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Bannai–Ito algebras and the universal R-matrix of \(\pmb {\mathfrak {osp}}(1|2)\)

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Abstract

The Bannai–Ito algebra BI(n) is viewed as the centralizer of the action of \(\mathfrak {osp}(1|2)\) in the n-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal R-matrix of \(\mathfrak {osp}(1|2)\). The specific structure of the \(\mathfrak {osp}(1|2)\) embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of BI(n) are derived from those of BI(3) by repeated action of the coproduct and using properties of the R-matrix and of the generators of the symmetric group \({\mathfrak {S}}_n\).

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Notes

  1. In what follows we shall keep using the inverse of \(\mathcal {R}\) even though \(\mathcal {R}^{-1}=\mathcal {R}\) (for \(\mathfrak {osp}(1|2)\)) to make clear that conjugations are involved.

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Acknowledgements

We have much benefited from discussions with L. Frappat, J. Gaboriaud and E. Ragoucy. N. Crampé is gratefully holding a CRM–Simons professorship. The research of L. Vinet is supported in part by a Discovery Grant from the Natural Science and Engineering Research Council (NSERC) of Canada. M. Zaimi holds a NSERC graduate scholarship.

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Authors and Affiliations

  1. Institut Denis-Poisson CNRS/UMR 7013, Université de Tours - Université d’Orléans, Parc de Grammont, 37200, Tours, France

    Nicolas Crampé

  2. Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7, Canada

    Luc Vinet & Meri Zaimi

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  1. Nicolas Crampé
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  2. Luc Vinet
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Correspondence to Nicolas Crampé.

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Crampé, N., Vinet, L. & Zaimi, M. Bannai–Ito algebras and the universal R-matrix of \(\pmb {\mathfrak {osp}}(1|2)\). Lett Math Phys 110, 1043–1055 (2020). https://doi.org/10.1007/s11005-019-01249-w

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