Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function \(f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)\). The k-th Yau algebra L^k(V ) is defined to be the Lie algebra of derivations of the k-th moduli algebra \(A^{k}(V) := \mathcal {O}_{n}/(f, m^{k}J(f))\), where k ≥ 0, m is the maximal ideal of \(\mathcal {O}_{n}\). The Generalized Cartan matrix C^k(V ) is an object associated to L^k(V ). We previously proposed a conjecture that ADE singularities can be completely characterized by C^k(V ), and verified it for k = 1 in our previous work. In this paper, we continue this work and verify this conjecture for k = 2.

令 ( V , 0) 是由全纯函数\(f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)\)定义的孤立超曲面奇点。第k个 Yau 代数L ^k ( V ) 被定义为第k个模代数的推导的李代数\(A^{k}(V) := \mathcal {O}_{n}/( f, m^{k}J(f))\)，其中k ≥ 0，m是\(\mathcal {O}_{n}\)的最大理想。广义嘉当矩阵C ^k ( V ) 是与L ^k ( V）。我们之前提出了一个猜想，即 ADE 奇点可以完全由C ^k ( V )表征，并在我们之前的工作中验证了k = 1。在本文中，我们继续这项工作并验证k = 2 的这个猜想。