Cyclic operads and algebra of chord diagrams

Cyclic operads and algebra of chord diagrams
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循环运算和弦图代数

DOI：
10.1007/s00029-002-8106-2
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发表时间：
2000
期刊：
Selecta Mathematica
影响因子：
--
通讯作者：
A. Vaintrob
A. Vaintrob
中科院分区：
--
文献类型：
--
作者：
V. Hinich;A. Vaintrob

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摘要。我们证明了代数 弦图$ \cal A $是Vassiliev结不变量的相关梯度代数的对偶，它与卡西米尔李代数在某张量范畴内的全称包络代数同构（卡西米尔李代数的PROP）。这为代数的已知陈述奠定了坚实的基础 $ \cal A $“看起来和行为都像一个通用的包络代数”。我们的结果的一个直接推论是关于弦图代数的Kirillov-Duflo同构的[BGRT]猜想。我们的主要工具是从类别构造函子 $ \tt CycOp $ of cyclic对类别进行操作 $ \tt ModOp $的模块操作符，它左伴随“树部分”函子 $ {\tt ModOp} \to {\tt CycOp} $。当将这种结构应用于操作符时，和弦图的代数就产生了 $ {\tt LIE} $。这种构造的另一个例子是Kontsevich的图复合体，它对应于操作符 对于同伦李代数$ {\tt LIE}_\infty $。
Abstract. We prove that the algebra $ \cal A $ of chord diagrams, the dual to the associated graded algebra of Vassiliev knot invariants, is isomorphic to the universal enveloping algebra of a Casimir Lie algebra in a certain tensor category (the PROP for Casimir Lie algebras). This puts on a firm ground a known statement that the algebra $ \cal A $ "looks and behaves like a universal enveloping algebra". An immediate corollary of our result is the conjecture of [BGRT] on the Kirillov-Duflo isomorphism for algebras of chord diagrams.¶ Our main tool is a general construction of a functor from the category $ \tt CycOp $ of cyclic operads to the category $ \tt ModOp $ of modular operads which is left adjoint to the "tree part" functor $ {\tt ModOp} \to {\tt CycOp} $. The algebra of chord diagrams arises when this construction is applied to the operad $ {\tt LIE} $. Another example of this construction is Kontsevich's graph complex which corresponds to the operad $ {\tt LIE}_\infty $ for homotopy Lie algebras.