Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

Arash Pourkia

doi:10.13189/ms.2015.030604

Abstract

First, referring to our previous work, 'Hopf cyclic cohomology in braided monoidal categories', we reduce the restriction of the ambient category C being symmetric. We let C to be non-symmetric but assume only the restriction, ψ 2 = id, on the braid map correspond to the Hopf algebra H, which is the main player in the theory. We define a family of examples of such desired braided Hopf algebras, H, living in the category of anyonic vector spaces. Next, on one hand, we will prove that these anyonic Hopf algebras are the enveloping (Hopf) algebras of particular quantum Lie algebras, which we will construct. On the other hand, we will show that, analogous to the non-super and the super case, the well known relationship between the periodic Hopf cyclic cohomology of an enveloping (super) algebra and the (super) Lie algebra homology also holds for these particular quantum Lie algebras.

Highlights

In [2, 3, 4], Connes and Moscovici, motivated by transverse index theory for foliations, defined a cohomology theory of cyclic type for Hopf algebras endowed with a modular pair in involution (MPI)
In [9] we extended all these formalisms of Hopf cyclic cohomology to the context of abelian braided monoidal categories
When the braiding is symmetric we associated a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution

Summary

Introduction

In [2, 3, 4], Connes and Moscovici, motivated by transverse index theory for foliations, defined a cohomology theory of cyclic type for Hopf algebras endowed with a modular pair in involution (MPI) This theory was later extended in [7, 8] to the more general case of Hopf cyclic cohomology with coefficients, by introducing the notion of stable anti Yetter-Drinfeld (SAYD) modules. It states that, analogous to the non-super [2, 3, 4, 5] and the super case [9], the well known relationship between the periodic Hopf cyclic cohomology of an enveloping algebra, HP(∗δ,1)(U (g)), and the Lie algebra homologies, Hi(g; Cδ), holds for those quantum Lie algebras of Section 4. To keep this note short we have not included preliminaries

Hopf cyclic theory in non-symmetric categories

Some quantum Lie algebras and their enveloping Hopf algebras