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Normalized Leonard pairs and Askey-Wilson relations | | Raimundas Vidunas
; | | Date: |
3 May 2005 | | Subject: | Rings and Algebras; Quantum Algebra MSC-class: 33C45, 33D45, 17B37, 05E35 | math.RA math.QA | | Abstract: | Let $V$ denote a vector space with finite positive dimension, and let $(A,B)$ denote a Leonard pair on $V$. As is known, the linear transformations $A,B$ satisfy the Askey-Wilson relations $ A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA +eI $, $ B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI $, for some scalars $b,g,h,r,s,w,e,f$. The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c*,t,t*$ are scalars and $t,t*$ are not zero, then $(tA+c,t*B+c*)$ is a Leonard pair on $V$ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit Askey-Wilson relations satisfied by them. | | Source: | arXiv, math.RA/0505041 | | Services: | Forum | Review | PDF | Favorites | |
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