A-Generalized Hessian pre-Lie algebras and A-Generalized Yang--Baxter Equations

Abstract

Inspired by the problem of constructing (ω-)pre-Lie algebra structures on the dual space of a pre-Lie algebra, we introduce the A-generalized Yang--Baxter equation as a generalization of the Yang--Baxter equation of pre-Lie algebras. We study its symmetric solutions through A-generalized Hessian pre-Lie algebras and split these solutions into two types. We further consider factorizable solutions of this equation and establish a one-to-one correspondence between them and generalized quadratic Rota--Baxter pre-Lie algebras of nonzero weight. By studying the structure of these algebras, we find all factorizable solutions. Finally, we study the structure of A-generalized Hessian pre-Lie algebras. In particular, we obtain a structural description via central and double extensions and classify low-dimensional non-trivial A-generalized Hessian pre-Lie algebras.

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BibTeX RIS

BibTeX

@misc{sun2026ageneralized,
  title = {{$A$-Generalized Hessian pre-Lie algebras and $A$-Generalized Yang--Baxter Equations}},
  author = {Yining Sun and Zeyu Hao and Ziyi Zhang and Liangyun Chen},
  howpublished = {arXiv},
  year = {2026},
  month = jun,
  doi = {10.48550/arxiv.2606.04041},
  eprint = {2606.04041},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04041},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - A-Generalized Hessian pre-Lie algebras and A-Generalized Yang--Baxter Equations
AU  - Yining Sun
AU  - Zeyu Hao
AU  - Ziyi Zhang
AU  - Liangyun Chen
PY  - 2026
DA  - 2026/06/02
PB  - arXiv
DO  - 10.48550/arxiv.2606.04041
UR  - https://arxiv.org/abs/2606.04041
AB  - Inspired by the problem of constructing (ω-)pre-Lie algebra structures on the dual space of a pre-Lie algebra, we introduce the A-generalized Yang--Baxter equation as a generalization of the Yang--Baxter equation of pre-Lie algebras. We study its symmetric solutions through A-generalized Hessian pre-Lie algebras and split these solutions into two types. We further consider factorizable solutions of this equation and establish a one-to-one correspondence between them and generalized quadratic Rota--Baxter pre-Lie algebras of nonzero weight. By studying the structure of these algebras, we find all factorizable solutions. Finally, we study the structure of A-generalized Hessian pre-Lie algebras. In particular, we obtain a structural description via central and double extensions and classify low-dimensional non-trivial A-generalized Hessian pre-Lie algebras.
ER  -

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