A survey of braided Hopf cyclic cohomology

Pourkia, Arash

doi:10.1090/fic/061/06

Cited by 2 publications

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References 26 publications

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“…The motivation for this came from the fact that we showed that the main theorems of braided Hopf cyclic cohomology in [8], stated for symmetric monoidal categories, are still valid in non-symmetric categories as long as we have ψ 2 H⊗H = id H⊗H (or for short ψ 2 = id) only for the object of interest, H. We should recall that the braided Hopf cyclic theory in [8,13] is a generalization of the Connes and Moscovici Hopf cyclic cohomology in [1,2,3], and of the more general case of Hopf cyclic cohomology with coefficients in [6,7], to the context of abelian braided monoidal categories. A short survey on the subject can be found in [12]. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Enveloping algebras of some quantum Lie algebras

Pourkia

2014

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We define a family of Hopf algebra objects, H, in the braided category of Z n -modules (known as anyonic vector spaces), for which the property ψ 2 H⊗H = id H⊗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 ψ 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.

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Section: Introductionmentioning

confidence: 99%

Enveloping algebras of some quantum Lie algebras

Pourkia

2014

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Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

Pourkia¹

2015

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In our previous work, "Hopf cyclic cohomology in braided monoidal categories", we extended the formalism of Connes and Moscovici's Hopf cyclic cohomology 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 C being symmetric. We let C to be non-symmetric but assume only the restriction on the braid map ψ H⊗H for the Hopf algebra H in C which is the main player in the theory. In the case of Hopf cyclic cohomology with coefficients, i.e., in general for the triple (H, C, M ), we also need the restriction on ψ H⊗M , ψ H⊗C and ψ M⊗C . We present a family of examples of non-symmetric categories in which many objects with the property ψ 2 = id exist (anyonic vector spaces).

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A survey of braided Hopf cyclic cohomology

Cited by 2 publications

References 26 publications

Enveloping algebras of some quantum Lie algebras

Enveloping algebras of some quantum Lie algebras

Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

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