Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 018, 17 pages      arXiv:1910.11446      https://doi.org/10.3842/SIGMA.2020.018

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

Hau-Wen Huang ^a and Sarah Bockting-Conrad ^b
a)  Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
^b)  Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

Received November 12, 2019, in final form March 16, 2020; Published online March 24, 2020

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.

Key words: Racah algebra; quadratic algebra; irreducible modules; tridiagonal pairs; universal property.

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