The quantum algebra Uq(sl2) and its equitable presentation

Tatsuro Ito, Paul Terwilliger*, Chih-wen Weng

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

61 Scopus citations


We show that the quantum algebra Uq(sl2) has a presentation with generators x±1,y, z and relations xx-1 = x-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.

Original languageEnglish
Pages (from-to)284-301
Number of pages18
JournalJournal of Algebra
Issue number1
StatePublished - 1 Apr 2006


  • Leonard pair
  • Quantum algebra
  • Quantum group
  • Tridiagonal pair


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