Enveloping algebras of some quantum Lie algebras

TL;DR

This paper introduces a family of Hopf algebras in anyonic vector spaces that serve as enveloping algebras for certain quantum Lie algebras, and explores their cyclic cohomology properties.

Contribution

It constructs new Hopf algebra objects in braided categories and establishes their relation to quantum Lie algebras and cyclic cohomology.

Findings

Identified Hopf algebra structures with involutive braiding in anyonic categories

Connected the cyclic cohomology of these algebras to quantum Lie algebra homology

Extended known relations from super and classical cases to anyonic quantum Lie algebras.

Abstract

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.