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

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.


Introduction
Throughout this paper, we adopt the following conventions. Let F denote an algebraically closed field and let char F denote the characteristic of F. Let Z denote the set of integers and let N denote the set of nonnegative integers. The bracket [ , ] stands for the commutator.
In this paper we consider the Racah algebra over F defined by generators and relations as follows. The generators are A, B, C, D and the relations assert that is central in . The Racah algebra is a universal analog of the original Racah algebras which first appeared in [17]. In that paper, the original Racah algebras were used to establish a link between representation theory and the quantum mechanical coupling of three angular momenta. Since that time, the connections between the Racah algebras and many other areas have been explored. We mention a few of them here. Their connections with the additive double-affine Hecke algebra of type C ∨ 1 , C 1 , the Bannai-Ito algebra, and the Lie algebras su(2), su (1,1) were investigated in [6,9,10,14]. Their realizations via the Racah polynomials, the intermediate Casimir operators, and the superintegrable models in two dimensions were presented in [5,7,8,9,11]. For information concerning the higher rank Racah algebras, see [3,4].
We now mention an error in the literature on Racah algebras. In [2], the authors considered the finite-dimensional irreducible modules of the original Racah algebras when char F = 0.
In [2,Lemma 5.6], it was claimed that the defining generators can be diagonalized on any finite-dimensional irreducible module of the Racah algebras. This result was then used in their classification of finite-dimensional irreducible modules of the Racah algebras in [2,Section 6]. It turns out that [2,Lemma 5.6] is conditional. We give the following example to help illustrate the issue arising in [2]. Therefore none of A, B, C is diagonalizable on V . We now show that V is in fact irreducible. Let W denote a nonzero -submodule of V . We will show that W = V . Observe that the element B has exactly three eigenvalues on V , namely 15 4 , 3 4 , − 1 4 . A direct calculation yields that the corresponding eigenspaces are each of dimension 1 and are spanned by respectively. Since W is nonzero, at least one of 15 4 , 3 4 , − 1 4 is an eigenvalue of B on W . Therefore W contains at least one of the elements listed in (1.1). Observe that the -module V is generated by and hence W = V . Therefore W = V . Since the -module V is irreducible, we now have a counterexample to [2, Lemma 5.6].
In light of the above example, we see that the finite-dimensional irreducible -modules are not yet completely classified. The goal of this paper is to provide such a classification. The idea of our classification comes from [13]. We mention that a similar issue arises in the case of the Bannai-Ito algebra BI [12] which is addressed by the first author in [16]. The result [15,Theorem 5.4] reveals that the Racah algebra is isomorphic to an F-subalgebra of BI. As an application of [16] and this result, the lattices of -submodules of finite-dimensional irreducible BI-modules are classified in [14].
The outline of this paper is as follows. In Section 2 we state our classification of finitedimensional irreducible -modules in Theorem 2.5. In Section 3 we display an infinite-dimensional -module and describe its universal property. In Section 4 we give necessary and sufficient conditions for the irreducibility of finite-dimensional -modules. In Section 5 we study the isomorphism classes of finite-dimensional irreducible -modules. In Section 6 we give our proof of Theorem 2.5.

Statement of results
In this section we more formally introduce the Racah algebra and state the main result of the paper which gives a classification of the finite-dimensional irreducible modules of the Racah algebra . This main result will be proved later in Section 6. It follows from the above definition that the element A + B + C is also central in . For notational convenience, we let (i) The Racah algebra is generated by the elements A, B, C.
(ii) The Racah algebra is generated by the elements A, B, δ.
Proof . (i) By (2.1) the element D can be expressed in terms of A, B. Hence (i) follows from Definition 2.1. (ii) By (2.5) the element C can be expressed in terms of A, B, δ. Hence (ii) follows from (i).
Lemma 2.3. The F-algebra has a presentation given by generators A, B, α, β, δ and relations Proof . We know from Lemma 2.2(ii) that A, B, α, β, δ generate . Observe that C = δ −A−B by (2.5) and D = 1 2 [A, B] by (2.1). The result can now be obtained by either using these two facts to eliminate C, D from the presentation of given in Definition 2.1 or by using D = 1 2 [A, B] to eliminate D from the presentation of given in [1,Proposition 3.4].
In the following proposition, we assert the existence of certain finite-dimensional -modules and describe the actions of the generators of on these modules. A reader familiar with the theory of tridiagonal pairs will immediately recognize the form of the matrices representing A and B as precisely those given in Terwilliger's 2001 seminal work on tridiagonal pairs [18,Theorem 3.2].
Proposition 2.4. For any a, b, c ∈ F and any d ∈ N there exists a (d+1)-dimensional -module R d (a, b, c) satisfying each of the following conditions: with respect to which the matrices representing A and B are (ii) The elements α, β, δ act on R d (a, b, c) as scalar multiplication by

respectively.
Proof . Using Lemma 2.3, this result can be verified through routine, though tedious, computations.
In order to state our main result more succinctly, we will use the following conventions and definitions. Let d ∈ N and let P = P d denote the set of all (a, b, c) ∈ F 3 that satisfy We define an action of the abelian group for all (a, b, c) ∈ P. We let P/{±1} 3 denote the set of the {±1} 3 -orbits of P. For (a, b, c) ∈ P, let [a, b, c] denote the {±1} 3 -orbit of P that contains (a, b, c). We are now ready to state the classification of finite-dimensional irreducible -modules.
Theorem 2.5. Assume that F is algebraically closed with char F = 0. Let d denote a nonnegative integer. Let M denote the set of all isomorphism classes of irreducible -modules that have dimension d + 1. Then there exists a bijection R : We will give a proof of Theorem 2.5 in Section 6.

An infinite-dimensional -module and its universal property
In this section we introduce an infinite-dimensional -module and its universal property in order to prove Theorem 2.5. For convenience the following conventions are used throughout the rest of this paper. We let a, b, c, ν denote any scalars in F. We define the following families of parameters associated with a, b, c, ν: Proposition 3.1. There exists an -module M ν (a, b, c) satisfying each of the following conditions: respectively.
Proof . Using Lemma 2.3, this result can be verified through routine computations.
Throughout the rest of this paper we will let {m i } ∞ i=0 denote the F-basis for M ν (a, b, c) from Proposition 3.1(i). The following result is an immediate consequence of Proposition 3.1(i).
Shortly we will describe the -module M ν (a, b, c) in an alternate way. To aid us in this endeavor, we first recall a Poincaré-Birkhoff-Witt basis for .
form an F-basis of .
Let I ν (a, b, c) denote the left ideal of generated by the elements We now consider certain cosets of /I ν (a, b, c).
Lemma 3.4. For each n ∈ N, each of the following holds: Proof . (i) We proceed by induction on n. Since I ν (a, b, c) contains the element (3.8), the statement holds for n = 0. Since I ν (a, b, c) contains both of the elements (3.8) and (3.9), the statement holds for n = 1. Now suppose n ≥ 2. Right multiplying each side of (2.6) by A n−2 yields that Since I ν (a, b, c) contains each of the elements listed in (3.10), it follows that BA n is congruent to modulo I ν (a, b, c). By the inductive hypothesis, the element (3.11) is congruent to an F-linear combination of A i , for all 0 ≤ i ≤ n, modulo I ν (a, b, c). Therefore (i) follows.
(ii) Observe that DA n = 1 2 ABA n − BA n+1 by (2.1). In light of this fact, the result now follows from Lemma 3.4(i).
(iii) We proceed by induction on n. The statement holds trivially for n = 0. Now suppose that n ≥ 1. By the inductive hypothesis, D n = DD n−1 is congruent to an F-linear combination of . Therefore the result follows.
Proof . By Lemma 3.3, the F-vector space /I ν (a, b, c) is spanned by Since I ν (a, b, c) contains the elements listed in (3.8) and (3.10), each of the elements listed in (3.13) can be expressed as an F-linear combination of The result now follows from these facts along with Lemma 3.4(iii).
Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero We are now ready to give our second description of M ν (a, b, c).
Theorem 3.6. There exists a unique -module homomorphism Proof . Consider the -module homomorphism Ψ : → M ν (a, b, c) that sends 1 to m 0 . By Proposition 3.1(i), the elements (3.8) and (3.9) are in the kernel of Ψ. By Proposition 3.1(ii), the elements listed in (3.10) are also in the kernel of Ψ. Hence I ν (a, b, c) is contained in the kernel of Ψ. It follows that Ψ induces an -module homomorphism Φ : Observe that Φ is the unique -module homomorphism with the desired property since /I ν (a, b, c) is generated by 1 + I ν (a, b, c) as an -module.
By Lemma 3.2 the homomorphism Φ sends are linearly independent, the cosets (3.14) are linearly independent. Combining this with Lemma 3.5, we see that the cosets (3.14) are an F-basis for /I ν (a, b, c). Therefore Φ is an isomorphism.
As a consequence of Theorem 3.6, the -module M ν (a, b, c) satisfies the following universal property.
then there exists a unique -module homomorphism M ν (a, b, c) → V that sends m 0 to v.
For the rest of the present paper, we will consider the case ν = d. Define N d (a, b, c) to be the A-cyclic F-subspace of M d (a, b, c) generated by the element m d+1 .
Proof . Recall from Lemma 3.2 that It follows from this fact that {m i } ∞ i=d+1 is an F-basis for N d (a, b, c). We now show that N d (a, b, c) is an -submodule of M d (a, b, c). By Proposition 3.1(i), By (3.4), the scalar ϕ d+1 = 0 when ν = d. Hence N d (a, b, c) is B-invariant. By Proposition 3.1(ii), the element δ acts on N d (a, b, c) as scalar multiplication by η. It now follows from Lemma 2.2(ii) that N d (a, b, c) is an -submodule of M d (a, b, c).
Recall the -module R d (a, b, c) from Proposition 2.4. In the sequel we display how themodule R d (a, b, c) is connected to M ν (a, b, c). For convenience we let {v i } d i=0 denote the F-basis for R d (a, b, c) from Proposition 2.4(i) in the rest of this paper. Lemma 3.9. There exists a unique -module isomorphism Proof . By Lemma 3.8, M d (a, b, c)/N d (a, b, c) is a (d+1)-dimensional -module with the F-basis Observe that the matrices representing A and B with respect to the F-basis {v i } d i=0 for R d (a, b, c) are identical with the matrices representing A and B with respect to the F-basis (3.15) for M d (a, b, c)/N d (a, b, c) by Propositions 2.4(i) and 3.1(i). By Propositions 2.4(ii) and 3.1(ii), the actions of δ on R d (a, b, c) and M d (a, b, c)/N d (a, b, c) are scalar multiplication by the same scalar η. In light of these comments, the result now follows from Lemma 2.2(ii).
Combining this with (3.16), we see that m d+1 is in the kernel of . Therefore N d (a, b, c) is contained in the kernel of . By Lemma 3.8, there exists an -module homomorphism The result follows from this fact along with Lemma 3.9.

Conditions for the irreducibility of R d (a, b, c)
In this section, we derive the necessary and sufficient conditions for R d (a, b, c) to be irreducible in terms of the parameters a, b, c, d. Throughout this section, we let a, b, c) is irreducible, then each of the following holds: (i) char F = 0 or char F > d,  (a, b, c), a contradiction to the irreducibility of R d (a, b, c). Therefore ϕ i = 0 for all 1 ≤ i ≤ d, which is equivalent to (i) and (ii) by (3.4).  R d (a, b, c).

Proof . It follows from Proposition 2.4(i) that
Comparing this with (4.1), the result now follows. − 1, b, c). Moreover, the matrices representing A and B with respect to the F-basis respectively.

Proof . By Proposition 2.4(i), there exists an F-basis {u
with respect to which the matrices representing A and B are equal to the matrices displayed in (4.2). By Lemma 4.2, it suffices to show that there is an - Observe that Bu 0 = θ * 0 u 0 and a direct calculation yields that By Proposition 2.4(ii), the elements α, β, δ act on R d (−a − 1, b, c) as scalar multiplication by ζ, ζ * , η, respectively. According to Proposition 3.7, there exists a unique -module homomorphism M d (a, b, c) → R d (−a − 1, b, c) that sends m 0 to u 0 . By inspecting the matrix representing A given in (4.2) we see that Hence there exists a -module homomorphism that maps v 0 to u 0 by Proposition 3.10. It now follows from (4.1) that this homomorphism sends w i to u i for all 0 ≤ i ≤ d. The result follows. (i) char F = 0 or char F > d, Proof . By Proposition 4.3, the -module R d (a, b, c) is isomorphic to R d (− a − 1, b, c). Hence the result follows by applying Lemma 4.1 to both R d (a, b, c) and R d (−a − 1, b, c).
Shortly we will show that the converse of Lemma 4.4 is also true. To aid us in doing so, we establish the following notation. We define It follows from Proposition 2.4(i) that Rv is a scalar multiple of v 0 for all v ∈ R d (a, b, c). Thus, for any integers i, j with 0 ≤ i, j ≤ d, there exists a unique L ij ∈ F such that By examining Proposition 2.4(i) further, we see that It follows from Proposition 4.3 that Solving the recurrence relation (4.5) with the initial conditions (4.4) and (4.6) yields that Theorem 4.5. The -module R d (a, b, c) is irreducible if and only if both of the following conditions hold: (i) char F = 0 or char F > d, (⇐) To see the irreducibility of R d (a, b, c), we assume that W is a nonzero -submodule of R d (a, b, c) and show that W = R d (a, b, c). Pick a nonzero vector w ∈ W . Since W is invariant under A and B, it follows that It now follows from (4.3) that Recall the parameters {φ i } i∈Z and {ϕ i } i∈Z from (3.3) and (3.4), respectively. It follows from our assumptions (i) and (ii) that the scalars ϕ i = 0 and φ i = 0 for all 1 ≤ i ≤ d. Let L denote the (d + 1) × (d + 1) matrix, indexed by 0, 1, . . . , d, with (i, j)-entry given by L ij for all 0 ≤ i, j ≤ d. By (4.4), the square matrix L is lower triangular. By (4.7), the diagonal entries of L are which we know to be nonzero. Therefore the matrix L is nonsingular. Since w is nonzero at least one of {a j } d j=0 is nonzero. Hence there exists an integer i with 0 ≤ i ≤ d such that Combining (4.10) with (4.8) and (4.9), we find that v 0 ∈ W . Since the -module R d (a, b, c) is generated by v 0 , it follows that W = R d (a, b, c) and so R d (a, b, c) is irreducible.

The isomorphism class of the -module R d (a, b, c)
In Proposition 4.3, we showed that the -module R d (a, b, c) is isomorphic to the -module R d (−a − 1, b, c). In this section, we discuss the isomorphism class of R d (a, b, c) in further detail.
Proof . By Proposition 2.4(i), there are F-bases for R d (a, b, c) and R d (a, b, −c − 1) with respect to which the matrices representing A and B are the same. By Proposition 2.4(ii), the actions of δ on R d (a, b, c) and R d (a, b, −c − 1) are both scalar multiplication by the same scalar η. Hence R d (a, b, c) is isomorphic to R d (a, b, −c − 1) by Lemma 2.2(ii).
Proof . By Proposition 2.4(i), there is an F-basis {u i } d i=0 for R d (a, −b − 1, c) with respect to which the matrices representing A and B are respectively. Since the -module R d (a, b, c) is irreducible, it follows from Theorem 4.5 that A direct calculation yields that Bv = θ * 0 v and By Proposition 2.4(ii), the elements α, β, δ act on R d (a, −b − 1, c) as scalar multiplication by ζ, ζ * , η, respectively. According to Proposition 3.7, there exists a unique -module homomorphism M d (a, b, c) → R d (a, −b − 1, c) that maps m 0 to v. By inspecting the matrix representing A given in (5.1), we see that Hence there exists an -module homomorphism that sends v 0 to v by Proposition 3.10. Since the -module R d (a, b, c) is irreducible, themodule R d (a, −b−1, c) is also irreducible by Theorem 4.5. Therefore (5.2) is an isomorphism.
We end this section with a simple combination of Propositions 4.3, 5.1, and 5.2.
The proof of Theorem 2.5 Theorems 4.5 and 5.3 indicate that the map R in Theorem 2.5 is well-defined. In this section, we shall show that R is a bijection.
Lemma 6.1. Assume that F is algebraically closed. If V is a finite-dimensional irreducible -module, then each central element of acts on V as scalar multiplication.
Proof . This result follows from applying Schur's lemma to .
Lemma 6.2. For any i ∈ Z, each of the following hold: Proof . The result can be routinely verified using (3.1).
Proof . Given any scalar κ ∈ F, we define Since char F = 0, for any distinct integers i, j, the scalars ϑ i (κ) and ϑ j (κ) are equal if and only if i + j = 2κ + 1. In particular {ϑ i (κ)} −∞ i=0 contains infinitely many values. Since F is algebraically closed, we may choose a scalar κ ∈ F such that ϑ 0 (κ) is an eigenvalue of A on V . Since V is of dimension d + 1, there are at most d + 1 distinct eigenvalues of A on V . Thus, there exists an integer j ≤ 0 such that ϑ j (κ) is an eigenvalue of A but ϑ j−1 (κ) is not an eigenvalue of A on V . Set Similarly, there exists a scalar λ ∈ F and an integer k ≤ 0 such that ϑ k (λ) is an eigenvalue of B but ϑ k−1 (λ) is not an eigenvalue of B in V . We set Observe that under these settings, we have By Lemma 6.1, the element δ acts on V as scalar multiplication. Since F is algebraically closed, there exists a scalar c ∈ F such that the action of δ on V is the scalar multiplication by To prove the theorem, it now suffices to show that there exists an -module isomorphism from Given any T ∈ and θ ∈ F, we let Pick any v ∈ V A (θ 0 ). Applying each side of (2.6) to v and using Lemma 6.2 to simplify the result, we obtain that Left multiplying each side of (6.3) by (A − θ 0 ), we obtain that By (6.1), the scalar θ −1 is not an eigenvalue of A in V . Hence In other words (A − θ 1 )Bv ∈ V A (θ 0 ) and therefore V A (θ 0 ) is invariant under (A − θ 1 )B. Since F is algebraically closed, there exists an eigenvector u of (A − θ 1 )B in V A (θ 0 ). Similarly, there exists an eigenvector w of (B − θ * 1 )A in V B (θ * 0 ). Define We now proceed by induction to show that Since u is an eigenvector of (A − θ 1 )B in V A (θ 0 ), the claim is true for i = 0, 1. Now suppose that i ≥ 2. Applying each side of (2.7) to u i−2 , we obtain that By Lemma 6.1, the right-hand side of (6.7) is a scalar multiple of u i−2 . Using the inductive hypothesis, (6.4), and Lemma 6.2(i), we find that the left-hand side of (6.7) is equal to plus an F-linear combination of u 0 , u 1 , . . . , u i−1 . Combining the above results, the claim (6.6) follows.
Next, we show that {u i } d i=0 is an F-basis for V . Suppose on the contrary that there is an integer h with 0 ≤ h ≤ d − 1 such that u h+1 is an F-linear combination of u 0 , u 1 , . . . , u h . Let W denote the F-subspace of V spanned by u 0 , u 1 , . . . , u h . Since W is B-invariant by (6.4) and A-invariant by (6.6), it follows that W is an -submodule of V by Lemma 2.2(ii). Since V is irreducible, this forces that W = V . By construction, W is of dimension at most d, a contradiction. Therefore {u i } d i=0 is an F-basis for V . By a similar argument, it follows that In other words Aw d = θ d w d by (6.5). Hence the matrix representing A with respect to the By (6.9), the matrix representing B with respect to the F-basis {w i } d i=0 is upper triangular with diagonal entries {θ * i } d i=0 . We let {ϕ i } d i=1 denote its superdiagonal entries as follows Applying each side of (2.6) to w i−1 , we obtain from the coefficients of w i that for all 1 ≤ i ≤ d, where ϕ 0 and ϕ d+1 are interpreted as zero. It is straightforward to verify that {ϕ i } d i=1 also satisfy the recurrence relation (6.12). Since char F = 0, the corresponding homogeneous recurrence relation with the initial values σ 0 = 0 and σ d+1 = 0, has the unique solution σ i = 0 for all 0 ≤ i ≤ d + 1. Therefore ϕ i = ϕ i for all 1 ≤ i ≤ d.
Up to this point, we have shown that 14) Applying each side of (2.7) to w 0 and using (6.14) to simplify the resulting equation, we find that Using logic similar to what was used to show (6.14), we obtain (A − θ 1 )(B − θ * 0 )u 0 = ϕ 1 u 0 . Now, by applying each side of (2.6) to u 0 and using the above equation to simplify the resulting equation, we find that αu 0 = ζu 0 . It follows from Lemma 6.1 that In view of (6.13)-(6.17), it follows from Proposition 3.7 that there exists a unique -module homomorphism M d (a, b, c) → V that sends m 0 to w 0 . By Proposition 3.1(i), the entries above the superdiagonal in (6.11) are zero. Combining the above -module homomorphism M d (a, b, c) → V with (6.10), there is an -module homomorphism

respectively.
Proof . To compute the traces of A, B on R d (a, b, c), use Proposition 2.4(i). By (2.5), the trace of C is equal to the trace of δ minus the trace of A + B on R d (a, b, c). Use the above facts along with Proposition 2.4(ii) to compute the trace of C on R d (a, b, c).
The following is a quick consequence of Theorems 5.3, 6.3 and Lemma 6.4. We are now ready to prove Theorem 2.5.
Proof of Theorem 2.5. By Theorems 4.5 and 5.3, the map R is well-defined. By Theorem 6.3, the map R is onto. Since any element of has the same trace on the isomorphic finitedimensional -modules, it follows from Lemma 6.4 that R is one-to-one.
In Example 1.1 we showed a five-dimensional irreducible -module on which none of A, B, C is diagonalizable. Note that the -module is isomorphic to R 4 − 1 2 , − 1 2 , − 1 2 . We finish this paper with the necessary and sufficient conditions for A, B, C to be diagonalizable on finitedimensional irreducible -modules. Theorem 6.6. Assume that F is algebraically closed with char F = 0. For any a, b, c ∈ F and d ∈ N satisfying the conditions (i) and (ii) of Theorem 4.5, the following statements are equivalent: (i) A (resp. B) (resp. C) is diagonalizable on R d (a, b, c),  d (a, b, c) if and only if b is not in (6.19).
To derive the condition for C as diagonalizable on R d (a, b, c), we consider the (d + 1)dimensional -module R d (b, c, a). By Theorem 4.5 the -module R d (b, c, a) is irreducible. By Definition 2.1 or [1, Proposition 4.1] there exists a unique F-algebra automorphism of that sends A, B, C, D to C, A, B, D, respectively. Let R d (b, c, a) denote the -module obtained by pulling back the -module R d (b, c, a) via . Observe that C is diagonalizable on R d (b, c, a) if and only if c is not in (6.19). Using Corollary 6.5 yields that R d (b, c, a) is isomorphic to the -module R d (a, b, c). The result follows.