We extend the formalism of Hopf cyclic cohomology to the context of braided categories. For a Hopf algebra in a braided monoidal abelian category we introduce the notion of stable anti-Yetter-Drinfeld module. We associate a para-cocyclic and a cocyclic object to a braided Hopf algebra endowed with a braided modular pair in involution in the sense of Connes and Moscovici. When the braiding is symmetric the full formalism of Hopf cyclic cohomology with coefficients can be extended to our categorical setting.
We present a formula for an infinite number of universal quantum logic gates, which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We obtain this family from a certain representation of the cyclic group of order $n$. We then show that this {\it discrete} family, parametrized by integers $n$, is in fact, a small sub-class of a larger {\it continuous} family, parametrized by real numbers $\theta$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.Comment: 12 pages, no figures. Submitted to Physical Review
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).
Braid groups and their representations are at the center of study, not only in low-dimensional topology, but also in many other branches of mathematics and theoretical physics. Burau representation of the Artin braid group which has two versions, reduced and unreduced, has been the focus of extensive study and research since its discovery in 1930's. It remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for knots and links. In the present work, we show that interesting representations of braid group could be achieved using a simple and intuitive approach, where we simply analyse the path of strands in a braid and encode the over-crossings, under-crossings or no-crossings into some parameters. More precisely, at each crossing, where, for example, the strand i crosses over the strand i + 1 we assign t to the top strand and b to the bottom strand. We consider the parameter t as a relative weight given to strand i relative to i + 1, hence the position i i + 1 for t in the matrix representation. Similarly, the parameter b is a relative weight given to strand i + 1 relative to i, hence the position i + 1 i for b in the matrix representation. We show this simple path analyzing approach leads us to an interesting simple representation. Next, we show that following the same intuitive approach, only by introducing an additional parameter, we can greatly improve the representation into the one with much smaller kernel. This more general representation includes the unreduced Burau representation, as a special case. Our new path analyzing approach has the advantage that it applies a very simple and intuitive method capturing the fundamental interactions of the strands in a braid. In this approach we intuitively follow each strand in a braid and create a history for the strand as it interacts with other strands via over-crossings, under-crossings or no-crossings. This, directly, leads us to the desired representations.
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