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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f: (\mathbb {C}^{n}, 0)\rightarrow (\mathbb {C}, 0)$\end{document}. The k-th Yau algebra Lk(V ) is defined to be the Lie algebra of derivations of the k-th moduli algebra Ak(V):=On/(f,mkJ(f))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A^{k}(V) := \mathcal {O}_{n}/(f, m^{k}J(f))$\end{document}, where k ≥ 0, m is the maximal ideal of On\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {O}_{n}$\end{document}. I.e., Lk(V ) := Der(Ak(V ),Ak(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 Ck(V ) associated to k-th Yau algebras Lk(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 ).