process

Equivariant cyclic homology for quantum groups 167d is an AYD-map and that the operator b H is yH -linear. Moreover we computeb H .t .x ˝ !da// D . 1/ j!j .t .4/ xS.t .1/ / ˝ .t .2/ !/.t .3/ a/t .6/ x .2/ S.t .1/ / ˝ .S 1 .t .5/ x .1/ S.t .2/ //t .4/ a/.t .3/ !//D . 1/ j!j .t .3/ xS.t .1/ / ˝ t .2/ .!a/ t .4/ x .2/ S.t .1/ / ˝ .t .2/ S 1 .x .1/ / a/.t .3/ !//D t b H .x ˝ !da/and deduce that b H is an AYD-map as well.Similar to the non-equivariant case we use d and b H to define an equivariant Karoubioperator H and an equivariant Connes operator B H byand H D 1 .b H d C db H /B H DnX j H don H n .A/. Let us record the following explicit formulas. For n>0we havej D0 H .x ˝ !da/ D . 1/ n1 x .2/ ˝ .S 1 .x .1/ / da/!on H n .A/ and in addition H .x ˝ a/ D x .2/ ˝ S 1 .x .1/ / a on H 0 .A/. For theConnes operator we computenXB H .x˝a 0 da 1 :::da n /D . 1/ ni x .2/˝S 1 .x .1/ /.da nC1 i :::da n /:da 0 :::da n iiD0Furthermore, the operator T is given byT.x˝ !/ D x .2/ ˝ S 1 .x .1/ / !on equivariant differential forms. Observe that all operators constructed so far areAYD-maps and thus commute with T according to Proposition 5.5.Lemma 6.1. On H n .A/ the following relations hold:a) nC1Hd D Td,b) H n D T C b H H n d,c) n H b H D b H T ,d) nC1H D .id db H /T ,e) . nC1HT /. n HT/D 0,f) B H b H C b H B H D id T .