Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

Abstract

Assume that $\mathbb{F}$ is an algebraically closed field with characteristic zero. The Racah algebra $\Re$ is the unital associative $\mathbb{F}$-algebra defined by generators and relations in the following way. The generators are $A$, $B$, $C$, $D$ and the relations assert that $[A,B]=[B,C]=[C,A]=2D$ and that each of $[A,D]+AC-BA$, $[B,D]+BA-CB$, $[C,D]+CB-AC$ is central in $\Re$. In this paper we discuss the finite-dimensional irreducible $\Re$-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional $\Re$-module and its universal property. We additionally give the necessary and sufficient conditions for $A$, $B$, $C$ to be diagonalizable on finite-dimensional irreducible $\Re$-modules.

References in corpus (10)

• Superintegrability in two dimensions and the Racah-Wilson algebra
• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• The equitable Racah algebra from three su(1,1) algebras
• The Bannai-Ito algebra and some applications
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• The Casimir elements of the Racah algebra
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• A discrete realization of the higher rank Racah algebra

Cited by in corpus (9)

• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• The Clebsch--Gordan coefficients of $U({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Johnson graphs
• An imperceptible connection between the Clebsch--Gordan coefficients of $U_q({\mathfrak{sl}}_2)$ and the Terwilliger algebras of Grassmann graphs
• Bethe ansatz diagonalization of the Heun-Racah operator
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• The universal enveloping algebra of ${\mathfrak{sl}}_2$ and the Racah algebra
• The Clebsch-Gordan Rule for $U({\mathfrak{sl}}_2)$, the Krawtchouk Algebras and the Hamming Graphs
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions