In this paper, we introduce the notion of a pre-Lie 2-algebra, which is the 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 group 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. 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.
Keywords: 2-algebras, pre-Lie$_\infty$-algebras, Lie 2-algebras, O-operators, 2-graded classical Yang-Baxter equations
2010 MSC: 17B99, 55U15
Theory and Applications of Categories, Vol. 34, 2019, No. 11, pp 269-294.
Published 2019-04-05.
TAC Home