### Journals Information

**
Mathematics and Statistics Vol. 3(6), pp. 157 - 163 DOI: 10.13189/ms.2015.030604 Reprint (PDF) (119Kb) **

## Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

**Arash Pourkia ^{1}^{,2}^{,*}**

^{1}Huron University College, Western University, Canada

^{2}American University of Middle East, Kuwait

**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.

**KEYWORDS**

Hopf Cyclic Cohomology, Braided Monoidal Categories, Braided Hopf Algebras, Quantum Lie Algebras

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Arash Pourkia , "Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories," Mathematics and Statistics, Vol. 3, No. 6, pp. 157 - 163, 2015. DOI: 10.13189/ms.2015.030604.

(b). APA Format:

Arash Pourkia (2015). Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories. Mathematics and Statistics, 3(6), 157 - 163. DOI: 10.13189/ms.2015.030604.