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Solutions to generalized Yang-Baxter equations via ribbon fusion categories

Kitaev, Alexei and Wang, Zhenghan (2012) Solutions to generalized Yang-Baxter equations via ribbon fusion categories. . (Submitted) http://resolver.caltech.edu/CaltechAUTHORS:20120713-102318475

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Abstract

Inspired by quantum information theory, we look for representations of the braid groups B_n on V^(⊗(n+m−2)) for some fixed vector space V such that each braid generator σ_i, i = 1, ..., n−1, acts on m consecutive tensor factors from i through i +m−1. The braid relation for m = 2 is essentially the Yang-Baxter equation, and the cases for m > 2 are called generalized Yang-Baxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m = 3. Examples are given from the Ising theory (or the closely related SU(2)_2), SO(N)_2 for N odd, and SU(3)_3. The solution from the Jones-Kauffman theory at a 6th root of unity, which is closely related to SO(3)_2 or SU(2)_4, is explicitly described in the end.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1203.1063v2arXivUNSPECIFIED
Additional Information:The second author is partially supported by NSF DMS 1108736 and would like to thank E. Rowell for observing (3) of Thm. 2.5, S. Hong for helping on 6j symbols, and R. Chen for numerically testing the solutions.
Group:IQIM, Institute for Quantum Information and Matter
Funders:
Funding AgencyGrant Number
NSFDMS 1108736
Record Number:CaltechAUTHORS:20120713-102318475
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120713-102318475
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:32421
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:19 Jul 2012 21:45
Last Modified:26 Dec 2012 15:31

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