Algebras whose Tits form accepts a maximal omnipresent root

TL;DR

This paper investigates the roots of the Tits form for certain finite-dimensional algebras, establishing a correspondence between maximal omnipresent roots and indecomposable modules, and characterizing the algebra's type in specific cases.

Contribution

It proves that maximal omnipresent roots correspond to indecomposable modules and characterizes strongly simply connected algebras as tame of tilted type when such roots exist.

Findings

Maximal omnipresent roots correspond to indecomposable modules.

Existence of such roots implies the algebra is tame of tilted type if strongly simply connected.

The Tits form's roots provide insight into the algebra's representation type.

Abstract

Let k be an algebraically closed field and A be a finite-dimensional associative basic k-algebra of the form A=kQ/I where Q is a quiver without oriented cycles or double arrows and I is an admissible ideal of kQ. We consider roots of the Tits form q_A, in particular in case q_A is weakly non-negative. We prove that for any maximal omnipresent root v of q_A, there exists an indecomposable A-module X such that v is the dimension vector of X. Moreover, if A is strongly simply connected, the existence of a maximal omnipresent root of q_A implies that A is tame of tilted type.