#### Vol. 7, No. 1, 2013

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On the invariant theory for tame tilted algebras

### Calin Chindris

Vol. 7 (2013), No. 1, 193–214
##### Abstract

We show that a tilted algebra $A$ is tame if and only if for each generic root $d$ of $A$ and each indecomposable irreducible component $C$ of $mod\left(A,d\right)$, the field of rational invariants $k{\left(C\right)}^{GL\left(d\right)}$ is isomorphic to $k$ or $k\left(x\right)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $d$ of $A$ and each indecomposable irreducible component $C\subseteq mod\left(A,d\right)$, the moduli space $\mathsc{ℳ}{\left(C\right)}_{\theta }^{ss}$ is either a point or just ${ℙ}^{1}$ whenever $\theta$ is an integral weight for which ${C}_{\theta }^{s}\ne \varnothing$. We furthermore show that the tameness of a tilted algebra is equivalent to the moduli space $\mathsc{ℳ}{\left(C\right)}_{\theta }^{ss}$ being smooth for each generic root $d$ of $A$, each indecomposable irreducible component $C\subseteq mod\left(A,d\right)$, and each integral weight $\theta$ for which ${C}_{\theta }^{s}\ne \varnothing$. 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 $A$ implies the tameness of $A$.

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.

##### Keywords
exceptional sequences, moduli spaces, rational invariants, tame and wild algebras, tilting
##### Mathematical Subject Classification 2010
Primary: 16G10
Secondary: 16R30, 16G60, 16G20
##### Milestones
Received: 17 September 2011
Revised: 2 January 2012
Accepted: 20 February 2012
Published: 28 March 2013
##### Authors
 Calin Chindris Department of Mathematics University of Missouri Columbia, MO 65211 United States http://www.math.missouri.edu/~chindrisc