The Racah algebra as a subalgebra of the Bannai--Ito algebra

Assume that $\mathbb F$ is a field with ${\rm char\,}\mathbb F\not=2$. The Racah algebra $\Re$ is a unital associative $\mathbb F$-algebra defined by generators and relations. The generators are $A,B,C,D$ and the relations assert that $$ [A,B]=[B,C]=[C,A]=2D $$ and each of \begin{gather*} [A,D]+AC-BA, \qquad [B,D]+BA-CB, \qquad [C,D]+CB-AC \end{gather*} is central in $\Re$. 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}\label{kappa&lambda&mu} \{X,Y\}-Z, \qquad \{Y,Z\}-X, \qquad \{Z,X\}-Y \end{gather} is central in $\mathfrak{BI}$. It was discovered that there exists an $\mathbb F$-algebra homomorphism $\zeta:\Re\to \mathfrak{BI}$ that sends \begin{eqnarray*} A&\mapsto&\frac{(2X-3)(2X+1)}{16}, \\ B&\mapsto&\frac{(2Y-3)(2Y+1)}{16}, \\ C&\mapsto&\frac{(2Z-3)(2Z+1)}{16}. \end{eqnarray*} We show that $\zeta$ is injective and therefore $\Re$ can be considered as an $\mathbb F$-subalgebra of $\mathfrak{BI}$. Moreover we show that any Casimir element of $\Re$ can be uniquely expressed as a polynomial in the elements (\ref{kappa&lambda&mu}) and $X+Y+Z$ with coefficients in $\mathbb F$.


Introduction
Throughout this paper we adopt the following conventions: Assume that F is a field with char F = 2. Let N denote the set of all nonnegative integers. The bracket [ , ] stands for the commutator and the curly bracket { , } stands for the anticommutator. An algebra is meant to be an associative algebra with unit 1 and a subalgebra is a subset of the parent algebra which is closed under the operations and has the same unit.
The Racah algbra [15,19] and the Bannai-Ito algebra [22] are the F-algebras defined by generators and relations to give the algebraic interpretations of the Racah polynomials and the Bannai-Ito polynomials, respectively. At first, the description of those relations involved several parameters. In recent papers [3,6,13,14] the role of the parameters is replaced by the central elements. The contemporary Racah and Bannai-Ito algebras are defined as follows: The Racah algebra ℜ is an F-algebra generated by A, B, C, D and the relations assert that is central in ℜ. Note that δ = A + B + C is also central in ℜ. The Bannai-Ito algebra BI is an F-algebra generated by X, Y, Z and the relations assert that each of The applications to the Racah problems for su (2), su(1, 1), sl −1 (2) and the connections to the Laplace-Dunkl and Dirac-Dunkl equations on the 2-sphere have been explored in [4, 5, 7-12, 15, 16, 19]. For more information and recent progress, see [2,3,6,18]. A result of [13] made the following link between the Racah algebra ℜ and the Bannai-Ito algebra BI. The standard realization for BI is a representation π : BI → End(F[x]) given in [22,Section 4]. Inspired by π, a representation τ : ℜ → End(F[x]) was constructed in [13,Section 2] as well as an F-algebra homomorphism ζ : ℜ → BI that sends Briefly τ is the composition of ζ followed by π. The main result of this paper is to prove that ζ is injective. To see this we derive the following results. We show that the monomials A i B j C k D ℓ α r β s for all i, j, k, ℓ, r, s ∈ N (2) are an F-basis for ℜ and the monomials are an F-basis for BI. We consider the following F-subspaces of BI induced from the basis (3) for BI: Let w X , w Y , w Z , w κ , w λ , w µ ∈ N be given. For each n ∈ N let BI n denote the F-subspace of BI spanned by X i Y j Z k κ r λ s µ t for all i, j, k, r, s, t ∈ N with We show that the sequence {BI n } n∈N is an N-filtration of BI if and only if We apply the basis (2) for ℜ and the N-filtration {BI n } n∈N of BI associated with (w X , w Y , w Z , w κ , w λ , w µ ) = (4, 4, 6, 8,9,9) to conclude the injectivity of ζ.
We regard the Racah algebra ℜ as an F-subalgebra of BI via ζ. Let C denote the commutative F-subalgebra of ℜ generated by α, β, γ, δ. Extending the setting [7, Section 2], each element of is called a Casimir element of ℜ [18]. Each Casimir element of ℜ is central in ℜ. We locate the expressions for the D 6 -symmetric Casimir elements [18,Section 5] of ℜ in terms of and κ, λ, µ. Note that ι, κ, λ, µ are in the centralizer of ℜ in BI. Furthermore we apply the N-filtration {BI n } n∈N of BI associated with to prove that for any Casimir element Ω of ℜ there exists a unique four-variable polynomial The outline of this paper is as follows: In §2 and §3 we present the required backgrounds on ℜ and BI, especially the basis (2) for ℜ and the criterion for {BI n } n∈N as an N-filtration of BI. In §4 we review the homomorphism ζ : ℜ → BI and evaluate the image of D under ζ. In §5 we give the proof for the injectivity of ζ. In §6 we show that each Casimir element of ℜ can be uniquely expressed as a polynomial in ι, κ, λ, µ over F. We define α, β, γ, δ as the following elements of ℜ: [18]). The following (i)-(iii) hold: (i) The F-algebra ℜ is generated by A, B, C.
(iii) The sum of α, β, γ is equal to zero.

Theorem 2.4. The elements
are an F-basis ℜ.
Proof. To prove the result we invoke the diamond lemma [1, Theorem 1.2]. The relations (9)-(16) are regarded as a reduction system. The F-linear combinations of (17) are exactly the irreducible elements under the reduction system. There are no inclusion ambiguities in the reduction system. The nontrivial overlap ambiguities involve the words CBA, DBA, DCB, DCA. In any reduction ways, we eventually obtain that Hence each of the overlap ambiguities is resolvable. Let M denote the free monoid with the alphabet set S = {A, B, C, D, α, β}. Let ℓ : M → N denote the length function of M. Consider an element w = s 1 s 2 · · · s n ∈ M where s 1 , s 2 , . . . , s n ∈ S. An operation on w is called an elementary operation if it is one of the following actions on w: • We interchange s i and s j where 1 ≤ i < j ≤ n and the position of s j is left to the position of s i in the list A, B, C, D, α, β.
• Choose s i ∈ {B, C, D} and replace s i by the left neighbor of s i in the list We define a binary relation on M as follows: For any u, w ∈ M we say that u → w whenever ℓ(u) < ℓ(w) or u is obtained from w by an elementary operation. For any u, w ∈ M we define u w if there exist u 0 , u 1 , . . . , u k ∈ M with k ∈ N such that By construction is a partial order relation on M satisfying the descending chain condition. Moreover is a monoid partial order on M compatible with the reduction system (9)- (16). Therefore, by diamond lemma the monomials (17) form an F-basis for ℜ.
Recall that the dihedral group D 6 has a presentation with generators σ, τ and relations Proposition 2.5 (Propositions 4.1 and 4.3, [18]). There exists a unique D 6 -action on ℜ such that (i), (ii) hold: (i) σ acts on ℜ as an F-algebra antiautomorphism of ℜ given in the following way: (ii) τ acts on ℜ as an F-algebra antiautomorphism of ℜ given in the following way: Moreover the D 6 -action on ℜ is faithful.
Let C denote the F-subalgebra of ℜ generated by α, β, γ, δ. It follows from Lemma 2.2(ii) that C is commutative. Definition 2.6 (Definition 5.2, [18]). The coset is called the Casimir class of ℜ. Each element of the Casimir class of ℜ is called a Casimir element of ℜ. Define Note that Ω A , Ω B , Ω C are mutually distinct [18, Corollary 6.5].

The Bannai-Ito algebra BI
Definition 3.1. The Bannai-Ito algebra BI is an F-algebra defined by generators and relations. The generators are X, Y, Z and the relations assert that each of We define ι, κ, λ, µ as the following elements of BI: Proposition 3.2. There exists a unique D 6 -action on BI such that (i), (ii) hold: (i) σ acts on BI as an F-algebra antiautomorphism of BI given in the following way: (ii) τ acts on BI as an F-algebra antiautomorphism of BI given in the following way: Moreover the D 6 -action on BI is faithful.
Proof. It is straightforward to verify the existence of the D 6 -action on BI by using (18) and Definition 3.1. Since D 6 is generated by σ and τ the uniqueness follows. The F-algebra antiautomorphism of BI given in (ii) is of order 6. It follows from [18, Lemma 4.2] that the D 6 -action on BI is faithful.
The F-algebra BI has a presentation with generators X, Y, Z, κ, λ, µ and relations Proof. Immediate from Definition 3.1.
Applying the diamond lemma to Proposition 3.3, we obtain the following Poincaré-Birkhoff-Witt basis for BI and the argument is similar to the proof of Theorem 2.4. Thus we omit the proof here.
form an F-basis for BI.
Let A denote an F-algebra and let H, K denote two F-subspaces of A. The product H · K is meant to be the F-subspace of A spanned by h · k for all h ∈ H and all k ∈ K. Recall that an N-filtration of A is a sequence {A n } n∈N of F-subspaces of A satisfies the following conditions: For convenience we always let A −1 denote the zero subspace of A.
We consider the following F-subspaces of BI induced from Theorem 3.4: Let w X , w Y , w Z , w κ , w λ , w µ ∈ N. For each n ∈ N let BI n denote the F-subspace of BI spanned by X i Y j Z k κ r λ s µ t for all i, j, k, r, s, t ∈ N with We call {BI n } n∈N the F-subspaces of BI associated with (w X , w Y , w Z , w κ , w λ , w µ ). In what follows we give a simple criterion for the above F-subspaces of BI to be an N-filtration of BI.
Proof. (⇒): By the construction of {BI n } n∈N and Theorem 3.4 the element On the other hand, by (N3) we have {X, Y } ∈ BI w X +w Y . The equation (23) implies By the above comments we see that BI w X +w Y contains Z +κ which is not in BI max{w Z ,wκ}−1 . Combined with (N2) the inequality (27) follows. The inequalities (28) and (29) follow by similar arguments.
(⇐): Condition (N1) is immediate from Theorem 3.4. Condition (N2) is immediate from the construction of {BI n } n∈N . Set S = {X, Y, Z, κ, λ, µ}. For all n ∈ N, let I n denote the set of all (i, j, k, r, s, t) ∈ N 6 with w X i+w Y j +w Z k +w κ r +w λ s+w µ t ≤ n. Let M denote the free monoid with the alphabet set S. There exists a unique monoid homomorphismw : M → N such thatw (u) = w u for all u ∈ S.
By (27)-(29), for each relation of Proposition 3.3, the value ofw on the monomial in the left-hand side is greater than or equal to those in the right-hand side. Thus, for all m, n ∈ N and for all (i ′ , j ′ , k ′ , r ′ , s ′ , t ′ ) ∈ I m and (i ′′ , j ′′ , k ′′ , r ′′ , s ′′ , t ′′ ) ∈ I n the product is equal to an F-linear combination of X i Y j Z k κ r λ s µ t for all (i, j, k, r, s, t) ∈ I m+n . In other words (N3) holds. The theorem follows.

The homomorphism ζ : ℜ → BI
According to [13,Section 2] there exists an F-algebra homomorphism ζ : ℜ → BI and the images of A, B, C, α, β, γ, δ under ζ are as follows: Theorem 4.1 ( [13]). There exists a unique F-algebra homomorphism ζ : ℜ → BI that sends We are now going to evaluate the image of D under ζ. (ii) The following elements of BI are equal: Proof. (i): Since κ is central in BI and by (23) it follows that Applying Proposition 3.2 to (31) yields the remaining equations in (i).
(ii): By Proposition 3.2(ii) it suffices to show that With trivial cancellations we obtain Since κ is central in BI and by (23) the element Z commutes with {Y, X}. Hence the right-hand side of (34) is zero. Therefore (32) follows. Using (23) twice we find that By Proposition 3.2(ii), τ 3 is an F-algebra antiautomorphism of BI that fixes X, Y, Z. Thus, applying τ 3 to (35) yields that Subtracting (36) from (35) yields (33). Hence (ii) follows.
For convenience we let L denote the common element of BI from Lemma 4.2(ii). Proof. By (4) we have 2D ζ = [A ζ , B ζ ]. A direct calculation yields that [A ζ , B ζ ] is equal to  Proof. It is routine to verify the corollary by using Propositions 2.5, 3.2 and Theorem 4.1.
We end this section with a comment: Recall from [14, 17] that a universal analogue of the additive DAHA (double affine Hecke algebra) of type (C ∨ 1 , C 1 ), denoted by H here, is an F-algebra generated by t 0 , t 1 , t ∨ 0 , t ∨ 1 and the relations assert that Proposition 2] there exists an F-algebra isomorphism ♮ : BI → H that sends The universal Askey-Wilson algebra [20] and the universal DAHA of type (C ∨ 1 , C 1 ) [21] are the q-analogues of ℜ and H, respectively. Therefore [21, Theorem 4.1] is a q-analogue of the homomorphism ♮ • ζ :
(ii) For any odd integer n ≥ 1 the following equations hold: Proof. All equations are established by routine inductions and using (23)-(25).
Lemma 5.2. (i) For any integer n ≥ 0 the following equations hold: (ii) For any even integer n ≥ 0 the following equation holds: (iii) For any odd integer n ≥ 1 the following equation holds: Proof. Evaluating L mod BI 13 by using Lemma 4.2(ii) and Lemma 5.1(ii) yields that Squaring the equation (39) a direct calculation shows that It follows from Lemma 5.1(i) that Now it is routine to derive (ii) by using (40) and (41).
(iii): To get (iii), one may multiply (39) by the equation from (ii) and simplify the resulting equation by using Lemma 5.1(i).
(iii) The coefficient of Proof. Using Lemma 5.1(i) and Lemma 5.2 one may express The lemma follows from the expression.
Proof. Suppose on the contrary that there exists a nonzero element I in the kernel of ζ. For all i, j, k, ℓ, r, s ∈ N let c(i, j, k, ℓ, r, s) denote the coefficient of in I with respect to the F-basis (17) for ℜ. Let S denote the set of all (i, j, k, ℓ, r, s) ∈ N 6 with c(i, j, k, ℓ, r, s) = 0. For each n ∈ N we let S(n) denote the set of all (i, j, k, ℓ, r, s) ∈ S with 8i + 8j + 12k + 14ℓ + 18r + 18s = n. We may write Applying ζ to (42) we have Since I = 0 there exists at least one n ∈ N with S(n) = ∅. Set Among the elements in S(N) we choose a 6-tuple (i, j, k, ℓ, r, s) that has the maximum value at ℓ. In what follows we evaluate the coefficient of in the right-hand side of (43) with respect to the F-basis (26) for BI. Denote by c the coefficient. Suppose that (i ′ , j ′ , k ′ , ℓ ′ , r ′ , s ′ ) is a 6-tuple in S(n) for some n ∈ N such that contributes to the coefficient c. By Theorem 3.4 the monomial (44) lies in BI N not in BI N −1 . By Lemma 5.2 the term (45) lies in BI n . It follows from (N2) that n ≥ N and the maximality of N implies n = N. By Lemma 5.3(i) we have ℓ ′ ≥ ℓ and the maximality of ℓ forces that ℓ ′ = ℓ. Combined with Lemma 5.3(ii) this yields that (i ′ , j ′ , k ′ , r ′ , s ′ ) = (i, j, k, r, s). Therefore is the only summand in the right-hand side of (43) contributes to the coefficient c. By Lemma 5.3(iii) the coefficient c is the nonzero scalar (−1) s · 4 −i−j−k−2ℓ−3r−3s · c(i, j, k, ℓ, r, s).
It follows from Theorem 3.4 that the right-hand side of (43) is nonzero, a contradiction. The theorem follows.
As a consequence of Theorem 5.4 the F-algebra homomorphism ♮ • ζ : ℜ → H described in §4 is injective. Note that [21, Theorem 4.5] is a q-analogue of the injectivity for ♮ • ζ.

The images of the Casimir elments of ℜ under ζ
In light of Theorem 5.4 the Racah algebra ℜ can be viewed as an F-subalgebra of the Bannai-Ito algebra BI via ζ.
Lemma 6.1. The element ι is in the centralizer of ℜ in BI.

Proof. By Theorem 4.1 and (22) the commutator [ι, A] is equal to
Simplifying (46) by using Lemma 4.2(i) yields that (46) is zero. Therefore ι commutes with A. Similarly ι commutes with B and C. Combined with Lemma 2.2(i) the lemma follows. By Lemma 6.1 each of ι, κ, λ, µ lies in the centralizer of ℜ in BI. The intention of the final section is to show that each Casimir element of ℜ can be uniquely expressed as a polynomial in ι, κ, λ, µ with coefficients in F.
Proof. Proceed by induction on n. It is trivial for n = 0. By (22) we have Hence the lemma holds for n = 1. Suppose that n ≥ 2. We divide ι n − Z n into Since Z ∈ BI 2 and by induction hypothesis, the first summand of (49) is in BI 2n−1 . By (22) the element ι ∈ BI 2 and hence ι n−1 ∈ BI 2n−2 . Combined with (48) the second summand of (49) is in BI 2n−1 . The lemma follows.
for all i, j, k, r, s, t ∈ N with i + j + 2k = n are an F-basis for BI n /BI n−1 . (ii) For all n ∈ N the elements X i Y j ι k κ r λ s µ t + BI n−1 for all i, j, k, r, s, t ∈ N with i + j + 2k = n are an F-basis for BI n /BI n−1 . (iii) For all n ∈ N the elements X i Y j ι k κ r λ s µ t for all i, j, k, r, s, t ∈ N with i + j + 2k ≤ n are an F-basis for BI n .
Proof. (i): Immediate from Theorem 3.4 and the construction of {BI n } n∈N . (ii): Immediate from Lemma 6.2 and (i). (iii): Using (ii) the statement (iii) follows by a routine induction on n.
Theorem 6.4. The elements X i Y j ι k κ r λ s µ t for all i, j, k, r, s, t ∈ N (50) are an F-basis for BI.
Proof. This is a reformulation of Proposition 3.3 by using (22).
Proposition 6.7. The D 6 -symmetric Casimir elements Ω A , Ω B , Ω C of ℜ have the following expressions: Proof. Applying Theorem 4.1 and Proposition 4.3 to (19) and replacing Z by ι − X − Y , we may express Ω A in terms of X, Y, ι, κ, λ, µ. To get (51) we apply Lemma 6.6 to express the resulting expression as an F-linear combination of (50). Combined with Lemma 2.8 and Proposition 3.2 we obtain (52) and (53).