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 thatand that each ofIn 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.

The universal DAHA of type (C1∨,C1) and Leonard triples

“…Proposition 2.1 (Proposition 2.4, [9]). For any scalars a, b, c ∈ F and any integer d ≥ 0, there exists a (d + 1)-dimensional ℜ-module R d (a, b, c) satisfying the following conditions:…”

“…An algebra is meant to be a unital associative algebra over F. An algebra homomorphism is meant to be a unital algebra homomorphism. Definition 1.1 (Definition 2.1, [9]). The universal Racah algebra ℜ is an algebra defined by generators and relations in the following way: The generators are A, B, C, D and the relations state that Using (1) yields that δ is central in ℜ.…”

“…Theorem 6.6,[9]). Let d ≥ 0 denote an integer and let a, b, c denote scalars in F. Suppose that the ℜ-module R d (a, b, c) is irreducible.…”

The Clebsch--Gordan rule and the Hamming graphs

“…Proof. The relation (3) is immediate from the setting of C. Using (3) the relations (1) and (2) can be written as (4) and (5), respectively. The lemma follows.…”

“…Theorem 2.5(iii) is immediate from Lemma 1.4 and Theorem 2.3. To see Theorem 2.5(i), (ii) one may apply the method similar to [3][4][5]. We omit the proofs for Theorem 2.5(i), (ii) because they are not related to the main results of this paper.…”