Let 𝔽 denote a field with char 𝔽≠2. The Racah algebra ℜ is the unital associative 𝔽-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 ℜ. Additionally, the element δ = A + B + C is central in ℜ. We call each element in D2 + A2 + B2 + (δ + 2){A,B}−{A2,B}−{A,B2} 2 + A(β − δ) + B(δ − α) + ℭ a Casimir element of ℜ, where ℭ is the commutative subalgebra of ℜ generated by α, β, γ, δ. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of ℜ: ΩA = D2 + BAC + CAB 2 + A2 + Bγ − Cβ − Aδ, ΩB = D2 + CBA + ABC 2 + B2 + Cα − Aγ − Bδ, ΩC = D2 + ACB + BCA 2 + C2 + Aβ − Bα − Cδ. The set {ΩA, ΩB, ΩC} is invariant under a faithful D6-action on ℜ. Moreover, we show that any Casimir element Ω is algebraically independent over ℭ; if char 𝔽 = 0, then the center of ℜ is ℭ[Ω].

中文翻译：

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拉卡代数的卡西米尔元素

让𝔽用字符𝔽≠2. 拉卡代数ℜ是单位联想𝔽- 由生成器和关系以下列方式定义的代数。发电机是一种,乙,C,D. 关系断言 [一种,乙] = [乙,C] = [C,一种] = 2D 和每个元素 α = [一种,D] + 一种C - 乙一种,β = [乙,D] + 乙一种 - C乙,γ = [C,D] + C乙 - 一种C 位于中心ℜ. 此外，该元素δ = 一种 + 乙 + C位于中心ℜ. 我们将每个元素称为 D2 + 一种2 + 乙2 + (δ + 2){一种,乙}-{一种2,乙}-{一种,乙2} 2 + 一种(β - δ) + 乙(δ - α) + ℭ 卡西米尔元素ℜ， 在哪里ℭ是的交换子代数ℜ由产生α,β,γ,δ. 本文的主要结果如下。以下每个不同的元素都是ℜ： Ω一种 = D2 + 乙一种C + C一种乙 2 + 一种2 + 乙γ - Cβ - 一种δ, Ω乙 = D2 + C乙一种 + 一种乙C 2 + 乙2 + Cα - 一种γ - 乙δ, ΩC = D2 + 一种C乙 + 乙C一种 2 + C2 + 一种β - 乙α - Cδ. 套装{Ω一种, Ω乙, ΩC}在忠实的情况下是不变的D6- 采取行动ℜ. 此外，我们证明了任何卡西米尔元素Ω是代数独立的ℭ; 如果字符𝔽 = 0, 那么中心ℜ是ℭ[Ω].