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The Bannai-Ito polynomials as Racah coefficients of the $ sl_{-1}(2)$ algebra


Authors: Vincent X. Genest, Luc Vinet and Alexei Zhedanov
Journal: Proc. Amer. Math. Soc. 142 (2014), 1545-1560
MSC (2010): Primary 16T05, 17B80, 33C45, 33C47, 81R05
DOI: https://doi.org/10.1090/S0002-9939-2014-11970-8
Published electronically: February 18, 2014
MathSciNet review: 3168462
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Abstract: The Bannai-Ito polynomials are shown to arise as Racah coefficients for $ sl_{-1}(2)$. This Hopf algebra has four generators, including an involution, and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for $ sl_{-1}(2)$. The Racah coefficients are recovered from a related Leonard pair.


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Vincent X. Genest
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal, Québec, Canada, H3C 3J7
Email: genestvi@crm.umontreal.ca

Luc Vinet
Affiliation: Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal, Québec, Canada, H3C 3J7
Email: luc.vinet@umontreal.ca

Alexei Zhedanov
Affiliation: Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine
Email: zhedanov@yahoo.com

DOI: https://doi.org/10.1090/S0002-9939-2014-11970-8
Received by editor(s): May 18, 2012
Received by editor(s) in revised form: May 31, 2012, and June 13, 2012
Published electronically: February 18, 2014
Communicated by: Sergei K. Suslov
Article copyright: © Copyright 2014 American Mathematical Society