Higher enveloping algebras
Higher enveloping algebras
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高包络代数
作者：
Ben Knudsen
对于任意选择的维数$n$和结构群$G$，我们给出了具有包络代数的小$G$-框架$n$维圆盘上的谱李代数，并以两种互补的方式描述了这些对象。第一个描述是一个抽象的表征，通过一个普遍的映射性质，证明了高包络代数是一个附加中的左伴随的值。第二部分是对庞加莱-伯克霍夫-维特定理的推广，给出了一个关于李代数同调的具体公式。我们的构造将Koszul对偶理论和Day卷积理论结合起来，将Beilinson-Drinfeld手性代数理论的基本组合学提升到高等代数的世界。就像那个理论一样，我们的理论与构型空间的几何密切相关，并在其应用中对这些空间进行了研究。本文用它证明了位形空间的稳定同伦类型是适当的同伦不变量。
We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincare-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.