Communications in Theoretical Physics

Braiding and Fusion Matrices for Quantum Octet Representation

Zhong-Qi Ma, Yu-Dong Yin

Published Jun 1, 1990• 1 citation

Abstract

The braiding and fusion matrices based on the octet representation of the quantum sl(3) enveloping algebra are computed. The multiplicity in the decomposition of the coproduct 8 × 8 manifests itself in those matrices.

DOI: 10.1088/0253-6102/13/4/509

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Communications in Theoretical Physics

Braiding and Fusion Matrices for Quantum Octet Representation

Zhong-Qi Ma, Yu-Dong Yin

Published Jun 1, 1990• 1 citation

Publisher

Abstract

DOI: 10.1088/0253-6102/13/4/509

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a \times b = \sum_c N_{ab}^c c

, where

N_{ab}^c

are non-negative integers (fusion coefficients). Braid group

B_n

generators

\sigma_i

satisfy

\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}

(Yang-Baxter equation). For Fibonacci anyons: fusion rule

\tau \times \tau = 1 + \tau

, with quantum dimension

d_\tau = \phi = (1+\sqrt{5})/2 \approx 1.618

. Braid matrices yield

R

-matrix eigenvalues:

e^{\pm i\pi/5}

e^{\pm i3\pi/5}

. CONNECTION: Fibonacci anyon quantum dimension

d_\tau = \phi = 1.618

(golden ratio). Fusion matrix eigenvalues involve

\phi^{-1} = 0.618

. Braid group representations correspond to

SU(2)_3

Chern-Simons theory, linked to

E_8

root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com