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Categorification of pre-Lie Algebras and solutions of 2-graded classical
Yang-Baxter equations

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Yunhe Sheng

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.

http://www.tac.mta.ca/tac/volumes/34/11/34-11.pdf

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