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

Abstract

Assume that 𝔽 is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative 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 ℜ. In this paper, we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules.