Quasi-triangular pre-Lie bialgebras, factorizable pre-Lie bialgebras and Rota-Baxter pre-Lie algebras

In this paper, first we introduce the notions of quasi-triangular pre-Lie bialgebras and factorizable pre-Lie bialgebras. A factorizable pre-Lie bialgebra leads to a factorization of the underlying pre-Lie algebra. We show that the symplectic double of a pre-Lie bialgebra naturally enjoys a factorizable pre-Lie bialgebra structure. Then we give the Rota-Baxter characterization of factorizable pre-Lie bialgebras. More precisely, we introduce the notion of quadratic Rota-Baxter pre-Lie algebras and show that there is a one-to-one correspondence between factorizable pre-Lie bialgebras and quadratic Rota-Baxter pre-Lie algebras. Finally, we develop the theories of matched pairs, bialgebras and Manin triples of Rota-Baxter pre-Lie algebras. In particular, a factorizable pre-Lie bialgebra gives rise to a Rota-Baxter pre-Lie bialgebra, and conversely a Rota-Baxter pre-Lie bialgebra gives rise to a factorizable pre-Lie bialgebra structure on the double space. (c) 2024 Elsevier B.V. All rights reserved.