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The Casimir elements of the Racah algebra

Let $\mathbb{F}$ denote a field with ${\rm char\,}\mathbb{F}\not=2$. The Racah algebra $\mathfrak{R}$ is the unital associative $\mathbb{F}$-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 \begin{gather*} α=[A,D]+AC-BA, \qquad β=[B,D]+BA-CB, \qquad γ=[C,D]+CB-AC \end{gather*} is central in $\mathfrak{R}$. Additionally the element $δ=A+B+C$ is central in $\mathfrak{R}$. The algebra $\mathfrak{R}$ was introduced by Genest-Vinet-Zhedanov. We consider a mild change in their setting to call each element in \begin{equation*} D^2+A^2+B^2 +\frac{(δ+2)\{A,B\}-\{A^2,B\}-\{A,B^2\}}{2} +A (β-δ) +B (δ-α)+\mathfrak{C} \end{equation*} a Casimir element of $\mathfrak{R}$, where $\mathfrak{C}$ is the commutative subalgebra of $\mathfrak{R}$ 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 $\mathfrak{R}$: \begin{align*} Ω_A = D^2 + \frac{B A C +C A B}{2} + A^2 +B γ-C β-A δ, Ω_B = D^2 + \frac{C B A +A B C}{2} + B^2 +C α-A γ-Bδ, Ω_C = D^2 + \frac{A C B +B C A}{2} + C^2 +A β-Bα-Cδ. \end{align*} The set $\{Ω_A,Ω_B,Ω_C\}$ is invariant under a faithful $D_6$-action on $\mathfrak{R}$. Moreover we show that any Casimir element $Ω$ is algebraically independent over $\mathfrak{C}$; if ${\rm char\,}\mathbb{F}=0$ then the center of $\mathfrak{R}$ is $\mathfrak{C}[Ω]$.