On the new k-th Yau algebras of isolated hypersurface singularities

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f: (Cⁿ, 0) → (C, 0). The Yau algebra L(V) is defined to be the Lie algebra of derivations of the moduli algebra A(V):=On/(f,∂f∂x1,⋯,∂f∂xn), i.e., L(V) = Der (A(V) , A(V)) and plays an important role in singularity theory. It is known that L(V) is a finite dimensional Lie algebra and its dimension λ(V) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras L^k(V) which are defined to be the Lie algebras of derivations of the moduli algebras A^k(V) = O_n/ (f, m^kJ(f)) , k≥ 0 , i.e., L^k(V) = Der (A^k(V) , A^k(V)) and where m is the maximal ideal of O_n. In particular, it is Yau algebra when k= 0. The dimension of L^k(V) is denoted by λ^k(V). These numbers i.e., k-th Yau numbers λ^k(V) , are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras L^k(V) and also compute the Lie algebras L¹(V) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the λ^k(V) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of k= 1 for large class of singularities.