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

Abstract

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $$f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)$$. The Yau algebra L(V) is defined to be the Lie algebra of derivations of the moduli algebra $$A(V):= {\mathcal {O}}_n/(f, \frac{\partial f}{\partial x_1},\cdots , \frac{\partial f}{\partial x_n})$$, i.e., $$L(V)=\text {Der}(A(V), A(V))$$ and plays an important role in singularity theory. It is known that L(V) is a finite dimensional Lie algebra and its dimension $$\lambda (V)$$ 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 $$A^k(V) = {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0$$, i.e., $$L^k(V)=\text {Der}(A^k(V), A^k(V))$$ and where m is the maximal ideal of $${\mathcal {O}}_n$$. In particular, it is Yau algebra when $$k=0$$. The dimension of $$L^k(V)$$ is denoted by $$\lambda ^k(V)$$. These numbers i.e., k-th Yau numbers $$\lambda ^k(V)$$, 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 $$L^1(V)$$ for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the $$\lambda ^k(V)$$ of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of $$k=1$$ for large class of singularities.