We define a family of Hopf algebra objects, H, in the braided category of
Znβ-modules (known as anyonic vector spaces), for which the property
ΟHβH2β=idHβHβ 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 Ο2=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