The Casimir elements of the Racah algebra

TL;DR

This paper characterizes the Casimir elements of the Racah algebra, showing their explicit forms, invariance properties, and the structure of the algebra's center, especially over fields of characteristic zero.

Contribution

It introduces explicit Casimir elements for the Racah algebra and proves their algebraic independence and invariance under a dihedral group action.

Findings

Explicit formulas for Casimir elements , , .

Invariance of these Casimir elements under a D6 group action.

The center of the Racah algebra is generated by the Casimir element  in characteristic zero.

Abstract

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*} \alpha=[A,D]+AC-BA, \qquad \beta=[B,D]+BA-CB, \qquad \gamma=[C,D]+CB-AC \end{gather*} is central in $\mathfrak{R}$. Additionally the element $\delta=A+B+C$ 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{(\delta+2)\{A,B\}-\{A^2,B\}-\{A,B^2\}}{2} +A (\beta-\delta) +B (\delta-\alpha)+\mathfrak{C} \end{equation*} a Casimir element of $\mathfrak{R}$, where $\mathfrak{C}$ is the commutative subalgebra of $\mathfrak{R}$ generated by $\alpha$, $\beta$, $\gamma$, $\delta$. The main results of this paper are as follows. Each of the following distinct elements is a Casimir element of $\mathfrak{R}$: \begin{align*} \Omega_A = D^2 + \frac{B A C +C A B}{2} + A^2 +B \gamma -C \beta -A \delta, \Omega_B = D^2 + \frac{C B A +A B C}{2} + B^2 +C \alpha -A \gamma -B\delta, \Omega_C = D^2 + \frac{A C B +B C A}{2} + C^2 +A \beta -B\alpha -C\delta. \end{align*} The set $\{\Omega_A,\Omega_B,\Omega_C\}$ is invariant under a faithful $D_6$-action on $\mathfrak{R}$. Moreover we show that any Casimir element $\Omega$ is algebraically independent over $\mathfrak{C}$; if ${\rm char\,}\mathbb{F}=0$ then the center of $\mathfrak{R}$ is $\mathfrak{C}[\Omega]$.