Abstract: 
A relation between ADE singularities and Lie algebras is established by the famous theory by GrothendieckBrieskorn: For a simple Lie algebra of type $ADE$ the adjoint orbit of codimension two in the nilpotent variety, which is the inverse image of the origin by the adjoint quotient map, has an ADE singularity of the same type as the Lie algebra. The simplest singularities of surfaces following $ADE$ singularities are the simple elliptic singularities defined by K. Saito and named as $\tilde E_6$, $\tilde E_7$, $\tilde E_8$ and $\tilde D_5$. Slodowy and Helmke associated an infinite dimensional Lie algebra with the singularities of type $\tilde D_5$. In my talk, we will briefly explain the results on ADEsingularities, extend it to the simple elliptic singularities of type $\tilde D_5$ and associate a finite dimensional Lie algebra to that family. This is a joint work with K.Nakamoto.
