论文标题
扭曲前代代数和Quasi-pre-lie bialgebras
Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras
论文作者
论文摘要
我们研究（Quasi-）围绕的前代代数和相关的$ l_ \ infty $ - 代数和分级分级谎言代数。然后，我们表明（Quasi）上的某些扭曲转换可以以相关差分等级lie代数（$ l_ \ infty $ -Algebras）的Maurer-Cartan方程的溶液为特征。此外，我们表明$ \ MATHCAL {O} $ - 运算符和扭曲的$ \ Mathcal {O} $ - 运营商是Maurer-Cartan方程的解决方案。作为应用，我们使用相关的差分级别的代数（$ l_ \ infty $ - 代数）和（Quasi-）扭曲的前代代代代数的扭曲理论研究（Quasi）（Quasi）lie bialgebras。特别是，我们使用Symblectic Lie代数对准式lie bialgebras进行了构造，该代数与半简单的lie代数上的cartan $ 3 $形式相似，给出了准lie bialgebra。
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.
