Two Relations That Generalize the Q-Serre Relations and the Dolan-Grady Relations

Terwilliger, Paul

doi:10.1142/9789812810199_0013

Cited by 115 publications

(262 citation statements)

References 48 publications

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“…To obtain the converse, assume (27), (28). We show L = R. By (29), it suffices to show (27), (30) and (29) we find L = R. We now have (26). ✷ Corollary 5.2 Let β, γ, γ * , ̺, ̺ * , ω, η, η * denote scalars in K. Then with reference to Definition 3.2 and Definition 4.2, the following (i), (ii) are equivalent.…”

Section: General Settingmentioning

confidence: 66%

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Leonard Pairs and the Askey–wilson Relations

2004

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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 * , ̺, ̺ * , ω, η, η * taken from K such that bothThe sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations.

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Section: General Settingmentioning

confidence: 66%

“…Concerning (28), for 0 ≤ i ≤ d we have E i LE i = E i RE i . Evaluating this using (30), (31) and Lemma 3.4(ii) we find a * i P (θ i , θ i ) = γ * θ 2 i + ωθ i + η. We now have (28).…”

Section: General Settingmentioning

confidence: 99%

Leonard Pairs and the Askey–wilson Relations

2004

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“…it has the same type (2, 3) as the DG-algebra. In a special case γ = γ 1 = 0 we obtain the so-called q-deformation of the Dolan-Grady relations [24,25]:…”

Section: Beyond the Aw-algebramentioning

confidence: 99%

Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Vinet

Zhedanov²

2008

SIGMA

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Abstract. We study Poisson and operator algebras with the "quasi-linear property" from the Heisenberg picture point of view. This means that there exists a set of one-parameter groups yielding an explicit expression of dynamical variables (operators) as functions of "time" t. We show that many algebras with nonlinear commutation relations such as the Askey-Wilson, q-Dolan-Grady and others satisfy this property. This provides one more (explicit Heisenberg evolution) interpretation of the corresponding integrable systems. Classical versionAssume that there is a classical Poisson manifold with the Poisson brackets (PB) {x, y} defined for all dynamical variables x, y belonging to this manifold. Of course, it is assumed that the PB satisfy the standard conditions: (i) {x, α 1 y + α 2 z} = α 1 {x, y} + α 2 {x, z} -linearity (α 1 , α 2 are arbitrary constants);(ii) {x, y} = −{y, x} -antisymmetricity; (iii) {x, yz} = z{x, y} + y{x, z} -the Leibnitz rule; (iv) {{x, y}, z} + {{y, z}, x} + {{z, x}, y} = 0 -the Jacobi identity.If one chooses the "Hamiltonian" H (i.e. some dynamical variable belonging to this manifold), then we have the standard Hamiltonian dynamics: all variables x(t) become depending on an additional time variable t and the equation of motion is defined aṡ x(t) = {x(t), H}.(1.1) Of course, the Hamiltonian H is independent of t, becauseḢ = {H, H} = 0. More generally, a dynamical variable Q is called the integral of motion ifQ = 0. Clearly, this is equivalent to the condition {Q, H} = 0. From (1.1) we haveẍ = {{x, H}, H} and more generally d n x dt n = {. . . {x, H}, H}, . . . H} = {x, H (n) }, This paper is a contribution to the

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“…Leonard pairs have been explored as linear algebraic objects, in connection with orthogonal polynomials, and as representations of certain algebras [17][18][19][20][21][22][23][24][25][26][27]. The notion of a Leonard triple was introduced by Curtin in [5].…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, results concerning Leonard triples also have interpretations as results concerning such polynomials. Leonard pairs play a role in representation theory [12,14,20,21,28] and combinatorics [4,7,8,10,12,[17][18][19]23]. Consequently, also Leonard triples play a role in representation theory and combinatorics.…”

Section: Introductionmentioning

confidence: 99%

Leonard triples and hypercubes

Miklavič

2007

J Algebr Comb

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Let V denote a vector space over C with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear operators on V such that for each of these operators there exists a basis of V with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal.Let D denote a positive integer and let Q D denote the graph of the D-dimensional hypercube. Let X denote the vertex set of Q D and let A ∈ Mat X (C) denote the adjacency matrix of Q D . Fix x ∈ X and let A * ∈ Mat X (C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat X (C) generated by A, A * . We refer to T as the Terwilliger algebra of Q D with respect to x. The matrices A and A * are related by the fact that 2iA = A * A ε − A ε A * and 2iA * = A ε A − AA ε , where 2iA ε = AA * − A * A and i 2 = −1.We show that the triple A, A * , A ε acts on each irreducible T -module as a Leonard triple. We give a detailed description of these Leonard triples.

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Two Relations That Generalize the Q-Serre Relations and the Dolan-Grady Relations

Cited by 115 publications

References 48 publications

Leonard Pairs and the Askey–wilson Relations

Leonard Pairs and the Askey–wilson Relations

Quasi-Linear Algebras and Integrability (the Heisenberg Picture)

Leonard triples and hypercubes

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