论文标题
对称同源性是代表同源性
Symmetric Homology is Representation Homology
论文作者
论文摘要
对称同源性是循环同源性的自然概括，其中对称组起着环状基团的作用。在关联代数的情况下，Z. fiedorowicz \ cite {f}引入了对称同源理论，并在S. ault \ cite \ cite {au1，au2}的工作中进一步开发。在本文中，我们表明，对于在特征$ 0 $的领域定义的代数，对称同源性理论自然等同于作者（与G. khachatryan共同）在\ cite {bkr}中引入的（一维）表示同源理论。使用代表同源性的已知结果，我们明确计算基本代数的对称同源性，例如多项式代数和（DG）的通用包围代数。作为一种应用，我们证明了Ault和Fiedorowicz的两个猜想，包括\ cite {Af07}的主要猜想在多项式代数的对称同源性的拓扑解释中。
Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz \cite{F} and was further developed in the work of S. Ault \cite{Au1, Au2}. In this paper, we show that, for algebras defined over a field of characteristic $0$, the symmetric homology theory is naturally equivalent to the (one-dimensional) representation homology theory introduced by the authors (jointly with G. Khachatryan) in \cite{BKR}. Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including the main conjecture of \cite{AF07} on topological interpretation of symmetric homology of polynomial algebras.
