The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra

Abstract

Assume that $\mathbb{F}$ is a field with $\mathrm{char}\mathbb{F}\ne 2$. The Racah algebra $\Re$ is a unital associative $\mathbb{F}$-algebra defined by generators and relations. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. The Bannai-Ito algebra $\mathfrak{BI}$ is a unital associative $\mathbb{F}$-algebra generated by $X$, $Y$, $Z$ and the relations assert that each of $\{X,Y\}-Z$, $\{Y,Z\}-X$, $\{Z,X\}-Y$ is central in $\mathfrak{BI}$. It was discovered that there exists an $\mathbb{F}$-algebra homomorphism $\zeta :\Re \to \mathfrak{BI}$ that sends $A\mapsto \frac{(2X-3)(2X+1)}{16}$, $B\mapsto \frac{(2Y-3)(2Y+1)}{16}$, $C\mapsto \frac{(2Z-3)(2Z+1)}{16}$. We show that $\zeta$ is injective and therefore $\Re$ can be considered as an $\mathbb{F}$-subalgebra of $\mathfrak{BI}$. Moreover we show that any Casimir element of $\Re$ can be uniquely expressed as a polynomial in $\{X,Y\}-Z$, $\{Y,Z\}-X$, $\{Z,X\}-Y$ and $X+Y+Z$ with coefficients in $\mathbb{F}$.

References in corpus (8)

• The Universal Askey-Wilson Algebra
• The Bannai-Ito algebra and some applications
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• $\mathfrak{osp}(1,2)$ and generalized Bannai-Ito algebras
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• Bannai-Ito algebras and the $osp(1,2)$ superalgebra

Cited by in corpus (2)

• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero