Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations

Abstract

In this paper, we introduce the notion of a pre-Lie 2-algebra, which is a categorification of a pre-Lie algebra. We prove that the category of pre-Lie 2-algebras and the category of 2-term pre-Lie${}_{\infty }$-algebras are equivalent. We classify skeletal pre-Lie 2-algebras by the third cohomology of a pre-Lie algebra. We prove that crossed modules of pre-Lie algebras are in one-to-one correspondence with strict pre-Lie 2-algebras. $\mathcal{O}$-operators on Lie 2-algebras are introduced, which can be used to construct pre-Lie 2-algebras. As an application, we give solutions of 2-graded classical Yang-Baxter equations in some semidirect product Lie 2-algebras.

22 pages

References in corpus (1)

• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation

Cited by in corpus (5)

• Representations and cohomologies of relative Rota-Baxter Lie algebras and applications
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Cohomologies and crossed modules for pre-Lie Rinehart algebras
• Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras