Methods and Applications of Analysis

Volume 25 (2018)

Number 4

In Memory of Professor John N. Mather: Part 2 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

Survey on derivation Lie algebras of isolated singularities

Pages: 307 – 322



Naveed Hussain (Huashang College Guangdong University of Finance and Economics, Guangzhou Guangdong, China; and Department of Mathematical Sciences, Tsinghua University, Beijing, China)


Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f : (\mathbb{C}^n, 0) \to (\mathbb{C}, 0)$. Let $L(V)$ be the Lie algebra of derivations of the moduli algebra $A(V) := \mathcal{O}_n / (f, \partial f / \partial x_1 , \dotsc , \partial f / \partial x_n)$, i.e., $L(V) = \operatorname{Der} (A(V) , A(V))$. The Lie algebra $L(V)$ is finite dimensional solvable algebra and plays an important role in singularity theory. According to Elashvili and Khimshiashvili ([15], [23]), $L(V)$ is called Yau algebra and the dimension of $L(V)$ is called Yau number. The studies of finite dimensional Lie algebras L(V) that arising from isolated singularities was started by Yau [44] and has been systematically studied by Yau, Zuo and their coauthors. Most studies of Lie algebras $L(V)$ were oriented to classify the isolated singularities. This work surveys the researches on Yau algebras $L(V)$ of isolated singularities.


fewnomial, Lie algebra, isolated singularity

2010 Mathematics Subject Classification

14B05, 32S05

Received 31 May 2018

Accepted 8 August 2018

Published 1 November 2019