论文标题
适当的扭曲
Twisting of properads
论文作者
论文摘要
我们研究了（强均同型）在代数的dg proporads p类中，研究托马斯·威尔瓦赫（Thomas Willwacher）的扭曲的内压函数TW。事实证明，如果p在lie bialgebras的适当级别下是适当的，那么相关的扭曲的poserad tw（p）通常会在准lie bialgebras（而不是在lieb下）下成为适当的。该结果意味着，任何环状同谱联想代数的循环共同体总体上都具有准lie bialgebra的诱导结构。我们表明，扭曲的Porporad TW（Lieb）的共同体学高度不平凡 - 它包含了最近在长结理论和代数曲线模量空间理论的背景下引入和研究所谓的Heired Graph复合物的同谋。 使用来自Prop类别的多差异函数到Operads类别，我们介绍了强烈同型lie Bialgebra的Maurer-Cartan元素的概念，并使用它来构建一个新的扭曲的内式函数TW，类别DG Prop（ERAD）S p（ERAD）s p在Holieb下，Liemal liebal liebal of Lieblieb的最小值。我们证明TW（Holieb）是lieb的准同构，并建立了与三角形谎言bialgebras的同型理论的关系。事实证明，DG谎言代数控制地图的变形，从Lieb到P的PACT在TW（P）上是通过派生的。在一些重要的例子中，该DG谎言代数具有丰富而有趣的共同体学（例如，Grothendieck-Teichmueller Lie代数）。 最后，我们介绍了钻石版本的the Endofunctor TW，该版本在涉及的（完全同质）的DG Poyrad类别中起作用，并讨论其在弦拓扑中的应用。
We study Thomas Willwacher's twisting endofunctor tw in the category of dg properads P under the operad of (strongly homotopy) Lie algebras. It is proven that if P is a properad under properad Lieb of Lie bialgebras , then the associated twisted properad tw(P) becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad tw(Lieb) is highly non-trivial -- it contains the cohomology of the so called haired graph complex introduced and studied recently in the context of the theory of long knots and the theory of moduli spaces of algebraic curves. Using a polydifferential functor from the category of props to the category of operads, we introduce the notion of a Maurer-Cartan element of a strongly homotopy Lie bialgebra, and use it to construct a new twisting endofunctor Tw in the category dg prop(erad)s P under HoLieb, the minimal resolution of Lieb. We prove that Tw(Holieb) is quasi-isomorphic to Lieb, and establish its relation to the homotopy theory of triangular Lie bialgebras. It is proven that the dg Lie algebra controlling deformations of the map from Lieb to P acts on Tw(P) by derivations. In some important examples this dg Lie algebra has a rich and interesting cohomology (containing, for example, the Grothendieck-Teichmueller Lie algebra). Finally, we introduce a diamond version of the endofunctor Tw which works in the category of dg properads under involutive (strongly homotopy) Lie bialgebras, and discuss its applications in string topology.
