LEONARD PAIRS AND THE ASKEY-WILSON RELATIONS

By Paul M. Terwilliger, Raimundas Vidunas. 2003

179 citations recorded by the source corpus.

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we show there exists a sequence of scalars β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both \begin{eqnarray*} A^2 A^*-\beta A A^*A+A^* A^2-\gamma (A A^*+A^*A) -\varrho\,A^* & = & \gamma^* A^2+\omega A+\eta\,I,\\[5pt] A^*{}^2A-\beta A^*AA^*+AA^*{}^2-\gamma^* (A^*A+A A^*)-\varrho^*A & = & \gamma A^*{}^2+\omega A^*+\eta^*I\,. \end{eqnarray*} The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least...

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/0a4e4920e2552e9c9bdc1cd4613efcf01498872e

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Citations in the corpus

Two linear transformations each tridiagonal with respect to an eigenbasis of the other: An Algebraic approach to the Askey scheme of orthogonal polynomials (2004, unknown): “ . . 63 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 1 Leonard pairs These notes are based on the papers [46], [47], [48], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100]. We begin by recalling the notion of a Leonard pair. We will use the following terms. Let X de”
The Tetrahedron algebra, the Onsager algebra, and the sl2 loop algebra (2007, background): “Over the years O has been investigated many times in connection with solvable lattice models [1], [2], [3], [5], [6], [7], [8], [15], [16], [22], [26], [27], [28], [30], [39] representation theory [13], [14], [21] Kac-Moody algebras [12], [32], [33] tridiagonal pairs [23], [24], [36], [37], [38] and”
TWO NON-NILPOTENT LINEAR TRANSFORMATIONS THAT SATISFY THE CUBIC q-SERRE RELATIONS (2005, unknown): “A recent topic of research concerns the Leonard pairs [23], [37], [39], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [53], [54] and the closely related tridiagonal pairs [1], [2], [24], [25], [26], [34], [35], [36].”
Tridiagonal pairs and the quantum affine algebra Uq(sl2) (代数的組合せ論) (2005, unknown): “…for 0 ≤ i ≤ d. [0∗0]: Apply the present Lemma, row [0∗D], with Vi replaced by Vd−i for 0 ≤ i ≤ d. [D∗0]: Apply the present Lemma, row [0∗D], with Vi replaced by Vd−i and V ∗i replaced by V ∗d−i for 0 ≤ i ≤ d. [D∗D]: Apply the present Lemma, row [0∗D], with V ∗i replaced by V ∗ d−i for 0 ≤ i ≤ d.”
The classification of Leonard triples of QRacah type (2011, unknown): “(6) Setting i = 0 in (3) we find U0 = E∗ 0V .”
Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair (2006, unknown): “We refer the reader to [3], [9], [12], [13], [14], [15], [16], [17], [18], [19], [21], [22], [23], [24], [25], [26], [27], [28], [30], [31] for background on Leonard pairs.”
Tridiagonal pairs of q-Racah type (2008, background): “See [9, 35, 52, 54, 59] for results on the tridiagonal relations.”
The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case (2006, background): “Terwilliger & Vidunas [15] showed that every Leonard pair satisfies the AW (3) relations for a suitable choice of the structure constants.”
Finite-Dimensional Irreducible Modules for the Three-Point 𝔰𝔩2 Loop Algebra (2007, background): “This has to do with the tridiagonal pairs (Ito and Terwilliger, 2007a; Ito et al., 2001) and the closely related Leonard pairs (Terwilliger, 2001, 2004; Terwilliger and Vidunas, 2004).”
An Algebraic Approach to the Askey Scheme of Orthogonal Polynomials (2006, unknown): “These notes are based on the papers [45], [46], [47], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88].”
The Universal Askey-Wilson Algebra (2011, methodology): “It is particularly useful in the theory of Leonard pairs [44, 54, 56, 57, 59, 60] and Leonard triples [19, 20, 40].”