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VOL. 46 | 2017 About 30 years of integrable Chiral Potts model, quantum groups at roots of unity and cyclic hypergeometric functions
Helen Au-Yang, Jacques H. H. Perk

Editor(s) Scott Morrison, David Penneys

Abstract

In this paper we discuss the integrable chiral Potts model, as it clearly relates to how we got befriended with Vaughan Jones, whose birthday we celebrated at the Qinhuangdao meeting. Remarkably we can also celebrate the birthday of the model, as it has been introduced about 30 years ago as the first solution of the star-triangle equations parametrized in terms of higher genus functions. After introducing the most general checkerboard Yang–Baxter equation, we specialize to the star-triangle equation, also discussing its relation with knot theory. Then we show how the integrable chiral Potts model leads to special identities for basic hypergeometric series in the $q$ a root-of-unity limit. Many of the well-known summation formulae for basic hypergeometric series do not work in this case. However, if we require the summand to be periodic, then there are many summable series. For example, the integrability condition, namely, the star-triangle equation, is a summation formula for a well-balanced $_4 \Phi _3$ series. We finish with a few remarks about the relation with quantum groups at roots of unity.

Information

Published: 1 January 2017
First available in Project Euclid: 21 February 2017

zbMATH: 06990149
MathSciNet: MR3635665

Rights: Copyright © 2017, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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