Vol. 305, No. 1, 2020

Download this article
Download this article For screen
For printing
Recent Issues
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Vol. 299: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Editorial Board
Subscriptions
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Author Index
To Appear
 
Other MSP Journals
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 : (n,0) (,0). The Yau algebra L(V ) is defined to be the Lie algebra of derivations of the moduli algebra A(V ) := 𝒪n(f, f x1 ,, f xn), i.e., L(V ) = Der(A(V ),A(V )). It is known that L(V ) is finite dimensional and its dimension λ(V ) is called the Yau number. We introduced a new Lie algebra L(V ) which was defined to be the Lie algebra of derivations of

A(V ) = 𝒪 n(f, f x1,, f xn,Det( 2f xixj)i,j=1,,n),

i.e., L(V ) = Der(A(V ),A(V )). L(V ) is finite dimensional and λ(V ) is the dimension of L(V ). In this paper we compute the generalized Cartan matrix C(V ) and other various invariants arising from the new Lie algebra L(V ) 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
Authors
Naveed Hussain
Department of Mathematical Sciences
Tsinghua University
Beijing
China
School of Data Sciences
Huashang College Guangdong University of Finance and Economics
Guangzhou Guangdong
China
Stephen S.-T. Yau
Department of Mathematical Sciences
Tsinghua University
Beijing
China
Huaiqing Zuo
Department of Mathematical Sciences
Tsinghua University
Beijing
China