Deformations of Inhomogeneous Simple Singularities and Quiver Representations

Abstract

This article is a summary of the author's unpublished Ph.D thesis. Its purpose is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of type $A_r$, $D_r$, $E_6$, $E_7$ and $E_8$ to the inhomogeneous simple singularities of type $B_r$, $C_r$, $F_4$ and $G_2$. To a simple homogeneous singularity, one can associate the representation space of a particular quiver. This space is endowed with an action of the symmetry group of the Dynkin diagram associated to the simple singularity. From this we will construct and compute explicitly the semiuniversal deformations of the inhomogeneous simple singularities. By quotienting such maps, we obtain deformations of other simple singularities. In some cases, the discriminants of these last deformations will be computed.