The k-th Yau algebra <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^k(V), k\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> defined to be the Lie algebra of derivations of the k-th moduli algebras <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A^k(V)= \mathcal {O}_n/(f, m^k J(f)), k\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi mathvariant="script">O</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <msup> <mi>m</mi> <mi>k</mi> </msup> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and where m is the maximal ideal of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}_n$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>. The k-th Milnor number and k-th Tjurina number are defined as follows: <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu ^k:= \text {dim}\mathcal {O}_n/( {m}^kJ(f)), \;\tau ^k:=\text {dim} \mathcal {O}_n/(f, {m}^kJ(f)).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>μ</mi> <mi>k</mi> </msup> <mo>:</mo> <mo>=</mo> <mtext>dim</mtext> <msub> <mi mathvariant="script">O</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi>m</mi> </mrow> <mi>k</mi> </msup> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.277778em" /> <msup> <mi>τ</mi> <mi>k</mi> </msup> <mo>:</mo> <mo>=</mo> <mtext>dim</mtext> <msub> <mi mathvariant="script">O</mi> <mi>n</mi> </msub> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <msup> <mrow> <mi>m</mi> </mrow> <mi>k</mi> </msup> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The dimension of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^k(V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denoted as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda ^k(V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>λ</mi> <mi>k</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we propose two questions for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu ^k, \tau ^k$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>μ</mi> <mi>k</mi> </msup> <mo>,</mo> <msup> <mi>τ</mi> <mi>k</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\lambda ^k$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>λ</mi> <mi>k</mi> </msup> </math></EquationSource> </InlineEquation> (see Conjecture <InternalRef RefID="FPar10">1.10</InternalRef> and Question <InternalRef RefID="FPar8">1.8</InternalRef>) and answer these two questions for simple singularities when k is small. We also verified the sharp upper estimate Conjecture <InternalRef RefID="FPar3">1.3</InternalRef> and the inequality Conjecture <InternalRef RefID="FPar1">1.1</InternalRef> for binomial singularities.

New Invariants of Singularities in Terms of k-TH Moduli and Derivation Lie Algebras

期刊论文
发表日期：

2026年1月27日

查看全文

Naveed Hussain,
Huaiqing Zuo

摘要

The k-th Yau algebra $L^k(V), k\ge 0$ $L^{k} (V), k \geq 0$ defined to be the Lie algebra of derivations of the k-th moduli algebras $A^k(V)= \mathcal {O}_n/(f, m^k J(f)), k\ge 0$ $A^{k} (V) = O_{n} / (f, m^{k} J (f)), k \geq 0$ and where m is the maximal ideal of $\mathcal {O}_n$ $O_{n}$ . The k-th Milnor number and k-th Tjurina number are defined as follows: $\mu ^k:= \text {dim}\mathcal {O}_n/( {m}^kJ(f)), \;\tau ^k:=\text {dim} \mathcal {O}_n/(f, {m}^kJ(f)).$ $μ^{k} : = dim O_{n} / (m^{k} J (f)), τ^{k} : = dim O_{n} / (f, m^{k} J (f)) .$ The dimension of $L^k(V)$ $L^{k} (V)$ denoted as $\lambda ^k(V)$ $λ^{k} (V)$ . In this paper, we propose two questions for $\mu ^k, \tau ^k$ $μ^{k}, τ^{k}$ and $\lambda ^k$ $λ^{k}$ (see Conjecture 1.10 and Question 1.8) and answer these two questions for simple singularities when k is small. We also verified the sharp upper estimate Conjecture 1.3 and the inequality Conjecture 1.1 for binomial singularities.