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arXiv · 2606.04041

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

Yining Sun·Zeyu Hao·Ziyi Zhang·Liangyun Chen
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Abstract

Inspired by the problem of constructing ($\omega$-)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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Yining Sun, Zeyu Hao, Ziyi Zhang, Liangyun Chen. 2026-06-02. $A$-Generalized Hessian pre-Lie algebras and $A$-Generalized Yang--Baxter Equations. https://arxiv.org/abs/2606.04041

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