Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

Abstract

In our previous work, "Hopf cyclic cohomology in braided monoidal categories", we extended the formalism of Connes and Moscovici's Hopf cyclic cohomology and the more general case of Hopf cyclic cohomology with coefficients to the context of abelian braided monoidal categories. In this paper we go one step further in reducing the restriction of the ambient category C being symmetric. We let C to be non-symmetric but assume only the restriction on the braid map ψ H⊗H for the Hopf algebra H in C which is the main player in the theory. In the case of Hopf cyclic cohomology with coefficients, i.e., in general for the triple (H, C, M ), we also need the restriction on ψ H⊗M , ψ H⊗C and ψ M⊗C . We present a family of examples of non-symmetric categories in which many objects with the property ψ 2 = id exist (anyonic vector spaces).

Enveloping algebras of some quantum Lie algebras

“…In Section 4 we will use the results of this section, the results from [13], and an idea similar to the one used for super Hopf algebras in [8], to compute the braided periodic Hopf cyclic cohomology of H = U (g) from the above examples.…”

“…This category is the symmetric category of super vector spaces for n = 2, but is not symmetric when n > 2, [9,11]. However, as we proved in [13], for many values of n > 2, one can always find objects A and B in C such that "behave symmetric", i.e., ψ B⊗A ψ A⊗B = id A⊗B . The following is a family of such examples.…”

“…In [13], we proved that in the non-symmetric category of anyonic vector spaces there are families of examples in which, for any two objects A and B, we have ψ B⊗A ψ A⊗B = id A⊗B . The motivation for this came from the fact that we showed that the main theorems of braided Hopf cyclic cohomology in [8], stated for symmetric monoidal categories, are still valid in non-symmetric categories as long as we have ψ 2 H⊗H = id H⊗H (or for short ψ 2 = id) only for the object of interest, H. We should recall that the braided Hopf cyclic theory in [8,13] is a generalization of the Connes and Moscovici Hopf cyclic cohomology in [1,2,3], and of the more general case of Hopf cyclic cohomology with coefficients in [6,7], to the context of abelian braided monoidal categories. A short survey on the subject can be found in [12].…”

“…In [13], we showed that in some of the non-symmetric categories of anyonic vector spaces, there are families of objects in which for any two objects A and B we have ψ B⊗A ψ A⊗B = id A⊗B . In this section, after a quick review, we provide examples of Hopf algebra objects within one of those families.…”

“…In this section, after a quick review, we provide examples of Hopf algebra objects within one of those families. In the next sections we set out to compute the Hopf cyclic cohomology of these Hopf algebras using the braided Hopf cyclic theory in [8,13].…”