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Enveloping algebras of some quantum Lie algebras
We define a family of Hopf algebra objects, , in the braided category of
-modules (known as anyonic vector spaces), for which the property
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 . 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
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