On the invariant theory for tame tilted algebras. (English) Zbl 1297.16016

Chindris, Calin

Algebra Number Theory 7, No. 1, 193-214 (2013).

A finite dimensional associative \(k\)-algebra \(A\) (the base field \(k\) is assumed to be algebraically closed and have characteristic zero in the paper) is tilted if it is the endomorphism algebra of a multiplicity-free tilting module over a connected path algebra of an acyclic quiver. The present paper characterizes tameness of a tilted algebra in terms of the invariant theory related to \(\text{mod}(A,d)\), the affine varieties of \(d\)-dimensional \(A\)-modules for various dimension vectors \(d\).
In particular, the following are equivalent for a tilted algebra \(A\): (1) \(A\) is tame; (2) for each generic root \(d\) and each irreducible indecomposable component \(C\) of \(\text{mod}(A,d)\), the field \(k(C)^{\text{GL}(d)}\) of rational invariants on \(C\) is isomorphic to \(k\) or \(k(x)\) (the field of fractions of the univariate polynomial ring); (3) for each generic root \(d\) and each irreducible indecomposable component \(C\) of \(\text{mod}(A,d)\), the moduli space \(M(C)_\theta^{ss}\) of \(\theta\)-semistable \(A\)-modules is either a point or the projective line, where \(\theta\) is an integral weight for which \(C\) contains a \(\theta\)-stable point; (4) \(M(C)_\theta^{ss}\) is smooth for all \(d\), \(C\) and \(\theta\) as in (3). – Along the way some useful technical reductions are proved about the behaviour of these moduli spaces with respect to tilting functors and the \(\theta\)-stable decomposition.

Reviewer: Matyas Domokos (Budapest)

MSC:

16G10	Representations of associative Artinian rings
16G60	Representation type (finite, tame, wild, etc.) of associative algebras
16G20	Representations of quivers and partially ordered sets
14L24	Geometric invariant theory
16R30	Trace rings and invariant theory (associative rings and algebras)

Keywords:

exceptional sequences; moduli spaces; bound quivers; rational invariants; tame algebras; wild algebras; tilted algebras; tilting modules; connected path algebras