Pre-Lie 2-bialgebras and 2-grade classical Yang-Baxter equations
Jiefeng Liu, Tongtong Yue, Qi Wang
TL;DR
The paper advances the categorification of para-Kähler and pre-Lie bialgebra theories by developing para-Kähler strict Lie 2-algebras and strict pre-Lie 2-bialgebras. It builds a cohesive framework connecting symplectic and para-Kähler structures to pre-Lie 2-algebras through $\mathcal{O}$-operators, Rota-Baxter operators, and derivations, and elucidates their representations and cohomologies. A central achievement is establishing the equivalence among para-Kähler strict Lie 2-algebras, Manin triples, and matched pairs of strict pre-Lie 2-algebras, and proving that strict pre-Lie 2-bialgebras correspond to such matched pairs. The coboundary theory leads to the notion of the $2$-graded classical Yang–Baxter equation, giving operator formulations and constructive methods for generating strict pre-Lie 2-bialgebras from solutions, with implications for higher-geometric and higher-algebraic phase-space constructions.
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.