On the invariant theory for tame tilted algebras

Abstract

We show that a tilted algebra [math] is tame if and only if for each generic root [math] of [math] and each indecomposable irreducible component [math] of [math] , the field of rational invariants [math] is isomorphic to [math] or [math] . Next, we show that the tame tilted algebras are precisely those tilted algebras [math] with the property that for each generic root [math] of [math] and each indecomposable irreducible component [math] , the moduli space [math] is either a point or just [math] whenever [math] is an integral weight for which [math] . We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space [math] being smooth for each generic root [math] of [math] , each indecomposable irreducible component [math] , and each integral weight [math] for which [math] . As a consequence of this latter description, we show that the smoothness of the various moduli spaces of modules for a strongly simply connected algebra [math] implies the tameness of [math] .\n¶ Along the way, we explain how moduli spaces of modules for finite-dimensional algebras behave with respect to tilting functors, and to theta-stable decompositions.