Leibniz 2-代数和twisted Courant 代数胚.doc

Leibniz 2-代数和 twisted Courant 代数胚 生云鹤1，刘张炬2 1 2 吉林大学数学学院，长春 130012 北京大学数学学院，北京 100871 摘要：在这篇文章中，给出了 Leibniz 代数的范畴化 -Leibniz 2-代数，它等价于 2 项的强同伦 Leibniz 代数。Omni-Lie 2-代数以及 twisted Courant 代数胚都给出具体的例子。Twisted Courant 代数胚的 Dirac 结构正是 H-twisted 李代数胚，它的截面是一个李 2-代数。 关键词：Leibniz 2-代数，omni-Lie 2-代数，twisted Courant 代数胚，第一 Pontryagin 类， Dirac 结构 中图分类号： O154 Leibniz 2-algebras and twisted Courant algebroids Yunhe Sheng1,Zhangju Liu2 1 Department of Mathematics, Jilin University, Changchun 130012 2 Department of Mathematics, Peking University, Beijing 100871 Abstract: In this paper, the categori?cation of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras is given. They reveal the algebraic structure of omni-Lie 2-algebras as well as twisted Courant algebroids by closed 4-forms. Dirac structures of twisted Courant algebroids give rise to 2-term L∞-algebras and geometric structures behind them are exactly H-twisted Lie algebroids Key words: Leibniz 2-algebra, omni-Lie 2-algebras, twisted Courant algebroid, the ?rst Pontryagin class, Dirac structure 0 Introduction Recently, people have payed more attention to higher categorical structures by reasons in both mathematics and physics. One way to provide higher categorical structures is by categorifying existing mathematical concepts. One of the simplest higher structures is a 2- vector space, which is the categori?cation of a vector space. If we further put a compatible Lie algebra structure on a 2-vector space, then we obtain a Lie 2-algebra [1, 2]. The Jacobi 基金项目： NSF of China 11101179),Doctoral Fund. of MEC (20100061120096) 作者简介： Correspondence author:Yunhe Sheng(1983-),male,Pofessor,research direction:Mathematical Physics, Poisson Geometry and Higher structures. Zhangju Liu(1955),mail, Professor, research direction:Mathematical Physics Poisson Ge- ometry. -1- identity is replaced by a natural transformation, called the Jacobiator, which also satis?es some coherence laws of its own. Recently, t