\mathbf{d}$ of Q, the field of rational invariants $k(rep(Q,\mathbf{d}))^{GL(\mathbf{d})}$ is isomorphic to $k$ or $k(t)$. Next, we extend this latter description to canonical algebras. More precisely, we show that a canonical algebra $\Lambda$ is tame if and only if for each generic root $\mathbf{d}$ of $\Lambda$ and each indecomposable irreducible component C of $rep(\Lambda,\mathbf{d})$, the field of rational invariants $k(C)^{GL(\mathbf{d})}$ is isomorphic to $k$ or $k(t)$. Along the way, we establish a general reduction technique for studying fields of rational invariants on Schur irreducible components of representation varieties.