Let $K$ be a field and $Q$ a finite quiver.
For a weight $\lambda \in K^{|Q_0|}$, the deformed preprojective
algebra $\Pi^{\lambda}$ was introduced by Crawley-Boevey and Holland
to study deformations of Kleinian singularities.
If $\lambda = 0$, then $\Pi^{0}$ is the preprojective algebra
introduced by Gelfand-Ponomarev, and appears in many areas of
mathematics.
Among interesting properties of $\Pi^{0}$, the classification of
tilting ideals of $\Pi^{0}$, shown by Buan-Iyama-Reiten-Scott, is
fundamental and important.
They constructed a bijection between the set of tilting ideals of
$\Pi^{0}$ and the Coxeter group $W_Q$ of $Q$.
In this talk, when $Q$ is non-Dynkin, we see that $\Pi^{\lambda}$ is a
$2$-Calabi-Yau algebra, and show that there exists a bijection between
tilting ideals and a Coxeter group.
However $W_Q$ does not appear, since $\Pi^{\lambda}$ is not necessarily basic.
Instead of $W_Q$, we consider the Ext-quiver of rigid simple modules,
and use its Coxeter group.
When $Q$ is an extended Dynkin quiver, we see that the Ext-quiver is
finite and this has information of singularities of a
representation space of semisimple modules.
This is joint work with William Crawley-Boevey.