The Casimir elements of the Racah algebra

Let denote a field with char a 2. The Racah algebra a is the unital associative-Algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that [A,B] = [B,C] = [C,A] = 2D and each of the elements α = [A,D] + AC-BA,β = [B,D] + BA-CB,γ = [C,D] + CB-AC is central in a. Additionally, the element = A + B + C is central in a. We call each element in D2 + A2 + B2 + (I + 2){A,B}-{A2,B}-{A,B2} 2 + A(β-I) + B(I-α) + a Casimir element of a, where is the commutative subalgebra of a generated by α, β, γ,. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of a: ωA = D2 + BAC + CAB 2 + A2 + Bγ-Cβ-A, ωB = D2 + CBA + ABC 2 + B2 + Cα-Aγ-B, ωC = D2 + ACB + BCA 2 + C2 + Aβ-Bα-C. The set {ωA, ωB, ωC} is invariant under a faithful D6-Action on a. Moreover, we show that any Casimir element ω is algebraically independent over if char = 0, then the center of a is [ω].