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 $A\left(V\right):={\mathcal{𝒪}}_n∕\left(f,\frac{\partial f}{\partial x_1},\cdots ,\frac{\partial f}{\partial x_n}\right)$, 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

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.

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Mathematical Subject Classification 2010

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