process

174 C. Voigtand using ˇ.x; y/ D .S.y/x/ as well as the fact that is right invariant one checksthat is an algebra homomorphism. Now we can apply Theorem 8.4 to obtain theassertion.We deduce a simpler description of equivariant periodic cyclic homology in certaincases. A bornological quantum group H is said to be of compact type if the dualalgebra yH is unital. Moreover let us call H of semisimple type if it is of compact typeand the value of the integral for yH on 1 2 yH is nonzero. For instance, the dual of acosemisimple Hopf algebra yH is of semisimple type.Proposition 8.6. Let H be a bornological quantum group of semisimple type. Thenwe haveHP H .A; B/ Š H .Hom A.H / .X H .T A/; X H .T B///for all H -algebras A and B.Proof. Under the above assumptions the canonical bilinear pairing ˇ W yH yH ! C isadmissible since the element 1 2 yH is invariant.Finally we discuss excision in equivariant periodic cyclic homology. Consider anextensionK E Qof H -algebras equipped with an equivariant linear splitting W Q ! E for the quotientmap W E ! Q.Let X H .T E W T Q/ be the kernel of the map X H .T / W X H .T E/ ! X G .T Q/ inducedby . The splitting yields a direct sum decomposition X H .T E/ D X H .T E WT Q/ ˚ X H .T Q/ of AYD-modules. The resulting extensionX H .T E W T Q/ X H .T E/ X H .T Q/of paracomplexes induces long exact sequences in homology in both variables. Thereis a natural covariant map W X H .T K/ ! X H .T E W T Q/ of paracomplexes and wehave the following generalized excision theorem.Theorem 8.7. The map W X H .T K/ ! X H .T E W T Q/ is a homotopy equivalence.This result implies excision in equivariant periodic cyclic homology.Theorem 8.8 (Excision). Let A be an H -algebra and let .; /W 0 ! K ! E !Q ! 0 be an extension of H -algebras with a bounded linear splitting. Then there aretwo natural exact sequencesHP0 H .A; K/ HP 0 H .A; E/ HPH0.A; Q/HP1 H .A; Q/ HP1 H .A; E/ HP1 H .A; K/