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166 C. Voigt6 Equivariant differential formsIn this section we define equivariant differential forms and the equivariant X-complex.Moreover we discuss the properties of the periodic tensor algebra of an H -algebra.These are the main ingredients in the construction of equivariant cyclic homology.Let H be a bornological quantum group. If A is an H -algebra we obtain a leftaction of H on the space H y˝ n .A/ byt .x ˝ !/ D t .3/ xS.t .1/ / ˝ t .2/ !for t;x 2 H and ! 2 n .A/. Here n .A/ D A C y˝ A y˝n for n>0is the space ofnoncommutative n-forms over A with the diagonal H -action. For n D 0 one defines 0 .A/ D A. There is a left action of the dual quantum group yH on H y˝ n .A/ givenbyf .x ˝ !/ D .f * x/ ˝ ! D f.x .2/ /x .1/ ˝ !:By definition, the equivariant n-forms n H .A/ are the space H y˝ n .A/ together withthe H -action and the yH -action described above. We computet .f .x ˝ !// D t .f .x .2/ /x .1/ ˝ !/D f.x .2/ /t .3/ x .1/ S.t .1/ / ˝ t .2/ !D .S 2 .t .1/ /*f (S 1 .t .5/ // .t .4/ xS.t .2/ / ˝ t .3/ !/D .S 2 .t .1/ /*f (S 1 .t .3/ // .t .2/ .x ˝ !//and deduce that n H .A/ is an H -AYD-module. We let H .A/ be the direct sum of thespaces n H .A/.Now we define operators d and b H on H .A/ byd.x ˝ !/ D x ˝ d!andb H .x ˝ !da/ D . 1/ j!j .x ˝ !ax .2/ ˝ .S 1 .x .1/ / a/!/:The map b H should be thought of as a twisted version of the usual Hochschild operator.We computebH 2 .x ˝ !dadb/ D . 1/j!jC1 b H .x ˝ !dab x .2/ ˝ .S 1 .x .1/ / b/!da/D . 1/ j!jC1 b H .x ˝ !d.ab/ x ˝ !adb x .2/ ˝ .S 1 .x .1/ / b/!da/D .x ˝ !ab x .2/ ˝ S 1 .x .1/ / .ab/! x ˝ !ab C x .2/ ˝ .S 1 .x .1/ / b/!ax .2/ ˝ .S 1 .x .1/ / b/!a C x .2/ ˝ S 1 .x .1/ / .ab/!/ D 0which shows that bH 2 is a differential as in the nonequivariant situation. Let us discussthe compatibility of d and b H with the AYD-module structure. It is easy to check that