On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation):
Review:
Chen Ning Yang, M. L. Ge (eds.). Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) doi:10.1142/0796
(focus on quantum Yang-Baxter equation)
Camilo Arias Abad, Introduction to representations of braid groups, Rev. colomb. mat. vol.49 no.1 (2015) (arXiv:1404.0724, doi:10.15446/recolma.v49n1.54160)
Toshitake Kohno, Introduction to representation theory of braid groups, Peking 2018 (pdf, pdf)
in relation to modular tensor categories:
Braid representations from the monodromy of the Knizhnik-Zamolodchikov connection on bundles of conformal blocks over configuration spaces of points:
Ivan Todorov, Ludmil Hadjiivanov, Monodromy Representations of the Braid Group, Phys. Atom. Nucl. 64 (2001) 2059-2068; Yad.Fiz. 64 (2001) 2149-2158 arXiv:hep-th/0012099, doi:10.1134/1.1432899
Ivan Marin, Sur les représentations de Krammer génériques, Annales de l’Institut Fourier, 57 6 (2007) 1883-1925 numdam:AIF_2007__57_6_1883_0
and understood in terms of anyon statistics:
Braid representations seen inside the topological K-theory of the braid group‘s classifying space:
Alejandro Adem, Daniel C. Cohen, Frederick R. Cohen, On representations and K-theory of the braid groups, Math. Ann. 326 (2003) 515-542 (arXiv:math/0110138, doi:10.1007/s00208-003-0435-8)
Frederick R. Cohen, Section 3 of: On braid groups, homotopy groups, and modular forms, in: J.M. Bryden (ed.), Advances in Topological Quantum Field Theory, Kluwer 2004, 275–288 (pdf)
See also:
As quantum gates for topological quantum computation with anyons:
Louis H. Kauffman, Samuel J. Lomonaco, Braiding Operators are Universal Quantum Gates, New Journal of Physics, Volume 6, January 2004 (arXiv:quant-ph/0401090, doi:10.1088/1367-2630/6/1/134)
Samuel J. Lomonaco, Louis Kauffman, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (2006) (arXiv:quant-ph/0605004)
(braid group representation serving as a topological quantum gate to compute the Jones polynomial)
Louis H. Kauffman, Samuel J. Lomonaco, Topological quantum computing and braid group representations, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) (doi:10.1117/12.778068, rg:228451452)
C.-L. Ho, A.I. Solomon, C.-H.Oh, Quantum entanglement, unitary braid representation and Temperley-Lieb algebra, EPL 92 (2010) 30002 (arXiv:1011.6229)
Louis H. Kauffman, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (arXiv:1710.04650, doi:10.1142/S0217751X18300235)
David Lovitz, Universal Braiding Quantum Gates [arXiv:2304.00710]
Introduction and review:
Colleen Delaney, Eric C. Rowell, Zhenghan Wang, Local unitary representations of the braid group and their applications to quantum computing, Revista Colombiana de Matemáticas(2017), 50 (2):211 (arXiv:1604.06429, doi:10.15446/recolma.v50n2.62211)
Eric C. Rowell, Braids, Motions and Topological Quantum Computing arXiv:2208.11762
Realization of Fibonacci anyons on quasicrystal-states:
Realization on supersymmetric spin chains:
Last revised on February 14, 2024 at 07:05:12. See the history of this page for a list of all contributions to it.