Kleinian singularities and algebras generated by elements that have given spectra and satisfy a~scalar sum relation

Anton Mellit

Algebra and discrete mathematics, no. 3 (2004), pp. 89-110.

Résumé

We consider the algebras $e_i{\mathrm{\Pi }}^{\lambda }(Q)e_i$, where ${\mathrm{\Pi }}^{\lambda }(Q)$ is the deformed preprojective algebra of weight $\lambda$ and $i$ is some vertex of $Q$, in the case where $Q$ is an extended Dynkin diagram and $\lambda$ lies on the hyperplane orthogonal to the minimal positive imaginary root $\delta$. We prove that the center of $e_i{\mathrm{\Pi }}^{\lambda }(Q)e_i$ is isomorphic to ${\mathcal{O}}^{\lambda }(Q)$, a deformation of the coordinate ring of the Kleinian singularity that corresponds to $Q$. We also find a minimal $k$ for which a standard identity of degree $k$ holds in $e_i{\mathrm{\Pi }}^{\lambda }(Q)e_i$. We prove that the algebras $A_{P_1,\dots ,P_n;\mu }=\mathbb{C}⟨x_1,\dots ,x_n|P_i(x_i)=0,\sum _{i=1}^n{x}_i=\mu e⟩$ make a special case of the algebras $e_c{\mathrm{\Pi }}^{\lambda }(Q)e_c$ for star-like quivers $Q$ with the origin $c$.