Higher Nash blow-up local algebras of singularities and its derivation Lie algebras

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

doi:10.1016/j.jalgebra.2022.11.016

Higher Nash blow-up local algebras of singularities and its derivation Lie algebras

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

^*Corresponding author for this work

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

8 Scopus citations

Abstract

In this paper, we introduce new invariants to a singularity (V,0), i.e., the derivation Lie algebras L_k(V) of the higher Nash blow-up local algebra M_k(V). A new conjecture about the non-existence of negative weighted derivations of L_k(V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L₂(V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of L_k(V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities.

Original language	English
Pages (from-to)	165-194
Number of pages	30
Journal	Journal of Algebra
Volume	618
DOIs	https://doi.org/10.1016/j.jalgebra.2022.11.016
State	Published - 15 Mar 2023
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

Derivations
Hessian algebra
Weighted homogeneous isolated hypersurface singularity

ASJC Scopus subject areas

Algebra and Number Theory

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10.1016/j.jalgebra.2022.11.016

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title = "Higher Nash blow-up local algebras of singularities and its derivation Lie algebras",

abstract = "In this paper, we introduce new invariants to a singularity (V,0), i.e., the derivation Lie algebras Lk(V) of the higher Nash blow-up local algebra Mk(V). A new conjecture about the non-existence of negative weighted derivations of Lk(V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L2(V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of Lk(V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities.",

keywords = "Derivations, Hessian algebra, Weighted homogeneous isolated hypersurface singularity",

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

note = "Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2023",

month = mar,

day = "15",

doi = "10.1016/j.jalgebra.2022.11.016",

language = "English",

volume = "618",

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journal = "Journal of Algebra",

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TY - JOUR

T1 - Higher Nash blow-up local algebras of singularities and its derivation Lie algebras

AU - Hussain, Naveed

AU - Ma, Guorui

AU - Yau, Stephen S.T.

AU - Zuo, Huaiqing

PY - 2023/3/15

Y1 - 2023/3/15

N2 - In this paper, we introduce new invariants to a singularity (V,0), i.e., the derivation Lie algebras Lk(V) of the higher Nash blow-up local algebra Mk(V). A new conjecture about the non-existence of negative weighted derivations of Lk(V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L2(V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of Lk(V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities.

AB - In this paper, we introduce new invariants to a singularity (V,0), i.e., the derivation Lie algebras Lk(V) of the higher Nash blow-up local algebra Mk(V). A new conjecture about the non-existence of negative weighted derivations of Lk(V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L2(V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of Lk(V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities.

KW - Derivations

KW - Hessian algebra

KW - Weighted homogeneous isolated hypersurface singularity

UR - https://www.scopus.com/pages/publications/85144925915

U2 - 10.1016/j.jalgebra.2022.11.016

DO - 10.1016/j.jalgebra.2022.11.016

M3 - Article

AN - SCOPUS:85144925915

SN - 0021-8693

VL - 618

SP - 165

EP - 194

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Higher Nash blow-up local algebras of singularities and its derivation Lie algebras

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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