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Askey–Wilson relations and Leonard pairs
- Discrete Math
"... It is known that if (A,A ∗ ) is a Leonard pair, then the linear transformations A, A ∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A ∗ A 2 − γ (AA ∗ +A ∗ A) − ̺ A ∗ = γ ∗ A 2 + ωA + η I, A ∗2 A − βA ∗ AA ∗ + AA ∗2 − γ ∗ (A ∗ A+AA ∗ ) − ̺ ∗ A = γA ∗2 + ωA ∗ + η ∗ I, for some scalars ..."
Abstract
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Cited by 9 (0 self)
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It is known that if (A,A ∗ ) is a Leonard pair, then the linear transformations A, A ∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A ∗ A 2 − γ (AA ∗ +A ∗ A) − ̺ A ∗ = γ ∗ A 2 + ωA + η I, A ∗2 A − βA ∗ AA ∗ + AA ∗2 − γ ∗ (A ∗ A+AA ∗ ) − ̺ ∗ A = γA ∗2 + ωA ∗ + η ∗ I, for some
Leonard pairs and the Askey-Wilson relations
- Department of Computational Science Faculty of Science Kanazawa University Kakuma-machi Kanazawa
"... Let V denote a vector space with finite positive dimension, and let (A,A ∗) denote a Leonard pair on V. As is known, the linear transformations A, A ∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A ∗ A 2 − γ (AA ∗ +A ∗ A) − ̺ A ∗ = γ ∗ A 2 + ωA + η I, A ∗2 A − βA ∗ AA ∗ + AA ∗2 − γ ∗ ( ..."
Abstract
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Cited by 47 (22 self)
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Let V denote a vector space with finite positive dimension, and let (A,A ∗) denote a Leonard pair on V. As is known, the linear transformations A, A ∗ satisfy the Askey-Wilson relations A 2 A ∗ − βAA ∗ A + A ∗ A 2 − γ (AA ∗ +A ∗ A) − ̺ A ∗ = γ ∗ A 2 + ωA + η I, A ∗2 A − βA ∗ AA ∗ + AA ∗2 − γ
The Universal Askey–Wilson Algebra
- SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2011
"... Let F denote a field, and fix a nonzero q ∈ F such that q4 ̸ = 1. We define an associative F-algebra ∆ = ∆q by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A + qBC − q−1CB q2 − q−2, B + qCA − q−1AC q2 − q−2, C + qAB − q−1BA q2 − q−2 is ..."
Abstract
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Cited by 7 (2 self)
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is central in ∆. We call ∆ the universal Askey–Wilson algebra. We discuss how ∆ is related to the original Askey–Wilson algebra AW(3) introduced by A. Zhedanov. Multiply each of the above central elements by q + q −1 to obtain α, β, γ. We give an alternate presentation for ∆ by generators and relations
Leonard pairs and the Askey- Wilson relations
, 2008
"... 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 irr ..."
Abstract
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provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations.
The Universal Askey–Wilson Algebra and the Equitable Presentation of Uq(sl2)
- SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS SIGMA 7 (2011), 099, 26 PAGES;
, 2011
"... Let F denote a field, and fix a nonzero q ∈ F such that q 4 ̸ = 1. The universal Askey–Wilson algebra is the associative F-algebra ∆ = ∆q defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A + qBC − q−1CB q2 − q−2, B + qCA − q−1A ..."
Abstract
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Cited by 14 (3 self)
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Let F denote a field, and fix a nonzero q ∈ F such that q 4 ̸ = 1. The universal Askey–Wilson algebra is the associative F-algebra ∆ = ∆q defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A + qBC − q−1CB q2 − q−2, B + qCA − q−1
www.elsevier.nl/locate/cam The factorization method for the Askey{Wilson polynomials
, 1999
"... A special Infeld{Hull factorization is given for the Askey{Wilson second order q-dierence operator. It is then shown ..."
Abstract
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A special Infeld{Hull factorization is given for the Askey{Wilson second order q-dierence operator. It is then shown
From Quantum Affine Symmetry to Boundary Askey-Wilson Algebra and Reflection Equation
, 804
"... Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The gene ..."
Abstract
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of the model in the stationary state. We discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the Askey-Wilson algebra to the reflection equation. PACS numbers 02.30.Ik, 11.30.Na, 05.50.+k, 05.70.Ln 1
From Quantum Affine Symmetry to Boundary Askey-Wilson Algebra and Reflection Equation
, 804
"... Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The gene ..."
Abstract
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of the model in the stationary state. We discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the Askey-Wilson algebra to the reflection equation. PACS numbers 02.30.Ik, 11.30.Na, 05.50.+k, 05.70.Ln 1
From Quantum Affine Symmetry to Boundary Askey-Wilson Algebra and Reflection Equation
, 804
"... Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The gene ..."
Abstract
- Add to MetaCart
of the model in the stationary state. We discuss the possibility of a solution beyond the stationary state on the basis of the proposed relation of the Askey-Wilson algebra to the reflection equation. PACS numbers 02.30.Ik, 11.30.Na, 05.50.+k, 05.70.Ln 1
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