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

DOI: 10.2140/pjm.2020.305.189

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

$$A^{\ast }\left(V\right)={\mathcal{𝒪}}_n/\left(f,\frac{\partial f}{\partial x_1},\dots ,\frac{\partial f}{\partial x_n},Det{\left(\frac{{\partial }^{2}f}{\partial x_i\partial x_j}\right)}_{i,j=1,\dots ,n}\right),$$

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

Milestones

Received: 20 January 2019

Revised: 24 July 2019

Accepted: 6 October 2019

Published: 17 March 2020