2020
DOI: 10.1080/00927872.2020.1832105
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The universal DAHA of type (C1∨,C1) and Leonard triples

Abstract: Assume that F is an algebraically closed field with characteristic zero. The universal Racah algebra ℜ is a unital associative F-algebra generated by A, B, C, D and the relations state that [A, B] = [B, C] = [C, A] = 2D and each of Let V denote a finite-dimensional irreducible H-module. In this paper we show that A, B, C are diagonalizable on V if and only if A, B, C act as Leonard triples on all composition factors of the ℜ-module V .

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Cited by 6 publications
(6 citation statements)
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“…Proof. By Lemma 5.3 we may assume that A, A * is in normalized TD/D form, and A, A * are as in (24). By Lemma 7.4, F and A − F are as in (26).…”
Section: Bipartite and Essentially Bipartite Leonard Pairsmentioning
confidence: 99%
See 2 more Smart Citations
“…Proof. By Lemma 5.3 we may assume that A, A * is in normalized TD/D form, and A, A * are as in (24). By Lemma 7.4, F and A − F are as in (26).…”
Section: Bipartite and Essentially Bipartite Leonard Pairsmentioning
confidence: 99%
“…Define scalars {x i } d i=1 by (17). Define A ∈ Mat d+1 (F) as in (24). By Lemma 5.3 the pair A, A * is a Leonard pair that has parameter array (34).…”
Section: Bipartite and Essentially Bipartite Leonard Pairsmentioning
confidence: 99%
See 1 more Smart Citation
“…We just mentioned how Leonard pairs are related to orthogonal polynomials. Leonard pairs have applications to many other areas of mathematics and physics, such as Lie theory [4,21,22,25,29,39], quantum groups [1, 2, 12-14, 26, 28, 30], spin models [16][17][18]41], double affine Hecke algebras [23,24,32,33,40], partially ordered sets [35,36,45,55], and exactly solvable models in statistical mechanics [5][6][7][8][9][10][11]. For more information about Leonard pairs and related topics, see [39,42,44,46,48].…”
Section: Introductionmentioning
confidence: 99%
“…We just mentioned how Leonard pairs are related to orthogonal polynomials. Leonard pairs have applications to many other areas of mathematics and physics, such as Lie theory [25,39,4,21,22,29], quantum groups [1,12,28,30,13,2,14,26], spin models [17,41,18,16], double affine Hecke algebras [40,23,24,32,33], partially ordered sets [35,45,55,36], and exactly solvable models in statistical mechanics [5,6,7,8,9,10,11]. For more information about Leonard pairs and related topics, see [48,46,39,42,44].…”
mentioning
confidence: 99%