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.