The Askey–Wilson algebra is realized in terms of the elements of the quantum algebras [Formula: see text] or [Formula: see text]. A new realization of the Racah algebra in terms of the Lie algebras [Formula: see text] or [Formula: see text] is also given. Details for different specializations are provided. The advantage of these new realizations is that one generator of the Askey–Wilson (or Racah) algebra becomes diagonal in the usual representation of the quantum algebras whereas the second one is tridiagonal. This allows us to recover easily the recurrence relations of the associated orthogonal polynomials of the Askey scheme. These realizations involve rational functions of the Cartan generator of the quantum algebras, where they are linear with respect to the other generators and depend on the Casimir element of the quantum algebras.

中文翻译：

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根据李和量子代数对 Askey-Wilson 类型代数的新实现

Askey-Wilson 代数是根据量子代数 [公式：见正文] 或 [公式：见正文] 的元素来实现的。还给出了根据李代数[公式：见正文]或[公式：见正文]的拉卡代数的新实现。提供了不同专业的详细信息。这些新实现的优点是，Askey-Wilson（或 Racah）代数的一个生成器在量子代数的通常表示中变成对角线，而第二个生成器是三对角线。这使我们能够轻松地恢复 Askey 方案的相关正交多项式的递归关系。这些实现涉及量子代数的嘉当发生器的有理函数，其中它们相对于其他发生器是线性的并且取决于量子代数的卡西米尔元素。