On the non-existence of negative weight derivations of the new moduli algebras of singularities

Ma, Guorui; Yau, Stephen S.-T.; Zuo, Huancong

doi:10.1016/j.jalgebra.2020.07.023

Cited by 13 publications

(5 citation statements)

References 23 publications

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“…From a geometric point of view, these associations support understanding the solvable Lie algebra (nilpotent), [9]. Since the 1980s, Yau and their research fellows have provided much work on singularities [9,[13][14][15][16][17][18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 97%

The Sharp Upper Estimate Conjecture for the Dimension δk(V) of New Derivation Lie Algebra

et al. 2022

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Hussain, Yau, and Zuo introduced the Lie algebra Lk(V) from the derivation of the local algebra Mk(V):=On/(g+J1(g)+⋯+Jk(g)). To find the dimension of a newly defined algebra is an important task in order to study its properties. In this regard, we compute the dimension of Lie algebra L5(V) and justify the sharp upper estimate conjecture for fewnomial isolated singularities. We also verify the inequality conjecture: δ5(V)<δ4(V) for a general class of singularities. Our findings are novel and an addition to the study of Lie algebra.

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Section: Introductionmentioning

confidence: 97%

The Sharp Upper Estimate Conjecture for the Dimension δk(V) of New Derivation Lie Algebra

et al. 2022

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“…These constructions are useful to study the solvable (nilpotent) Lie algebras from the geometric point of view ( [16]). Yau, Zuo, and their collaborators have been systematically studying various derivation Lie algebras of isolated hypersurface singularities (see, e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]).…”

Section: Introductionmentioning

confidence: 99%

On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

2021

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Let (V,0)={(z1,…,zn)∈Cn:f(z1,…,zn)=0} be an isolated hypersurface singularity with mult(f)=m. Let Jk(f) be the ideal generated by all k-th order partial derivatives of f. For 1≤k≤m−1, the new object Lk(V) is defined to be the Lie algebra of derivations of the new k-th local algebra Mk(V), where Mk(V):=On/((f)+J1(f)+…+Jk(f)). Its dimension is denoted as δk(V). This number δk(V) is a new numerical analytic invariant. In this article we compute L4(V) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of δ4(V). We also verify a sharp upper estimate conjecture for the δ4(V) for large class of singularities. Furthermore, we verify another inequality conjecture: δ(k+1)(V)<δk(V),k=3 for low-dimensional fewnomial singularities.

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“…For recent progress on the local k-th Hessian Lie algebra, see [35]. We proposed a new conjecture about the non-existence of negative weight derivations of the new moduli algebras of weighted homogeneous hypersurface singularities and verify this conjecture up to dimension three.…”

Section: Torelli-type Theoremsmentioning

confidence: 82%

On derivation Lie algebras of singularities and Torelli-type theorems

Yau

Zuo

2022

Notices of the International Consortium of Chinese Mathematicians

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Since Brieskorn gave the connection between simple Lie algebras and ADE singularities in 1970, it has become an important problem to establish connections between singularities and solvable (nilpotent) Lie algebras. Recently, we have constructed some new natural maps between the set of complex analytic isolated singularities and the set of finite dimensional solvable (nilpotent) Lie algebras. Furthermore, we use these new Lie algebras to obtain Torelli-type theorems of simple elliptic singularities. The main purpose of this paper is to summarize the results that we have obtained on new Lie algebras arising from isolated hypersurface singularities.

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On the non-existence of negative weight derivations of the new moduli algebras of singularities

Cited by 13 publications

References 23 publications

The Sharp Upper Estimate Conjecture for the Dimension δk(V) of New Derivation Lie Algebra

The Sharp Upper Estimate Conjecture for the Dimension δk(V) of New Derivation Lie Algebra

On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

On derivation Lie algebras of singularities and Torelli-type theorems

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