The Casimir elements of the Racah algebra

Abstract

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].