Hopf cyclic cohomology in braided monoidal categories

Abstract

We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.

Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

“…Proof. As in the proof of Theorem 3.6 in [7]. Now if we put M = I and C = H Theorem 2.1 reduces the braided version of Connes-Moscovici's Hopf cyclic theory [2,3,4], in non-symmetric monoidal categories, as follows.…”

“…The proofs of the two theorems presented here are the same as the proofs of their analogous theorems in [7].…”

“…We assign a cocyclic object to a braided triple (H, C, M ) in an abelian braided monoidal category C but with the symmetric condition for C replaced by only restrictions on ψ H⊗M , ψ H⊗H , ψ H⊗C and ψ M ⊗C (Theorem 2.1). Then we let C = H and M = I, the identity object of C, and present an analogous of Theorem 3.6 in [7], but again with the symmetric condition for C replaced, in this case, by only one restriction on ψ H⊗H (Theorem 2.2). The latter case is in fact the braided version of Connes-Moscovici's Hopf cyclic cohomology in a non-symmetric category C in which only the relation ψ 2 H⊗H = id H holds.…”

“…In the most general case, for the triple (H, C, M ), we also need similar restrictions on ψ H⊗C and ψ M ⊗C . This is in fact because the main players in the theory are H, M and C and nowhere in any of the proofs in [7] we need more than the above relations. Therefore the proofs of the two theorems, Theorems 2.1 and 2.2, presented here are the same as the proofs of their analogous theorems, Theorems 3.6 and 7.1 in [7].…”

A “natural” graded Hopf-algebra and its graded Hopf-cyclic cohomology

“…This 'twisted' flip (flip with a sign) is in fact a special symmetric braiding. In [1], [6], [7] (co)cyclic modules in a symmetric braided tensor category are introduced, where the 'twisted' flip is replaced by a braid action.…”