Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities

Abstract

Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras $$L_{k}(V)$$ associated to (V, 0). Its dimension is denoted as $$\lambda _{k}(V)$$ . The $$L_{k}(V)$$ is a generalization of the Yau algebra L(V) and $$L_{0}(V)=L(V)$$ . These numbers $$\lambda _{k}(V)$$ are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute $$L_1(V)$$ for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of $$\lambda _{1}(V)$$ . We also formulate a sharp upper estimate conjecture for the $$L_k(V)$$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.