Enveloping algebras of some quantum Lie algebras

We define a family of Hopf algebra objects, $H$, in the braided category of $\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\psi^2_{H\otimes H}=id_{H\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $\psi^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analogous to the non-super and the super case, the well known relation 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, in the category of anyonic vector spaces

A note on Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

In our previous work, Hopf cyclic cohomology in braided monoidal categories, we extended the formalism of Hopf cyclic cohomology due to Connes and Moscovici 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 being symmetric. We let the ambient category to be non-symmetric but assume only the restriction on the braid map for the Hopf algebra object (in that category) which is the main player in the theory. In the case of Hopf cyclic cohomology with (nontrivial) coefficients we also need to have similar restrictions on the braid map for the object(s) providing the coefficients datum. We present a family of examples of non-symmetric categories in which many objects with such a restrictions on the braid map exist (anyonic vector spaces).Comment: The author would like to express his sincere appreciation to Masoud Khalkhali for illuminating discussions and encouragement

Hopf cyclic cohomology in braided monoidal categories

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.Comment: 50 pages. One reference added. Proofs are visualized through braiding diagrams. Final version to appear in `Homology, Homotopy and Applications