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• W1006833761 abstract "Let K denote an algebraically closed 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 that satisfy the following four 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 does not exist a 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; to avoid trivialities assume d⩾1. We assume that A,A⁎ belongs to a family of tridiagonal pairs said to have q-Racah type. This is the most general type of tridiagonal pair. Let {Ui}i=0d and {Ui⇓}i=0d denote the first and second split decompositions of V. In an earlier paper we introduced the double lowering operator ψ:V→V. One feature of ψ is that both ψUi⊆Ui−1 and ψUi⇓⊆Ui−1⇓ for 0⩽i⩽d, where U−1=0 and U−1⇓=0. Define linear transformations K:V→V and B:V→V such that (K−qd−2iI)Ui=0 and (B−qd−2iI)Ui⇓=0 for 0⩽i⩽d. Our results are summarized as follows. Using ψ, K, B we obtain two actions of Uq(sl2) on V. For each of these Uq(sl2)-module structures, the Chevalley generator e acts as a scalar multiple of ψ. For each of the Uq(sl2)-module structures, we compute the action of the Casimir element on V. We show that these two actions agree. Using this fact, we express ψ as a rational function of K±1,B±1 in several ways. Eliminating ψ from these equations we find that K and B are related by a quadratic equation." @default.
• W1006833761 created "2016-06-24" @default.
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• W1006833761 date "2014-03-01" @default.
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• W1006833761 title "Tridiagonal pairs of q -Racah type, the double lowering operator ψ , and the quantum algebra<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:msub><mml:mrow><mml:mi>U</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy=false>(</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant=fraktur>sl</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo stretchy=false>)</mml:mo></mml:math>" @default.
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• W1006833761 doi "https://doi.org/10.1016/j.laa.2013.12.007" @default.
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