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116
Tridiagonal pairs and the quantum affine algebra . . .
, 2003
"... Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A,A ∗ denote a tridiagonal pair on V. Let θ0,θ1,...,θd (resp. θ ∗ 0,θ ∗ 1,...,θ ∗ d) denote a standard ordering of ..."
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Cited by 58 (27 self)
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Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A,A ∗ denote a tridiagonal pair on V. Let θ0,θ1,...,θd (resp. θ ∗ 0,θ ∗ 1,...,θ ∗ d) denote a standard ordering of the eigenvalues of A (resp. A ∗). We assume there exist nonzero scalars a,a ∗ in K such that θi = aq 2i−d and θ ∗ i = a ∗ q d−2i for 0 ≤ i ≤ d. We display two irreducible Uq ( ̂ sl2)module structures on V and discuss how these are related to the actions of A and A∗.
Leonard pairs and the AskeyWilson relations
 Department of Computational Science Faculty of Science Kanazawa University Kakumamachi 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 AskeyWilson 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 ∗ ( ..."
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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 AskeyWilson 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 β, γ, γ ∗ , ̺, ̺ ∗ , ω,η, η ∗. 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 ∗ A ∗ +c ∗ ) is a Leonard pair on V as well. These affine transformations can be used to bring the Leonard pair or its AskeyWilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit AskeyWilson relations satisfied by them. 1
The quantum algebra Uq(sl2) and its equitable presentation
 J. Algebra
"... We show that the quantum algebra Uq(sl2) has a presentation with generators x ±1,y,z and relations xx −1 = x −1 x = 1, qxy − q−1yx = 1, q − q−1 qyz − q−1zy ..."
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Cited by 32 (18 self)
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We show that the quantum algebra Uq(sl2) has a presentation with generators x ±1,y,z and relations xx −1 = x −1 x = 1, qxy − q−1yx = 1, q − q−1 qyz − q−1zy
Balanced Leonard pairs
 Linear Algebra Appl. Submitted
"... Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A: V → V and A ∗ : V → V that satisfy the following two conditions: (i) There exists a basis for V with respect to which the matrix re ..."
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Cited by 28 (17 self)
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Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A: V → V and A ∗ : V → V that satisfy the following two conditions: (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. Let v ∗ 0, v ∗ 1,..., v ∗ d (respectively v0, v1,..., vd) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ≤ i ≤ d, let ai denote the coefficient of v ∗ i, when we write Av ∗ i as a linear combination of v ∗ 0, v ∗ 1,..., v ∗ d, and let a∗i denote the coefficient of vi, when we write A∗vi as a linear combination of v0, v1,..., vd.. Moreover we show that for d ≥ 1 the In this paper we show a0 = ad if and only if a ∗ 0 = a ∗ d
The Tetrahedron algebra, the Onsager algebra, and the sl2 loop algebra
, 2005
"... Let K denote a field with characteristic 0 and let T denote an indeterminate. We give a presentation for the threepoint loop algebra sl2 ⊗ K[T,T −1,(T − 1) −1] via generators and relations. This presentation displays S4symmetry. Using this presentation we obtain a decomposition of the above loop a ..."
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Cited by 27 (15 self)
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Let K denote a field with characteristic 0 and let T denote an indeterminate. We give a presentation for the threepoint loop algebra sl2 ⊗ K[T,T −1,(T − 1) −1] via generators and relations. This presentation displays S4symmetry. Using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras, each of which is isomorphic to the Onsager algebra.
Deformed DolanGrady relations in quantum integrable models, Nucl.Phys
 B
"... A new hidden symmetry is exhibited in the reflection equation and related quantum integrable models. It is generated by a dual pair of operators {A, A ∗ } ∈ A subject to q−deformed DolanGrady relations. Using the inverse scattering method, a new family of quantum integrable models is proposed. In ..."
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Cited by 25 (9 self)
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A new hidden symmetry is exhibited in the reflection equation and related quantum integrable models. It is generated by a dual pair of operators {A, A ∗ } ∈ A subject to q−deformed DolanGrady relations. Using the inverse scattering method, a new family of quantum integrable models is proposed. In the simplest case, the Hamiltonian is linear in the fundamental generators of A. For general values of q, the corresponding spectral problem is quasiexactly solvable. Several examples of twodimensional massive/massless (boundary) integrable models are reconsidered in light of this approach, for which the fundamental generators of A are constructed explicitly and exact results are obtained. In particular, we exhibit a dynamical AskeyWilson symmetry algebra in the (boundary) sineGordon model and show that asymptotic (boundary) states can be expressed in terms of q−orthogonal polynomials.
Commutative association schemes
 European J. Combin
"... Abstract. Association schemes were originally introduced by Bose and his coworkers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield ..."
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Cited by 22 (7 self)
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Abstract. Association schemes were originally introduced by Bose and his coworkers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the “commutative case, ” has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and grouptheoretic symmetry, culminating in Schrijver’s SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work. 1.
Irreducible modules for the quantum affine algebra Uq( ̂ sl2) and its Borel subalgebra Uq( ̂ sl2) ≥0
 J. Algebra
"... Let Uq ( ̂ sl2) ≥0 denote the Borel subalgebra of the quantum affine algebra Uq ( ̂ sl2). We show that the following hold for any choice of scalars ε0, ε1 from the set {1, −1}. (i) Let V be a finitedimensional irreducible Uq ( ̂ sl2) ≥0module of type (ε0, ε1). Then the action of Uq ( ̂ sl2) ≥0 ..."
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Cited by 21 (8 self)
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Let Uq ( ̂ sl2) ≥0 denote the Borel subalgebra of the quantum affine algebra Uq ( ̂ sl2). We show that the following hold for any choice of scalars ε0, ε1 from the set {1, −1}. (i) Let V be a finitedimensional irreducible Uq ( ̂ sl2) ≥0module of type (ε0, ε1). Then the action of Uq ( ̂ sl2) ≥0 on V extends uniquely to an action of Uq ( ̂ sl2) on V. The resulting Uq ( ̂ sl2)module structure on V is irreducible and of type (ε0, ε1). (ii) Let V be a finitedimensional irreducible Uq ( ̂ sl2)module of type (ε0, ε1). When the Uq ( ̂ sl2)action is restricted to Uq ( ̂ sl2) ≥0, the resulting Uq ( ̂ sl2) ≥0module structure on V is irreducible and of type (ε0, ε1). 1 The quantum affine algebra Uq ( ̂ sl2) The affine KacMoody Lie algebra ̂ sl2 has played an essential role in diverse areas of mathematics and physics. Elements of ̂ sl2 can be represented as vertex operators, which are certain generating functions that appear in the dual resonance models of particle physics (see [15] and [8]). The algebra ̂ sl2 also features prominently in the study of KnizhnikZamolodchikov equations Support from NSF grant #DMS–0245082 is gratefully acknowledged.