On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities

Let (V,0) be an isolated hypersurface singularity defined by the holomorphic function f: (ℂⁿ, 0) → (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) : = O_n/ (f, m^kJ(f)) , where k ≥ 0, m is the maximal ideal of O_n. I.e., L^k(V) := Der(A^k(V),A^k(V)). These new series of derivation Lie algebras are quite subtle invariants since they capture enough information about the complexity of singularities. In this paper we formulate a conjecture for the complete characterization of ADE singularities by using generalized Cartan matrix C^k(V) associated to k-th Yau algebras L^k(V), k ≥ 1. In this paper, we provide evidence for the conjecture and give a new complete characterization for ADE singularities. Furthermore, we compute their other various invariants that arising from the 1-st Yau algebra L¹(V).