process

Equivariant cyclic homology for quantum groups 157and f; g 2 yH the relation.id y˝ r / 12 .id y˝ r 1 /.a ˝ f ˝ g/ D .id y˝ r /..˛.a/ ˝ 1/.1 ˝ r1 .f ˝ g///D .id y˝ /.˛.a//.1 ˝ r r1 .f ˝ g//D .˛ y˝ id/.˛.a//.1 ˝ f ˝ g/D 12 13 .a ˝ f ˝ g/in M.A y˝ yH y˝ yH/. Using that A is regular we deduce that.id y˝ r / 12 .id y˝ 1r /.a ˝ f ˝ g/ D 12 13 .a ˝ f ˝ g/in A y˝ yH y˝ yH and hence defines a right yH -comodule structure on A. In additionwe have. y˝ id/ 13 23 .a ˝ b ˝ f/D . y˝ id/ 13 .a ˝ ˛.b/.1 ˝ f//D ˛.a/˛.b/.1 ˝ f/D ˛.ab/.1 ˝ f/D . y˝ id/.a ˝ b ˝ f/and it follows that A becomes an H -algebra using this coaction.Conversely, assume that A is an H -algebra implemented by the coactionW A y˝ yH ! A y˝ yH . Define bornological automorphisms l and r of A y˝ yH by l D .id y˝ S 1 / 1 .id y˝ S/; r D :The map l is left yH -linear for the action of yH on the second tensor factor and r isright yH -linear. Since is a compatible with the multiplication we have r . y˝ id/ D . y˝ id/ 13r 23 rand l . y˝ id/ D . y˝ id/ 23l 13l:In addition one has the equation.id y˝ / 12lD .id y˝ / 13rrelating l and r . These properties of the maps l and r imply that˛.a/.b ˝ f/D r .a ˝ f /.b ˝ 1/; .b ˝ f/˛.a/D .b ˝ 1/ l .a ˝ f/defines an algebra homomorphism ˛ from A to M.A y˝ yH/. As in the proof of Proposition7.3 in [27] one shows that ˛ is essential. Observe that we may identify the naturalmap A y˝A .A y˝ yH/ ! A y˝ yH induced by ˛ with 13r W A y˝A .A y˝ yH y˝ y yHH/ !.A y˝A A/ y˝ . yH y˝ y yHH/since A is essential.The maps ˛l and ˛r associated to the homomorphism ˛ can be identified with land r , respectively. Finally, the coaction identity .id y˝ r / 12 .id y˝ r 1 / D 12 13implies .˛ y˝ id/˛ D .id y˝ /˛. Hence ˛ defines an algebra coaction of yH on A.It follows immediately from the constructions that the two procedures describedabove are inverse to each other.