#### Vol. 305, No. 1, 2020

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Generalized Cartan matrices arising from new derivation Lie algebras of isolated hypersurface singularities

### Naveed Hussain, Stephen S.-T. Yau and Huaiqing Zuo

Vol. 305 (2020), No. 1, 189–217
##### Abstract

Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f:\left({ℂ}^{n},0\right)\to \left(ℂ,0\right)$. The Yau algebra $L\left(V\right)$ is defined to be the Lie algebra of derivations of the moduli algebra $A\left(V\right):={\mathsc{𝒪}}_{n}∕\left(f,\frac{\partial f}{\partial {x}_{1}},\cdots \phantom{\rule{0.3em}{0ex}},\frac{\partial f}{\partial {x}_{n}}\right)$, i.e., $L\left(V\right)=Der\left(A\left(V\right),A\left(V\right)\right)$. It is known that $L\left(V\right)$ is finite dimensional and its dimension $\lambda \left(V\right)$ is called the Yau number. We introduced a new Lie algebra ${L}^{\ast }\left(V\right)$ which was defined to be the Lie algebra of derivations of

${A}^{\ast }\left(V\right)={\mathsc{𝒪}}_{n}/\left(f,\frac{\partial f}{\partial {x}_{1}},\dots ,\frac{\partial f}{\partial {x}_{n}},Det{\left(\frac{{\partial }^{2}f}{\partial {x}_{i}\partial {x}_{j}}\right)}_{\phantom{\rule{-0.17em}{0ex}}i,j=1,\dots ,n}\right),$

i.e., ${L}^{\ast }\left(V\right)=Der\left({A}^{\ast }\left(V\right),{A}^{\ast }\left(V\right)\right)$. ${L}^{\ast }\left(V\right)$ is finite dimensional and ${\lambda }^{\ast }\left(V\right)$ is the dimension of ${L}^{\ast }\left(V\right)$. In this paper we compute the generalized Cartan matrix $C\left(V\right)$ and other various invariants arising from the new Lie algebra ${L}^{\ast }\left(V\right)$ for simple elliptic singularities and simple hypersurface singularities. We use the generalized Cartan matrix to characterize the ADE singularities.

 Dedicated to professor Shing Tung Yau on the occasion of his 70th birthday
##### Keywords
isolated singularity, Lie algebra, generalized Cartan matrix.
##### Mathematical Subject Classification 2010
Primary: 14B05, 32S05