3-Lie bialgebras and 3-pre-Lie algebras induced by involutive derivations

zbMATH Open1463.17006arXiv1906.06771MaRDI QIDQ5142785

Shuai Hou, Chuangchuang Kang, Ruipu Bai

Publication date: 14 January 2021

Abstract: In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A,{,,,}_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $Altime{s}_{a{d}^{*}}{A}^{*}$, which is associated to the adjoint representation $(A,ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra $Altime{s}_{a{d}^{*}}{A}^{*}$, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras are constructed.

Full work available at URL: https://arxiv.org/abs/1906.06771

zbMATH Keywords

3-Lie algebralocal cocycle 3-Lie bialgebrainvolutive derivation

Mathematics Subject Classification ID

Automorphisms, derivations, other operators for Lie algebras and super algebras (17B40) Lie bialgebras; Lie coalgebras (17B62) Ternary compositions (17A40)

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