process

Equivariant cyclic homology for quantum groups 171that this yields indeed a complex. Remark that both entries in the above Hom-complexare only paracomplexes.Let us consider the special case that H D D.G/ is the smooth group algebraof a locally compact group G. In this situation the definition of HPH reduces tothe definition of HPG given in [25]. This is easily seen using the Takesaki–Takaiisomorphism obtained in Proposition 3.7 and the results from [27].As in the group case HPH is a bifunctor, contravariant in the first variable andcovariant in the second variable. We define HP H .A/ D HP H .C;A/to be the equivariantperiodic cyclic homology of A and HPH .A/ D HP H .A; C/ to be equivariantperiodic cyclic cohomology. There is a natural associative productHP Hi.A; B/ HP Hj.B; C / ! HPHiCj.A; C /;.x; y/ 7! x yinduced by the composition of maps. Every equivariant homomorphism f W A ! Bdefines an element in HP0 H .A; B/ denoted by Œf . The element Œid 2 HP 0 H .A; A/is denoted 1 or 1 A . An element x 2 HP H .A; B/ is called invertible if there exists anelement y 2 HP H .B; A/ such that x y D 1 A and y x D 1 B . An invertible elementof degree zero is called an HP H -equivalence. Such an element induces isomorphismsHP H .A; D/ Š HP H .B; D/ and HP H .D; A/ Š HP H .D; B/ for all H -algebras D.8 Homotopy invariance, stability and excisionIn this section we show that equivariant periodic cyclic homology is homotopy invariant,stable and satisfies excision in both variables. Since the arguments carry over from thegroup case with minor modifications most of the proofs will only be sketched. Moredetails can be found in [25].We begin with homotopy invariance. Let B be a pro-H -algebra and consider theFréchet algebra C 1 Œ0; 1 of smooth functions on the interval Œ0; 1. We denote byBŒ0; 1 the pro-H -algebra B y˝ C 1 Œ0; 1 where the action on C 1 Œ0; 1 is trivial. A(smooth) equivariant homotopy is an equivariant homomorphism ˆW A ! BŒ0; 1 ofH -algebras. Evaluation at the point t 2 Œ0; 1 yields an equivariant homomorphismˆt W A ! B. Two equivariant homomorphisms from A to B are called equivariantlyhomotopic if they can be connected by an equivariant homotopy.Theorem 8.1 (Homotopy invariance). Let A and B be H -algebras and let ˆW A !BŒ0; 1 be a smooth equivariant homotopy. Then the elements Œˆ0 and Œˆ1 inHP0 H .A; B/ are equal. Hence the functor HP H is homotopy invariant in both variableswith respect to smooth equivariant homotopies.Recall that 2 H .A/ is the paracomplex 0 H .A/ ˚ 1 H .A/ ˚ 2 H .A/=b.3 H .A//with the usual differential B C b and the grading into even and odd forms for anypro-H -algebra A. There is a natural chain map 2 W 2 H .A/ ! X H .A/.Proposition 8.2. Let A be an equivariantly quasifree pro-H -algebra. Then the map 2 W 2 H .A/ ! X H .A/ is a homotopy equivalence.