Curved infinity-algebras and their characteristic classes
Curved infinity-algebras and their characteristic classes
复制标题
弯曲无穷代数及其特征类
DOI：
10.1112/jtopol/jts011
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发表时间：
2012
影响因子：
1.1
通讯作者：
Lazarev A
中科院分区：
文献类型：
--
作者：
Lazarev A
本文研究了Kontsevich关于A ∞-和L ∞-代数的特征类构造到曲代数的自然推广.这些定义的同源类的变体他的图同源性，允许顶点的价至少为1。我们计算这个图的同调，这是由具有奇价顶点的星星形图控制的。我们还对域上的非平凡曲循环A ∞-和L ∞-代数进行了分类，直到规范等价，并证明了这些代数本质上是至多2维的代数，只有偶数元运算。我们应用推理来计算形式向量场的李代数的同调的稳定映射。最后，我们解释了这些结果的推广到其他类型的代数，使用语言的操作。一个关键的观察是，控制曲代数的运算与那些控制酉代数的Koszul对偶密切相关。
In this paper, we study a natural extension of Kontsevich's characteristic class construction forA∞‐ andL∞‐algebras to the case of curved algebras. These define homology classes on a variant of his graph homology that allows vertices of valence at least 1. We compute this graph homology, which is governed by star‐shaped graphs with odd‐valence vertices. We also classify non‐trivially curved cyclicA∞‐ andL∞‐ algebras over a field up to gauge equivalence, and show that these are essentially reduced to algebras of dimension at most 2 with only even‐ary operations. We apply the reasoning to compute stability maps for the homology of Lie algebras of formal vector fields. Finally, we explain a generalization of these results to other types of algebras, using the language of operads. A key observation is that operads governing curved algebras are closely related to the Koszul dual of those governing unital algebras.