## Abstract

We show that the quantum algebra U_{q}(sl_{2}) has a presentation with generators x^{±1},y, z and relations xx^{-1} = x^{-1}x = 1, qxy - q^{-1}yx/q - q^{-1} = 1, qyz - q^{-1}zy/q - q^{-1} = 1, qzx - q^{-1}xz/q - q^{-1} = 1. We call this the equitable presentation. We show that y (respectively z) is not invertible in U_{q}(sl_{2}) by displaying an infinite-dimensional U_{q}(sl_{2})-module that contains a nonzero null vector for y(respectively z). We consider finite-dimensional U_{q}(sl_{2})-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 U_{q}(sl_{2})-module. We display a linear operator Ω that acts on finite-dimensional U_{q} (sl_{2})-modules, and satisfies Ω^{-1}xΩ = y, Ω^{-1}yΩ = z, Ω^{-1}zΩ = x on these modules. We define Ω using the q-exponential function.

Original language | English |
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Pages (from-to) | 284-301 |

Number of pages | 18 |

Journal | Journal of Algebra |

Volume | 298 |

Issue number | 1 |

DOIs | |

State | Published - 1 Apr 2006 |

## Keywords

- Leonard pair
- Quantum algebra
- Quantum group
- Tridiagonal pair