Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero

Abstract

Assume that $\mathbb{F}$ is an algebraically closed with characteristic $0$. 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 \begin{gather*} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather*} is central in $\mathfrak{BI}$. In this paper we classify the finite-dimensional irreducible $\mathfrak{BI}$-modules up to isomorphism. As we will see the elements $X,Y,Z$ are not always diagonalizable on finite-dimensional irreducible $\mathfrak{BI}$-modules.

The paper is to correct the main result of Communications in Algebra 44 (2016), 919-943; the paper arXiv:1910.11446 is to correct the main result of Communications in Algebra 47 (2019), 1869-1891

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
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• 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
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• $\mathfrak{osp}(1,2)$ and generalized Bannai-Ito algebras

Cited by in corpus (6)

• Finite-Dimensional Irreducible Modules of the Racah 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
• Finite-dimensional modules of the universal Askey--Wilson algebra and DAHA of type $(C_1^{\vee },C_1)$
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• The Clebsch-Gordan Rule for $U({\mathfrak{sl}}_2)$, the Krawtchouk Algebras and the Hamming Graphs