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68
Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition
, 2003
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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∗.
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
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.
A family of tridiagonal pairs related to the quantum affine algebra Uq( sl2
 Electron. J. Linear Algebra
"... Abstract. A type of tridiagonal pair is considered, said to be mild of qSerre type.It is shown that these tridiagonal pairs induce the structure of a quantum affine algebra Uq ( sl2)module on their underlying vector space.This is done by presenting an explicit basis for the underlying vector spa ..."
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Cited by 23 (1 self)
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Abstract. A type of tridiagonal pair is considered, said to be mild of qSerre type.It is shown that these tridiagonal pairs induce the structure of a quantum affine algebra Uq ( sl2)module on their underlying vector space.This is done by presenting an explicit basis for the underlying vector space and describing the Uq ( sl2)action on that basis.
An integrable structure related with tridiagonal algebras, Nuclear Phys
 B
"... The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the DolanGrady construction. The involution property relies on the tridiagonal algebraic structure associated with a defor ..."
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Cited by 21 (7 self)
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The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the DolanGrady construction. The involution property relies on the tridiagonal algebraic structure associated with a deformation parameter q. Representations are shown to be generated from a class of quadratic algebras, namely the reflection equations. The spectral problem is briefly discussed. Finally, related massive quantum integrable models are shown to be superintegrable.
A family of tridiagonal pairs and related symmetric functions
 J. Phys. A: Math. Gen
"... A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions be ..."
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Cited by 18 (6 self)
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A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete secondorder q−difference equations which generalize the ones associated with the AskeyWilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.
Some trace formulae involving the split sequences of a Leonard pair
 Linear Algebra Appl
"... Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A: V → V and A ∗ : V → V that satisfy (i), (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal a ..."
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Cited by 18 (16 self)
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Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A: V → V and A ∗ : V → V that satisfy (i), (ii) below: (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. Let diag(θ0, θ1,..., θd) denote the diagonal matrix referred to in (ii) above and let diag(θ ∗ 0, θ ∗ 1,...,θ ∗ d) denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u0, u1,..., ud for V and there exist scalars ϕ1, ϕ2,..., ϕd in K such that Aui = θiui + ui+1 (0 ≤ i ≤ d − 1), Aud = θdud, A ∗ ui = ϕiui−1 + θ ∗ i ui (1 ≤ i ≤ d), A ∗ u0 = θ ∗ 0u0. The sequence ϕ1, ϕ2,..., ϕd is called the first split sequence of the Leonard pair. It is known that there exists a basis v0, v1,...,vd for V and there exist scalars φ1, φ2,..., φd in K such