Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras

Jiefeng Liu

Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras

Quantum Algebra 2020-03-30 v2 Mathematical Physics math.MP

Authors: Jiefeng Liu

Abstract

We study (quasi-)twilled pre-Lie algebras and the associated $L_\infty$ -algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ( $L_\infty$ -algebras). Furthermore, we show that $\mathcal{O}$ -operators and twisted $\mathcal{O}$ -operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ( $L_\infty$ -algebras) and the twisting theory of (quasi-)twilled pre-Lie algebras. In particular, we give a construction of quasi-pre-Lie bialgebras using symplectic Lie algebras, which is parallel to that a Cartan $3$ -form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.

Keywords

pre-Lie algebras quasi-pre-Lie bialgebras twilled pre-Lie algebras L-infinity algebras Maurer-Cartan equations O-operators

Cite

@article{arxiv.2003.11926,
  title  = {Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras},
  author = {Jiefeng Liu},
  journal= {arXiv preprint arXiv:2003.11926},
  year   = {2020}
}

Comments

26 pages. The idea of this paper is similar to the paper arXiv:1902.03033 submitted by Yunhe Sheng and Rong Tang, and there is text overlap. The purpose of arXiv:1902.03033 is to study the Leibniz bialgebras

Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras

Abstract

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