论文标题
VB-LIE代数和VB-Courant代数的分类
Categorification of VB-Lie algebroids and VB-Courant algebroids
论文作者
论文摘要
在本文中，首先，我们介绍了$ \ vb $ -lie $ 2 $ -Algebroid的概念，可以将其视为$ \ vb $ -lie algebroid的分类。谎言$ 2 $ -Algebroid的切线延长是$ \ vb $ -lie $ 2 $ -2 $ -Algebroid自然。我们表明，选择分裂后，在$ \ vb $ -lie $ 2 $ -Algebroids和flat superconnections之间有一对一的对应关系，而在3项矢量捆绑板上的谎言2级偏用型。然后，我们介绍了$ \ vb $ - $ \ lwx $ 2-Algebroid的概念，可以将其视为$ \ vb $ courant-courant algebroid的分类。我们表明，分裂谎言3-晶格和$ \ vb $ - $ \ lwx $ 2-Algebroids之间有一对一的对应关系。引入了$ \ vb $ -lie $ 2 $ -BialgeBroid的概念，并将$ \ vb $ -lie $ 2 $ -BialgeBroid的双重概念是$ \ vb $ - $ \ lwx $ 2-algebroid。最后，我们介绍了$ e $ - $ \ lwx $ 2-Algebroid的概念，并显示与$ \ vb $ - $ \ lwx $ 2-algebroid相关的是，有一个$ e $ e $ - $ \ lwx $ 2-algebroid在毕业的脂肪束上自然自然。通过这个结果，我们从给定的3个代数中构建了一个谎言3个代数的谎言3-代数，该谎言提供了有趣的例子，包括谎言3-代数，包括较高的弦乐谎言2-代数的较高类似物。
In this paper, first we introduce the notion of a $\VB$-Lie $2$-algebroid, which can be viewed as the categorification of a $\VB$-Lie algebroid. The tangent prolongation of a Lie $2$-algebroid is a $\VB$-Lie $2$-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between $\VB$-Lie $2$-algebroids and flat superconnections of a Lie 2-algebroid on a 3-term complex of vector bundles. Then we introduce the notion of a $\VB$-$\LWX$ 2-algebroid, which can be viewed as the categorification of a $\VB$-Courant algebroid. We show that there is a one-to-one correspondence between split Lie 3-algebroids and split $\VB$-$\LWX$ 2-algebroids. The notion of a $\VB$-Lie $2$-bialgebroid is introduced and the double of a $\VB$-Lie $2$-bialgebroid is a $\VB$-$\LWX$ 2-algebroid. Finally, we introduce the notion of an $E$-$\LWX$ 2-algebroid and show that associated to a $\VB$-$\LWX$ 2-algebroid, there is an $E$-$\LWX$ 2-algebroid structure on the graded fat bundle naturally. By this result, we give a construction of a Lie 3-algebra from a given Lie 3-algebra, which provides interesting examples of Lie 3-algebras including the higher analogue of the string Lie 2-algebra.
