Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities

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

doi:10.1007/s00209-020-02688-1

Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities

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

^*Corresponding author for this work

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

21 Scopus citations

Abstract

Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras L_k(V) associated to (V, 0). Its dimension is denoted as λ_k(V). The L_k(V) is a generalization of the Yau algebra L(V) and L(V) = L(V). These numbers λ_k(V) are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute L₁(V) for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of λ₁(V). We also formulate a sharp upper estimate conjecture for the L_k(V) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.

Original language	English
Pages (from-to)	1813-1829
Number of pages	17
Journal	Mathematische Zeitschrift
Volume	298
Issue number	3-4
DOIs	https://doi.org/10.1007/s00209-020-02688-1
State	Published - Aug 2021
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

Keywords

Isolated hypersurface singularity
Lie algebra
Moduli algebra

ASJC Scopus subject areas

General Mathematics

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10.1007/s00209-020-02688-1

Cite this

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title = "Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities",

abstract = "Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras Lk(V) associated to (V, 0). Its dimension is denoted as λk(V). The Lk(V) is a generalization of the Yau algebra L(V) and L(V) = L(V). These numbers λk(V) are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute L1(V) for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of λ1(V). We also formulate a sharp upper estimate conjecture for the Lk(V) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.",

keywords = "Isolated hypersurface singularity, Lie algebra, Moduli 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-Verlag GmbH, DE part of Springer Nature.",

year = "2021",

month = aug,

doi = "10.1007/s00209-020-02688-1",

language = "English",

volume = "298",

pages = "1813--1829",

journal = "Mathematische Zeitschrift",

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TY - JOUR

T1 - Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities

AU - Hussain, Naveed

AU - Yau, Stephen S.T.

AU - Zuo, Huaiqing

PY - 2021/8

Y1 - 2021/8

N2 - Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras Lk(V) associated to (V, 0). Its dimension is denoted as λk(V). The Lk(V) is a generalization of the Yau algebra L(V) and L(V) = L(V). These numbers λk(V) are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute L1(V) for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of λ1(V). We also formulate a sharp upper estimate conjecture for the Lk(V) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.

AB - Let (V, 0) be an isolated hypersurface singularity. We introduce a series of new derivation Lie algebras Lk(V) associated to (V, 0). Its dimension is denoted as λk(V). The Lk(V) is a generalization of the Yau algebra L(V) and L(V) = L(V). These numbers λk(V) are new numerical analytic invariants of an isolated hypersurface singularity. In this article we compute L1(V) for fewnomial isolated singularities (Binomial, Trinomial) and obtain the formulas of λ1(V). We also formulate a sharp upper estimate conjecture for the Lk(V) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture and prove it for binomial and trinomial singularities.

KW - Isolated hypersurface singularity

KW - Lie algebra

KW - Moduli algebra

UR - https://www.scopus.com/pages/publications/85099083786

U2 - 10.1007/s00209-020-02688-1

DO - 10.1007/s00209-020-02688-1

M3 - Article

AN - SCOPUS:85099083786

SN - 0025-5874

VL - 298

SP - 1813

EP - 1829

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 3-4

ER -

Inequality conjectures on derivations of local k-th Hessain algebras associated to isolated hypersurface singularities

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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