• Journal of applied mathematics & informatics
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• v.40 no.5_6
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• pp.1137-1150
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• 2022

Consider the quantum algebra U_q(sl₂) over field 𝓕 (char(𝓕) = 0) with equitable generators x^±1, y and z, where q is fixed nonzero, not root of unity scalar in 𝓕. Let V denote a finite dimensional irreducible module for this algebra. Let Λ ∈ End(V), and let {A₁, A₂, A₃} = {x, y, z}. First we show that if Λ, A₁ is a Leonard pair, then this Leonard pair have four types, and we show that for each type there exists a Leonard pair Λ, A₁ in which Λ is a linear combination of 1, A₂, A₃, A₂A₃. Moreover, we use Λ to construct 𝚼 ∈ U_q(sl₂) such that 𝚼, A^-1₁ is a Leonard pair, and show that 𝚼 = I + A₁Φ + A₁ΨA₁ where Φ and Ψ are linear combination of 1, A₂, A₃.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.401-414
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• 2019

Let ${\mathcal{F}}$ denote an algebraically closed field with characteristic not two. Fix an integer $d{\geq}3$, let $Mat_{d+1}({\mathcal{F}})$ denote the ${\mathcal{F}}$-algebra of $(d+1){\times}(d+1)$ matrices with entries in ${\mathcal{F}}$. An ordered pair of matrices A, $A^*$ in $Mat_{d+1}({\mathcal{F}})$ is said to be LB-TD form whenever A is lower bidiagonal with subdiagonal entries all 1 and $A^*$ is irreducible tridiagonal. Let A, $A^*$ be a Leonard pair in $Mat_{d+1}({\mathcal{F}})$ with fundamental parameter ${\beta}=2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.