A characterization and solvability of quasihomogeneous singularities

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f :‫-ރ‬algebra and it depends only on the isomorphism class of the germ (V, 0).It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasihomogeneous in terms of its local k-th Hessian algebra H k ( f ).Xu and Yau proved that (V, 0) admits a quasihomogeneous structure if and only if H 0 ( f ) is isomorphic to a finite-dimensional nonnegatively graded algebra in the early 1980s.In this paper, on the one hand, we generalize Xu and Yau's result to H n+1 ( f ).On the other hand, a new series of finite-dimensional Lie algebras L k (V ) (resp.L k (V )) was defined to be the Lie algebra of derivations of the k-thand is finite-dimensional.We prove that (V, 0) is quasihomogeneous singularity if L n+1 (V ) (resp.L k (V ) := Der( A k (V ))) satisfies certain conditions.Moreover, we investigate whether the Lie algebras L k (V ) (resp.L k (V )) are solvable.