Localization of Unitary Braid Group Representations

@article{Rowell2012LocalizationOU,
  title={Localization of Unitary Braid Group Representations},
  author={Eric C. Rowell and Zhenghan Wang},
  journal={Communications in Mathematical Physics},
  year={2012},
  volume={311},
  pages={595-615}
}
Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations… 
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References

SHOWING 1-10 OF 59 REFERENCES
From Quantum Groups to Unitary Modular Tensor Categories
Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties nec- essary to produce 3-dimensional TQFTs. Although
Extraspecial two-Groups, generalized Yang-Baxter equations and braiding quantum gates
TLDR
It is suggested that through their connection with braiding gates, extraspecial 2-groups, unitary representations of the braid group and the GHZ states may play an important role in quantum error correction and topological quantum computing.
Braid group representations from twisted quantum doubles of finite groups
We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group
Braid representations from quantum groups of exceptional lie type
We study the problem of determining if the braid group representations obtained from quantum groups of types E, F and G at roots of unity have infinite image or not. In particular we show that when
A Modular Functor Which is Universal¶for Quantum Computation
Abstract:We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently
Quantum groups and subfactors of type B, C, and D
The main object of this paper is the study of a sequence of finite dimensional algebras, depending on 2 parameters, which appear in connection with the Kauffman link invariant and with Drinfeld's and
Extraspecial 2-groups and images of braid group representations
We investigate a family of (reducible) representations of the braid groups Bn corresponding to a specific solution to the Yang‐Baxter equation. The images of Bn under these representations are finite
Hecke algebra representations of braid groups and link polynomials
By studying representations of the braid group satisfying a certain quadratic relation we obtain a polynomial invariant in two variables for oriented links. It is expressed using a trace, discovered
Topological Quantum Computation
The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones poly-
...
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