The Casimir elements of the Racah algebra

Let [Formula: see text] denote a field with [Formula: see text]. The Racah algebra [Formula: see text] is the unital associative [Formula: see text]-algebra defined by generators and relations in the following way. The generators are [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The relations assert that [Formula: see text] and each of the elements [Formula: see text] is central in [Formula: see text]. Additionally, the element [Formula: see text] is central in [Formula: see text]. We call each element in [Formula: see text] a Casimir element of [Formula: see text], where [Formula: see text] is the commutative subalgebra of [Formula: see text] generated by [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of [Formula: see text]: [Formula: see text] [Formula: see text] [Formula: see text] The set [Formula: see text] is invariant under a faithful [Formula: see text]-action on [Formula: see text]. Moreover, we show that any Casimir element [Formula: see text] is algebraically independent over [Formula: see text]; if [Formula: see text], then the center of [Formula: see text] is [Formula: see text].

New realizations of algebras of the Askey–Wilson type in terms of Lie and quantum algebras

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.