nLab
braid representation
Contents
Context
Representation theory
representation theory

geometric representation theory

Ingredients
Definitions
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Theorems
Contents
Idea
A braid representation is a linear representation of a braid group .

In topological quantum computation on anyons , braid representations serve as quantum gates .

References
Braid group representations (as topological quantum gates)
On linear representations of braid groups (see also at braid group statistics and interpretation as quantum gates in topological quantum computation ):

Review:

in relation to modular tensor categories :

Colleen Delaney , Lecture notes on modular tensor categories and braid group representations , 2019 (pdf , pdf )
See also:

R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary $q$ and link polynomials , Journal of Mathematical Physics 33, 3918 (1992) (doi:10.1063/1.529840 )
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 (12 May 2006) (arXiv:quant-ph/0605004 )

(braid group representation serving as a topological quantum gate to compute the Jones polynomial )

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 )

Last revised on July 12, 2021 at 17:11:01.
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