线性化、分裂性质与同伦代数
Linearisation, splitting property and homotopy algebras
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中文总结 AI 辅助
本文在分次余代数框架下，通过显式递归构造证明形式向量场、形式流形态射、$L_\text{\infty}[1]$代数及$A_\text{\infty}[1]$代数的线性化可由对应分裂性质刻画，还简化了相关定理的证明。
中文摘要 AI 辅助
本文在分次余代数框架下研究向量场的形式线性化问题。我们证明形式向量场可线性化当且仅当满足分裂性质，通过提供线性化同构的显式递归构造实现该结论。该准则为Basto-Gonçalves关于可容许共振向量场的定理提供了简化证明。我们还建立了形式流形态射的对应分裂准则，证明态射可线性化当且仅当满足该性质。此外，我们得到Bandiera关于可线性化（等价于同伦交换）$L_\infty[1]$代数的刻画的初等显式证明。最后，我们将该框架扩展至$A_\infty[1]$代数，证明其线性化可由类似分裂性质刻画。
英文摘要
We study the linearisation of formal vector fields and homotopy algebras using graded coalgebras. We prove that a formal vector field is linearisable if and only if it satisfies a splitting property, and we give an explicit recursive construction of a linearising coordinate transformation. This provides a direct coalgebraic proof of Bandiera's characterisation of linearisable, equivalently homotopy abelian, $L_{\infty}[1]$ algebras. We establish an associative analogue, characterising linearisability of $A_{\infty}[1]$ algebras by a splitting property for one-sided Hochschild cochains. We also examine the corresponding splitting property for standard Hochschild cochains and show that it is equivalent to $A_{\infty}[1]$ commutativity in the sense of Briggs and Gélinas. Finally, we establish a corresponding criterion for formal endomorphisms and give sufficient conditions for splitting that recover Basto-Gonçalves' linearisation theorem for admissible formal vector fields.
发表机构
- Korea Institute for Advanced Study（韩国高等研究院）
- University of Science and Technology of China（中国科学技术大学）
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