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arXiv · 2503.14165

Pre-Lie 2-bialgebras and 2-grade classical Yang-Baxter equations

Jiefeng Liu·Tongtong Yue·Qi Wang
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Abstract

We introduce a notion of a para-Kähler strict Lie 2-algebra, which can be viewed as a categorification of a para-Kähler Lie algebra. In order to study para-Kähler strict Lie 2-algebra in terms of strict pre-Lie 2-algebras, we introduce the Manin triples, matched pairs and bialgebra theory for strict pre-Lie 2-algebras and the equivalent relationships between them are also established. By means of the cohomology theory of Lie 2-algebras, we study the coboundary strict pre-Lie 2-algebras and introduce 2-graded classical Yang-Baxter equations in strict pre-Lie 2-algebras. The solutions of the 2-graded classical Yang-Baxter equations are useful to construct strict pre-Lie 2-algebras and para-Kähler strict Lie 2-algebras. In particular, there is a natural construction of strict pre-Lie 2-bialgebras from the strict pre-Lie 2-algebras.

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Jiefeng Liu, Tongtong Yue, Qi Wang. 2025-03-18. Pre-Lie 2-bialgebras and 2-grade classical Yang-Baxter equations. https://arxiv.org/abs/2503.14165

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