Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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LEONARD PAIRS GENERATED FROM Uq(sl2)

  • Received : 2022.02.02
  • Accepted : 2022.07.29
  • Published : 2022.09.30
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Abstract

Consider the quantum algebra Uq(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 ∈ Uq(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.

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Acknowledgement

This work was supported by the research grant of the University of Jordan.

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