The classification of Leonard triples of QRacah type

Details The classification of Leonard triples of QRacah type The classification of Leonard triples of QRacah type

2011-08-02

arxiv.org/abs/1108.0458v1

Mathematics

Rings and Algebras

33 pages, 2 figures. arXiv admin note: text overlap with arXiv:math/0306301

Scientific paper

Let $\mathbb{K}$ denote an algebraically closed field. Let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. By a Leonard triple on $V$ we mean an ordered triple of linear transformations in ${\rm End}(V)$ such that for each of these transformations there exists a basis of $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. There is a family of Leonard triples said to have QRacah type. This is the most general type of Leonard triple. We classify the Leonard triples of QRacah type up to isomorphism. We show that any Leonard triple of QRacah type satisfies the $\mathbb{Z}_3$-symmetric Askey-Wilson relations.

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