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Linearisation, splitting property and homotopy algebras

  • Seokbong Seol
  • Kai Wang
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Abstract

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∞[1] algebras. We establish an associative analogue, characterising linearisability of A∞[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∞[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.

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Cite

BibTeXRIS
BibTeX
@misc{seol2026linearisation,
  title = {{Linearisation, splitting property and homotopy algebras}},
  author = {Seokbong Seol and Kai Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.05875},
  eprint = {2608.05875},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.05875},
}
RIS
TY  - UNPB
TI  - Linearisation, splitting property and homotopy algebras
AU  - Seokbong Seol
AU  - Kai Wang
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2608.05875
UR  - https://arxiv.org/abs/2608.05875
AB  - 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_∞[1] algebras. We establish an associative analogue, characterising linearisability of A_∞[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_∞[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.
ER  -