Cyclic Operads and Cyclic Homology

The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the non-commutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characterized by the presence of several algebraic operations. Such structures appear, for example, in homotopy theory [18], [3] and topological field theory [9]. In this paper, we extend the formalism of cyclic homology to this more general framework. This extension is only possible under certain conditions which are best explained using the concept of an operad. In this approach to universal algebra, an algebraic structure is described by giving, for each n ≥ 0, the space P(n) of all n-ary expressions which can be formed from the operations in the given algebraic structure, modulo the universally valid identities. Permuting the arguments of the expressions gives an action of the symmetric group Sn on P(n). The sequence P = {P(n)} of these Sn-modules, together with the natural composition structure on them, is the operad describing our class of algebras. In order to define cyclic homology for algebras over an operad P, it is necessary that P is what we call a cyclic operad : this means that the action of Sn on P(n) extends to an action of Sn+1 in a way compatible with compositions (see Section 2). Cyclic operads are a natural generalization of associative algebras with involution (see (2.2)). For associative algebras over a field of characteristic 0, it is a result of Feigin and Tsygan [6] that cyclic homology is the non-abelian derived functor of the functor A 7→ A/[A,A], the target of the universal trace on A. The notion of a trace does not make sense for more general algebras. However, for an associative algebra A with unit, there is a bijection between traces T and invariant bilinear forms B on A, obtained by setting T (x) = B(1, x). It turns out that the structure of a cyclic operad on P is precisely the data needed to speak about invariant bilinear forms on algebras over an operad P. The cyclic homology of an algebra over a cyclic operad P is now defined as the non-abelian derived functor of the target of the universal invariant bilinear form. In making this definition, we were strongly influenced by ideas of M. Kontsevich: he emphasized that invariant bilinear forms are more fundamental than traces [15]. Furthermore, he constructed an action of Sn+1 on the space Lie(n) of n-ary operations on Lie algebras. We construct an explicit complex CA(P, A) calculating the cyclic homology. There is a long exact sequence which involves not only the cyclic homology HA•(P, A) but two other functors HB•(P, A) and HC•(P, A):