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Ito, Tatsuro ; Terwilliger, Paul M.
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金沢大学大学院自然科学研究科計算科学<br />金沢大学理学部<br />As part of our study of the q-tetrahedron algebra {squared times}q we introduce the
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notion of a q-inverting pair. Roughly speaking, this is a pair of invertible semisimple linear transformations on a finite-dimensional vector space, each of which acts on the eigenspaces of the other according to a certain rule. Our main result is a bijection between the following two sets: (i) the isomorphism classes of finite-dimensional irreducible {squared times}q-modules of type 1; (ii) the isomorphism classes of q-inverting pairs. © 2007 Elsevier Inc. All rights reserved.
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Ito, Tatsuro ; Terwilliger, Paul
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金沢大学大学院自然科学研究科計算科学<br />金沢大学理学部<br />Let double-struck K denote an algebraically closed field and let q denote a nonzero
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scalar in double-struck K that is not a root of unity. Let V denote a vector space over double-struck K with finite positive dimension and let A, A* denote a tridiagonal pair on V. Let θ0, θ1,..., θd (resp. θ0*, θ1*,..., θd*) denote a standard ordering of the eigenvalues of A (resp. A*). We assume there exist nonzero scalars a, a* in double-struck K such that θi = aq 2i-d and θi* = a*qd-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*. © 2006 Springer Science + Business Media, LLC.
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Ito, Tatsuro ; Terwilliger, P.
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金沢大学大学院自然科学研究科計算科学<br />Recently, Hartwig and the second author found a presentation for the three-point 2 loop algebra
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via generators and relations. To obtain this presentation they defined an algebra by generators and relations, and displayed an isomorphism from to the three-point 2 loop algebra. We introduce a quantum analog of which we call q. We define q via generators and relations. We show how q is related to the quantum group Uq(2), the Uq(2) loop algebra, and the positive part of [image omitted]. We describe the finite dimensional irreducible q-modules under the assumption that q is not a root of 1, and the underlying field is algebraically closed.
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Ito, Tatsuro ; Terwilliger, Paul
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金沢大学大学院自然科学研究科計算科学<br />Let F denote an algebraically closed field with characteristic 0 and let V denote a vector space
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over F with finite positive dimension. Let A, A* denote a tridiagonal pair on V with diameter d. We say that A, A* has Krawtchouk type whenever the sequence {d - 2 i}i = 0d is a standard ordering of the eigenvalues of A and a standard ordering of the eigenvalues of A*. Assume A, A* has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form 〈, 〉 on V such that 〈 Au, v 〉 = 〈 u, Av 〉 and 〈 A* u, v 〉 = 〈 u, A* v 〉 for u, v ∈ V. We show that the following tridiagonal pairs are isomorphic: (i) A, A*; (ii) - A, - A*; (iii) A*, A; (iv) - A*, - A. We give a number of related results and conjectures. © 2007 Elsevier Inc. All rights reserved.
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Ito, Tatsuro ; Terwilliger, Paul ; Weng, Chih-wen
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金沢大学理学部<br />We show that the quantum algebra Uq(sl2) has a presentation with generators x±1,y, z and relations xx-1 = x
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-1x = 1, qxy - q-1yx/q - q-1 = 1, qyz - q-1zy/q - q-1 = 1, qzx - q-1xz/q - q-1 = 1. We call this the equitable presentation. We show that y (respectively z) is not invertible in Uq(sl2) by displaying an infinite-dimensional Uq(sl2)-module that contains a nonzero null vector for y(respectively z). We consider finite-dimensional Uq(sl2)-modules under the assumption that q is not a root of 1 and char (K) ≠ 2, where K is the underlying field. We show that y and z are invertible on each finite-dimensional Uq(sl2)-module. We display a linear operator Ω that acts on finite-dimensional Uq (sl2)-modules, and satisfies Ω-1xΩ = y, Ω-1yΩ = z, Ω-1zΩ = x on these modules. We define Ω using the q-exponential function. © 2005 elsevier Inc. All rigths reserved.
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Ito, Tatsuro ; Nomura, Kazumasa ; Terwilligerc, Paul
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金沢大学理工研究域数物科学系<br />Let F denote a field and let V denote a vector space over F with finite positive dimension. We consi
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der a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi}i = 0d of the eigenspaces of A such that A* Vi ⊆ Vi - 1 + Vi + Vi + 1 for 0 ≤ i ≤ d, where V- 1 = 0 and Vd + 1 = 0; (iii) there exists an ordering {Vi*}i = 0δ of the eigenspaces of A* such that AVi* ⊆ Vi - 1* + Vi* + Vi + 1* for 0 ≤ i ≤ δ, where V- 1* = 0 and Vδ + 1* = 0; (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ and for 0 ≤ i ≤ d the dimensions of Vi, Vd - i, Vi*, Vd - i* coincide. The pair A, A* is called sharp whenever dim V0 = 1. It is known that if F is algebraically closed then A, A* is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture. © 2011 Elsevier Inc. All rights reserved.
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| 7.
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Ito, Tatsuro ; Terwilliger, Paul M.
概要:
Let double-struck K denote an algebraically closed field and let q denote a nonzero scalar in double-struck K that is no
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t a root of unity. Let V denote a vector space over double-struck K with finite positive dimension and let A, A* denote a tridiagonal pair on V. Let θ0, θ1,..., θd (resp. θ0*, θ1*,..., θd*) denote a standard ordering of the eigenvalues of A (resp. A*). We assume there exist nonzero scalars a, a* in double-struck K such that θi = aq 2i-d and θi* = a*qd-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*. © 2006 Springer Science + Business Media, LLC.
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Ito, Tatsuro ; Sato, Jugo
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TD-pairs of type II over C with shape 1, 2, ..., 2, 1 are classified by constructing all of them explicitly as certain s
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ort of tensor product of two L-pairs in a matrix form involving 8 parameters in addition to the diameter. Six of the 8 parameters describe the eigenvalues and the other two give the zeros of the Drinfel'd polynomial. © 2014 Elsevier Inc.
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| 9.
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Ito, Tatsuro ; Terwilligerc, Paul
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金沢大学理工研究域数物科学系<br />We introduce the notion of a mock tridiagonal system. This is a generalization of a tridiagonal syst
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em in which the irreducibility assumption is replaced by a certain nonvanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism. © 2011 Elsevier Inc. All rights reserved.
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Ito, Tatsuro ; Terwilliger, Paul
概要:
We introduce the notion of a mock tridiagonal system. This is a generalization of a tridiagonal system in which the irre
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ducibility assumption is replaced by a certain nonvanishing condition. We show how mock tridiagonal systems can be used to construct tridiagonal systems that meet certain specifications. This paper is part of our ongoing project to classify the tridiagonal systems up to isomorphism. © 2011 Elsevier Inc. All rights reserved.
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| 11.
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Ito, Tatsuro ; Nomura, Kazumasa ; Terwilliger, Paul
概要:
Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear
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transformations A:V→V and A:V→V that satisfy the following conditions: (i) each of A, A is diagonalizable; (ii) there exists an ordering { Vi}i=0d of the eigenspaces of A such that AVi⊆Vi-1+Vi+Vi+1 for 0≤i≤d, where V-1=0 and Vd+1=0; (iii) there exists an ordering {Vi}i=0δ of the eigenspaces of A such that AVi⊆Vi-1+Vi+Vi+1 for 0≤i≤δ, where V-1=0 and Vδ+1=0; (iv) there is no subspace W of V such that AW⊆W, AW⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0≤i≤d the dimensions of Vi,Vd-i,Vi,Vd-i coincide. The pair A,A is called sharp whenever dimV0=1. It is known that if F is algebraically closed then A,A is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture. © 2011 Elsevier Inc. All rights reserved.
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| 12.
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Ito, Tatsuro ; Paul, Terwilliger
概要:
金沢大学理工研究域数物科学系<br />Recently Brian Hartwig and the second author found a presentation for the three-point 2 loop algebra
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by generators and relations. To obtain this presentation they defined a Lie algebra by generators and relations, and displayed an isomorphism from to the three-point 2 loop algebra. In this article, we describe the finite-dimensional irreducible -modules from multiple points of view.全文公開200912
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| 13.
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Ito, Tatsuro ; Paul, Terwilliger
概要:
金沢大学理工研究域数物科学系<br />Let denote a field and let V denote a vector space over F with finite positive dimension. We conside
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r a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi} di=0 of the eigenspaces of A such that A* V i ⊆ Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1 = 0 and Vd+1 = 0; (iii) there exists an ordering {Vi*}δi=0 of the eigenspaces of A* such that AV*i ⊆V* i-1+V*i+V*i+1 for 0 ≤ i ≤ δ, where V*-1=0 and V*δ+1=0 (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ, and for 0 ≤ i ≤ d the dimensions of Vi, V*i, Vd-i, V*d-i coincide. Denote this common dimension by ρi and call A, A* sharp whenever ρ0 = 1. Let T denote the -subalgebra of End (V) generated by A, A*. We show: (i) the center Z(T) is a field whose dimension over is ρ0; (ii) the field Z(T) is isomorphic to each of E 0TE0, EdTEd, E* 0TE*0, E*dTE*d, where Ei (resp. E*i) is the primitive idempotent of A (resp. A*) associated with Vi (resp. V*i); (iii) with respect to the Z(T)-vector space V the pair A, A* is a sharp tridiagonal pair. © 2010 World Scientific Publishing Company.
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| 14.
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Ito, Tatsuro ; Paul, Terwilliger
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金沢大学理工研究域数物科学系<br />Let F denote an algebraically closed field and let V denote a vector space over F with finite positi
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ve dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi}i = 0d of the eigenspaces of A such that A* Vi ⊆ Vi - 1 + Vi + Vi + 1 for 0 ≤ i ≤ d, where V- 1 = 0 and Vd + 1 = 0; (iii) there exists an ordering {Vi*}i = 0δ of the eigenspaces of A* such that A Vi* ⊆ Vi - 1* + Vi* + Vi + 1* for 0 ≤ i ≤ δ, where V- 1* = 0 and Vδ + 1* = 0; (iv) there is no subspace W of V such that A W ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ. For 0 ≤ i ≤ d let θi (resp. θi*) denote the eigenvalue of A (resp. A*) associated with Vi (resp. Vi*). The pair A, A* is said to have q-Racah type whenever θi = a + b q2 i - d + c qd - 2 i and θi* = a* + b* q2 i - d + c* qd - 2 i for 0 ≤ i ≤ d, where q, a, b, c, a*, b*, c* are scalars in F with q, b, c, b*, c* nonzero and q2 ∉ {1, - 1}. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over F that have q-Racah type. Our proof involves the representation theory of the quantum affine algebra Uq (over(sl, ̂)2). © 2009 Elsevier Inc. All rights reserved.
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