Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

Naveed Hussain; Stephen S.T. Yau; Huaiqing Zuo

doi:10.1007/s10468-021-10074-6

Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

Naveed Hussain
, Stephen S.T. Yau^*
, Huaiqing Zuo^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f: (ℂⁿ, 0) → (ℂ, 0). The k-th Yau algebra L^k(V) is defined to be the Lie algebra of derivations of the k-th moduli algebra A^k(V) : = O_n/ (f, m^kJ(f)) , where k ≥ 0, m is the maximal ideal of O_n. The Generalized Cartan matrix C^k(V) is an object associated to L^k(V). We previously proposed a conjecture that ADE singularities can be completely characterized by C^k(V), and verified it for k = 1 in our previous work. In this paper, we continue this work and verify this conjecture for k = 2.

Original language	English
Pages (from-to)	1461-1492
Number of pages	32
Journal	Algebras and Representation Theory
Volume	25
Issue number	6
DOIs	https://doi.org/10.1007/s10468-021-10074-6
State	Published - Dec 2022
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature B.V.

Keywords

Generalized Cartan matrix
Isolated singularity
Lie algebra

ASJC Scopus subject areas

General Mathematics

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10.1007/s10468-021-10074-6

Cite this

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title = "Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem",

abstract = "Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f: (ℂn, 0) → (ℂ, 0). The k-th Yau algebra Lk(V) is defined to be the Lie algebra of derivations of the k-th moduli algebra Ak(V) : = On/ (f, mkJ(f)) , where k ≥ 0, m is the maximal ideal of On. The Generalized Cartan matrix Ck(V) is an object associated to Lk(V). We previously proposed a conjecture that ADE singularities can be completely characterized by Ck(V), and verified it for k = 1 in our previous work. In this paper, we continue this work and verify this conjecture for k = 2.",

keywords = "Generalized Cartan matrix, Isolated singularity, Lie algebra",

author = "Naveed Hussain and Yau, \{Stephen S.T.\} and Huaiqing Zuo",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Nature B.V.",

year = "2022",

month = dec,

doi = "10.1007/s10468-021-10074-6",

language = "English",

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TY - JOUR

T1 - Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

AU - Hussain, Naveed

AU - Yau, Stephen S.T.

AU - Zuo, Huaiqing

PY - 2022/12

Y1 - 2022/12

N2 - Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f: (ℂn, 0) → (ℂ, 0). The k-th Yau algebra Lk(V) is defined to be the Lie algebra of derivations of the k-th moduli algebra Ak(V) : = On/ (f, mkJ(f)) , where k ≥ 0, m is the maximal ideal of On. The Generalized Cartan matrix Ck(V) is an object associated to Lk(V). We previously proposed a conjecture that ADE singularities can be completely characterized by Ck(V), and verified it for k = 1 in our previous work. In this paper, we continue this work and verify this conjecture for k = 2.

AB - Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function f: (ℂn, 0) → (ℂ, 0). The k-th Yau algebra Lk(V) is defined to be the Lie algebra of derivations of the k-th moduli algebra Ak(V) : = On/ (f, mkJ(f)) , where k ≥ 0, m is the maximal ideal of On. The Generalized Cartan matrix Ck(V) is an object associated to Lk(V). We previously proposed a conjecture that ADE singularities can be completely characterized by Ck(V), and verified it for k = 1 in our previous work. In this paper, we continue this work and verify this conjecture for k = 2.

KW - Generalized Cartan matrix

KW - Isolated singularity

KW - Lie algebra

UR - https://www.scopus.com/pages/publications/85108839324

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DO - 10.1007/s10468-021-10074-6

M3 - Article

AN - SCOPUS:85108839324

SN - 1386-923X

VL - 25

SP - 1461

EP - 1492

JO - Algebras and Representation Theory

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IS - 6

ER -

Generalized Cartan Matrices Associated to k-th Yau Algebras of Singularities and Characterization Theorem

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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