process

170 C. Voigtb) There exists an equivariant pro-linear map rW 1 .R/ ! 2 .R/ satisfyingfor all a 2 R and ! 2 1 .R/.r.a!/ D ar.!/; r.!a/ Dr.!/a !dac) There exists a projective resolution 0 ! P 1 ! P 0 ! R C of the R-bimodule R Cof length 1 in pro.H -Mod/.A map rW 1 .R/ ! 2 .R/ satisfying condition b) in Theorem 6.6 is also calledan equivariant graded connection on 1 .R/.We have the following basic examples of quasifree pro-H -algebras.Proposition 6.7. Let A be any H -algebra. The periodic tensor algebra T A is H -equivariantlyquasifree.An important result in theory of Cuntz and Quillen relates the X-complex of theperiodic tensor algebra T A to the standard complex of A constructed using noncommutativedifferential forms. The comparison between the equivariant X-complex andequivariant differential forms is carried out in the same way as in the group case [25].Proposition 6.8. There is a natural isomorphism X H .T A/ Š H .A/ such that thedifferentials of the equivariant X-complex correspond to@ 1 D b .id C/d on oddH .A/@ 0 DXn 1 2j b C Bj D0on 2nH .A/:Theorem 6.9. Let H be a bornological quantum group and let A be an H -algebra.Then the paracomplexes H .A/ and X H .T A/ are homotopy equivalent.For the proof of Theorem 6.9 it suffices to observe that the corresponding argumentsin [25] are based on the relations obtained in Proposition 6.1.7 Equivariant periodic cyclic homologyIn this section we define equivariant periodic cyclic homology for bornological quantumgroups.Definition 7.1. Let H be a bornological quantum group and let A and B be H -algebras.The equivariant periodic cyclic homology of A and B isHP H .A; B/ D H .Hom A.H / .X H .T .A Ì H Ì yH //; X H .T .B Ì H Ì yH ///:We write Hom A.H / for the space of AYD-maps and consider the usual differential fora Hom-complex in this definition. Using Proposition 5.5 it is straightforward to check