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Deformations of inhomogeneous simple singularities and quiver representations. (English) Zbl 1468.14005

The article gives a summary of the author’s unpublished Ph.D thesis. It is known that Dynkin diagrams can be separated in two classes: the simply laced (or homogeneous) ones \(A_k\) (\(k\geq 1\)), \(D_k\) (\(k\geq 4\)), \(E_6\), \(E_7\) and \(E_8\), and the non-simply laced (or inhomogeneous) ones \(B_k\) (\(k\geq 2\)), \(C_k\) (\(k\geq 3\)), \(F_4\) and \(G_2\).
The aim of the article is to generalise a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the homogeneous simple singularities to the inhomogeneous ones.
To a homogeneous simple 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 which allows the construction and explicit computation of the semiuniversal deformations of the inhomogeneous simple singularities.
By quotienting such maps, deformations of other simple singularities are obtained.
In some cases, the discriminants of these last deformations are computed.

MSC:

14B07 Deformations of singularities
16G20 Representations of quivers and partially ordered sets
17B22 Root systems
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