Askey-Wilson relations and Leonard pairs
Quantum Algebra
2013-10-04 v2
Abstract
It is known that if is a Leonard pair, then the linear transformations , satisfy the Askey-Wilson relations A^2 B - b A B A + B A^2 - g (A B+B A) - r B = h A^2 + w A + e I, B^2 A - b B A B + A B^2 - h (A B+B A) - s A = g B^2 + w B + f I, for some scalars . The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations.
Cite
@article{arxiv.math/0511509,
title = {Askey-Wilson relations and Leonard pairs},
author = {Raimundas Vidunas},
journal= {arXiv preprint arXiv:math/0511509},
year = {2013}
}
Comments
22 pages; corrected version; the example of Section 2 has the normalization consistent with the rest of the paper