Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 3

Classification of the nilradical of $k$-th Yau algebras arising from singularities

Pages: 835 – 862

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n3.a2

Authors

Naveed Hussain (School of Data Sciences, Guangzhou Huashang College, Guangzhou, China; and Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Punjab, Pakistan)

Stephen S.-T. Yau (Department of Mathematical Sciences, Tsinghua University, Beijing, China; and Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou, China)

Huaiqing Zuo (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

Every Lie algebra is a semi-direct product of semisimple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Classification of nilpotent Lie algebras with dimension up to $7$ is known, but not for dimension greater than $7$. Therefore it is important to establish connections between theory of singularities and theory of nilpotent Lie algebras. Let $(V, 0)$ be an isolated hypersurface singularity defined by the holomorphic function $f : (\mathbb{C}^n, 0) \to (\mathbb{C}, 0)$. The $k$‑th Yau algebras $L^k (V), k \geq 0$ were introduced by the authors. It was defined to be the Lie algebra of derivations of the $k$‑th moduli algebra $A^k (V)$. These Lie algebras are solvable in general and play an important role in the study of singularities. In this paper, we investigate the new connection between the nilpotent Lie algebras of dimension less than or equal to $7$ and the nilradical of $k$‑th Yau algebras.

Keywords

derivation, nilpotent Lie algebra, isolated singularity, $k$-th Yau algebras

2010 Mathematics Subject Classification

Primary 14B05. Secondary 17B66, 32S05.

Full Text (PDF format)

Both Yau and Zuo are supported by NSFC Grant 11961141005. Zuo is supported by Tsinghua University Initiative Scientific Research Program and NSFC Grant 11771231. Yau is supported by Tsinghua University start-up fund and Tsinghua University Education Foundation fund (042202008).

Received 4 November 2021

Received revised 27 December 2021

Accepted 30 December 2021

Published 24 July 2022