arXiv 1302.65342013-02math.ATmath.AGmath.QA

关于Hochschild复形的代数结构

On algebraic structures of the Hochschild complex

Hossein Abbaspour

备注：: This is the final version. Many improvements and corrections have been made.To appear in Free Loop Spaces in Geometry and Topology, IRMA Lectures in Mathematicsand Theoretical Physics, to be published by EMS-PH

摘要：

本文首先回顾了弱Poincar对偶假设下微分分次代数的Hochschild（余）同调的各种已知代数结构，如Calabi-Yau代数、导出Poincar对偶代数和闭Frobenius代数.这包括$HH^*(A,A^\vee)$或$HH^*(A,A)$上的BV-代数结构，在后一种情况下，它是$HH^*(A,A)$上的自然Gerstenhaber结构的扩展。作为例子，在证明了流形$M$的Moore闭路空间的链复形是CY-代数后，利用Burghelea-Fiedorowicz-Goodwillie定理得到了自由空间同调上的BV-结构.在第六节中，我们证明了这些BV/coBV结构确实可以定义为对称开Frobenius DG-代数的霍赫希尔德同调。特别证明了对称开Frobenius代数的霍赫希尔德同调和上同调是BV和coBV-代数。第七节给出了对称交换Frobenius代数的移位相对霍赫希尔德同调上的BV结构。根据Chas-Sullivan和Goresky-Hingston关于自由闭路空间的结果，我们预期在相对霍赫希尔德同调上存在BV-结构.第八节给出了Sullivan图在闭Frobenius DG-代数的Hochschild（co）链复形上的作用。利用同构$C^*(A ,A) \simeq C^*(A,A^\vee)$恢复了闭Froebenius代数的Tradler-Zeinalian \cite{TZ}结果.

英文摘要：

We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincar{é} duality hypothesis, such as Calabi-Yau algebras, derived Poincar{é} duality algebras and closed Frobenius algebras. This includes a BV-algebra structure on $HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of the natural Gerstenhaber structure on $HH^*(A,A)$. As an example, after proving that the chain complex of the Moore loop space of a manifold $M$ is a CY-algebra and using Burghelea-Fiedorowicz-Goodwillie theorem we obtain a BV-structure on the homology of the free space. In Sections 6 we prove that these BV/coBVstructures can be indeed defined for the Hochschild homology of a symmetric open Frobenius DG-algebras. In particular we prove that the Hochschild homology and cohomology of a symmetric open Frobenius algebra is a BV and coBV-algebra. In Section 7 we exhibit a BV structure on the shifted relative Hochschild homology of a symmetric commutative Frobenius algebra. The existence of a BV-structure on the relative Hochschild homology was expected in the light of Chas-Sullivan and Goresky-Hingston results for free loop spaces. In Section 8 we present an action of Sullivan diagrams on the Hochschild (co)chain complex of a closed Frobenius DG-algebra. This recovers Tradler-Zeinalian \cite{TZ} result for closed Froebenius algebras using the isomorphism $C^*(A ,A) \simeq C^*(A,A^\vee)$.

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摘要：

英文摘要：

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