The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
Hau-Wen Huang









The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra

The Racah Algebra as a Subalgebra
 of the Bannai–Ito Algebra

Hau-Wen HUANG

H.-W. Huang

Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
mailto:hauwenh@math.ncu.edu.twhauwenh@math.ncu.edu.tw

Received May 22, 2020, in final form July 31, 2020; Published online August 10, 2020

Assume that 𝔽 is a field with char𝔽≠2. The Racah algebra  is a unital associative 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 [A,D]+AC-BA, [B,D]+BA-CB, [C,D]+CB-AC is central in . The Bannai–Ito algebra 𝔅ℑ is a unital associative 𝔽-algebra generated by X, Y, Z and the relations assert that each of {X,Y}-Z, {Y,Z}-X, {Z,X}-Y is central in 𝔅ℑ. It was discovered that there exists an 𝔽-algebra homomorphism ζ→𝔅ℑ that sends A ↦(2X-3)(2X+1)/16, B ↦(2Y-3)(2Y+1)/16, C ↦(2Z-3)(2Z+1)/16. We show that ζ is injective and therefore  can be considered as an 𝔽-subalgebra of 𝔅ℑ. Moreover we show that any Casimir element of  can be uniquely expressed as a polynomial in {X,Y}-Z, {Y,Z}-X, {Z,X}-Y and X+Y+Z with coefficients in 𝔽.

Bannai–Ito algebra; Racah algebra; Casimir elements

81R10; 81R12



§ INTRODUCTION


Throughout this paper we adopt the following conventions:
Assume that 𝔽 is a field with char𝔽≠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 algebra <cit.> and the Bannai–Ito algebra <cit.> 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 <cit.> 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  is an 𝔽-algebra generated by A, B, C, D and the relations assert that

    [A,B]=[B,C]=[C,A]=2D

and each of

    α=[A,D]+AC-BA,
        β=[B,D]+BA-CB,
        γ=[C,D]+CB-AC

is central in . Note that

    δ=A+B+C

is also central in .
The Bannai–Ito algebra 𝔅ℑ is an 𝔽-algebra generated by X, Y, Z and the relations assert that each of

    κ={X,Y}-Z,
        λ={Y,Z}-X,
        μ={Z,X}-Y

is central in 𝔅ℑ. 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 <cit.>. For more information and recent progress, see <cit.>.

A result of <cit.> made the following link between the Racah algebra  and the Bannai–Ito algebra 𝔅ℑ. The standard realization for 𝔅ℑ is a representation π𝔅ℑ→ End(𝔽[x])
given in <cit.>.
Inspired by π, a representation τ→ End(𝔽[x]) was constructed in <cit.> as well as an 𝔽-algebra homomorphism ζ→𝔅ℑ that sends

    A ↦(2X-3)(2X+1)/16,
        
    B ↦(2Y-3)(2Y+1)/16,
        
    C ↦(2Z-3)(2Z+1)/16.

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^ℓα^r β^s
        for all i,j,k,ℓ,r,s∈ℕ

are an 𝔽-basis for  and the monomials

    X^i Y^j Z^k κ^r λ^s μ^t
        for all i,j,k,r,s,t∈ℕ

are an 𝔽-basis for 𝔅ℑ. We consider the following 𝔽-subspaces of 𝔅ℑ induced from the basis (<ref>) 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

    w_X i+w_Y j+w_Z k+w_κ r+w_λ s+w_μ t≤ n.

We show that the sequence {𝔅ℑ_n}_n∈ℕ is an ℕ-filtration of 𝔅ℑ if and only if

    max{w_Z, w_κ}≤ w_X+w_Y,
        max{w_X, w_λ}≤ w_Y+w_Z,
        max{w_Y, w_μ}≤ w_Z+w_X.

We apply the basis (<ref>) for  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  as an 𝔽-subalgebra of 𝔅ℑ via ζ.
Let ℭ denote the commutative 𝔽-subalgebra of  generated by α, β, γ, δ. Extending the setting <cit.>, each element of

    D^2+A^2+B^2
    +(δ+2){A,B}-{A^2,B}-{A,B^2}/2
    +A (β-δ)
    +B (δ-α)+ℭ

is called a Casimir element of  <cit.>. Each Casimir element of  is central in . We locate the expressions for the D_6-symmetric Casimir elements <cit.>
of  in terms of

    ι=X+Y+Z

and κ, λ, μ. Note that ι, κ, λ, μ are in the centralizer of  in 𝔅ℑ.
Furthermore we apply the ℕ-filtration {𝔅ℑ_n}_n∈ℕ of 𝔅ℑ associated with

    (w_X,w_Y,w_Z,w_κ ,w_λ,w_μ)=(1,1,2,0,0,0)

to prove that for any Casimir element Ω of  there exists a unique four-variable polynomial P(x_1,x_2,x_3,x_4) over 𝔽 such that

    Ω=P(ι,κ,λ,μ).


The outline of this paper is as follows: In Sections <ref> and <ref> we present the required backgrounds on  and 𝔅ℑ, especially the basis (<ref>) for  and the criterion for {𝔅ℑ_n}_n∈ℕ as an ℕ-filtration of 𝔅ℑ.
In Section <ref> we review the homomorphism ζ→𝔅ℑ and evaluate the image of D under ζ.
In Section <ref> we give the proof for the injectivity of ζ.
In Section <ref> we show that each Casimir element of  can be uniquely expressed as a polynomial in ι, κ, λ, μ over 𝔽.



§ THE RACAH ALGEBRA 


[<cit.>]
The Racah algebra  is an 𝔽-algebra defined by generators and relations in the following way. The generators are A, B, C, D. The relations assert that

    [A,B]=[B,C]=[C,A]=2D

and each of

    [A,D]+AC-BA,
        
    [B,D]+BA-CB,
        
    [C,D]+CB-AC

is central in .


We define α, β, γ, δ as the following elements of :

    α = [A,D]+AC-BA, 
    β = [B,D]+BA-CB,
    γ = [C,D]+CB-AC,
    δ = A+B+C.


[<cit.>] 
The following (i)–(iii) hold:
=0pt

  (i) The 𝔽-algebra  is generated by A, B, C.

  (ii) Each of α, β, γ, δ is central in .

  (iii) The sum of α, β, γ is equal to zero.




The 𝔽-algebra  has a presentation with generators A, B, C, D, α, β and relations

    BA = AB-2D, 
    
    CB = BC-2D, 
    
    CA = AC+2D, 
    
    DA = AD-AB+AC+2D-α, 
    
    DB = BD-BC+AB-β, 
    
    DC = CD-AC+BC-2D+α+β, 
    α A = Aα,    α B = Bα,     α C = Cα,     α D = Dα,
    β A = Aβ,    β B = Bβ,    β C = Cβ,    β D = Dβ,    βα=αβ.



Relations (<ref>)–(<ref>) are immediate from (<ref>).
Relation (<ref>) follows from (<ref>), (<ref>). Relation (<ref>) follows from (<ref>), (<ref>) and (<ref>). Relation (<ref>) follows from (<ref>), (<ref>) and Lemma <ref>(iii). Relations (<ref>) and (<ref>) follow from Lemma <ref>(ii).


The elements

    A^i B^j C^k D^ℓα^r β^s
        for all i,j,k,ℓ, r,s∈ℕ

are an 𝔽-basis .

To prove the result we invoke the diamond lemma <cit.>.
The relations (<ref>)–(<ref>) are regarded as a reduction system.
The 𝔽-linear combinations of (<ref>) 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

    CBA=ABC+2AB-2BC-2AD+2BD-2CD-2β,
    
    DBA=ABD-2D^2+A^2B-AB^2-2AD+2BD+2AB-2BC-Aβ-Bα-2β,
    
    DCB=BCD-2D^2+B^2C-BC^2-2BD+2CD-2AC+2BC+Bα+Bβ-Cβ
    -4D+2α+2β,
    
    DCA=ACD+2D^2-A^2C+AC^2-2AD-2AC+2BC+2CD+Aα+Aβ-Cα
     -4D+2α+2β.

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 . An operation on w is called an elementary operation if it is one of the following actions on w:
=0pt

  ∙ 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

    A,    B,    C,    D.


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

    u=u_0→ u_1→⋯→ u_k-1→ u_k=w.

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 (<ref>)–(<ref>). Therefore, by diamond lemma the monomials (<ref>) form an 𝔽-basis for .


Recall that the dihedral group D_6 has a presentation with generators σ, τ and relations

    σ^2=1,    τ^6=1,     (στ)^2=1.



[<cit.>] There exists a unique D_6-action on  such that (i), (ii) hold:
=0pt

  (i) σ acts on  as an 𝔽-algebra antiautomorphism of  given in the following way:




  (ii) τ acts on  as an 𝔽-algebra antiautomorphism of  given in the following way:


Moreover the D_6-action on  is faithful.


Let ℭ denote the 𝔽-subalgebra of  generated by α, β, γ, δ. It follows from Lemma <ref>(ii) that ℭ is commutative.

[<cit.>]
The coset

    D^2+A^2+B^2+(δ+2){A,B}-{A^2,B}-{A,B^2}/2 +A (β-δ) +B (δ-α)+ℭ

is called the Casimir class of . Each element of the Casimir class of  is called a Casimir element of .


Define

    Ω_A=D^2+B A C+C A B/2+ A^2+B γ-C β-A δ,
    Ω_B=D^2+C B A+A B C/2+ B^2+C α-A γ-Bδ,
    Ω_C=D^2+A C B+B C A/2+ C^2+A β-Bα-Cδ.

Note that Ω_A, Ω_B, Ω_C are mutually distinct <cit.>.

[<cit.>]
Each of Ω_A, Ω_B, Ω_C is a Casimir element.


[<cit.>]
The set {Ω_A,Ω_B,Ω_C} is invariant under the D_6-action on .
Moreover the restrictions of σ and τ to {Ω_A,Ω_B,Ω_C} are as follows:



[<cit.>]
The elements Ω_A, Ω_B, Ω_C are called the D_6-symmetric Casimir elements of .




§ THE BANNAI–ITO ALGEBRA 𝔅ℑ



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
{X,Y}-Z, {Y,Z}-X, {Z,X}-Y is central in 𝔅ℑ.


We define ι, κ, λ, μ as the following elements of 𝔅ℑ:

    ι = X+Y+Z, 
    κ = {X,Y}-Z, 
    λ = {Y,Z}-X, 
    μ = {Z,X}-Y.



There exists a unique D_6-action on 𝔅ℑ such that (i), (ii) hold:
=0pt

  (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.


It is straightforward to verify the existence of the D_6-action on 𝔅ℑ by using (<ref>) and Definition <ref>. 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 <cit.> 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

    YX=-XY+Z+κ,    
    ZY=-YZ+X+λ,    
    ZX=-XZ+Y+μ,
    κ X=Xκ,    κ Y=Yκ,    κ Z=Zκ,
    λ X=Xλ,    λ Y=Yλ,    λ Z=Zλ,    λκ=κλ,
    μ X=Xμ,    μ Y=Yμ,    μ Z=Zμ,    μκ=κμ,    μλ=λμ.


Immediate from Definition <ref>.


Applying the diamond lemma to Proposition <ref>, we obtain the following Poincaré–Birkhoff–Witt basis for 𝔅ℑ. Since the argument is similar to the proof of Theorem <ref>, we omit the proof here.


The elements

    X^i Y^j Z^k κ^r λ^s μ^t
        for all i,j,k,r,s,t∈ℕ

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:
=0pt

  (N1) ⋃_n∈ℕ𝒜_n=𝒜.

  (N2) 𝒜_n⊆𝒜_n+1 for all n∈ℕ.

  (N3) 𝒜_m·𝒜_n⊆𝒜_m+n for all m,n∈ℕ.

For convenience we always let 𝒜_-1 denote the zero subspace of 𝒜.

We consider the following 𝔽-subspaces of 𝔅ℑ induced from Theorem <ref>: Let w_X, w_Y, w_Z, w_κ, w_λ, w_μ denote the nonnegative integers. 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

    w_Xi+w_Yj+w_Zk+w_κ r+w_λ s+w_μ t≤ n.

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.


Let w_X, w_Y, w_Z, w_κ, w_λ, w_μ∈ℕ.
Let {𝔅ℑ_n}_n∈ℕ denote the 𝔽-subspaces of 𝔅ℑ associated with (w_X, w_Y, w_Z, w_κ, w_λ, w_μ). Then {𝔅ℑ_n}_n∈ℕ is an ℕ-filtration of 𝔅ℑ
if and only if

    max{w_Z, w_κ}≤ w_X+w_Y,
    max{w_X, w_λ}≤ w_Y+w_Z,
    max{w_Y, w_μ}≤ w_Z+w_X.


 (⇒) By the construction of {𝔅ℑ_n}_n∈ℕ and Theorem <ref> the element

    Z+κ∉𝔅ℑ_max{w_Z, w_κ}-1.

On the other hand, by (N3) we have {X,Y}∈𝔅ℑ_w_X+w_Y. The equation (<ref>) implies

    Z+κ={X,Y}.

By the above comments we see that 𝔅ℑ_w_X+w_Y contains Z+κ which is not in 𝔅ℑ_max{w_Z, w_κ}-1. Combined with (N2) the inequality (<ref>) follows. The inequalities (<ref>) and (<ref>) follow by similar arguments.

(⇐) Condition (N1) is immediate from Theorem <ref>. 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 homomorphism w̃ M→ℕ such that

    w̃(u)=w_u     for all u∈ S.

By (<ref>)–(<ref>), for each relation of Proposition <ref>, the value of w̃ 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

    X^i' Y^j' Z^k'κ^r'λ^s'μ^t'·
    X^i” Y^j” Z^k”κ^r”λ^s”μ^t”

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→𝔅ℑ


According to <cit.> there exists an 𝔽-algebra homomorphism ζ→𝔅ℑ and the images of A, B, C, α, β, γ, δ under ζ are as follows:
[<cit.>]

There exists a unique 𝔽-algebra homomorphism ζ→𝔅ℑ that sends

    A ↦(2X-3)(2X+1)/16,    
    B ↦(2 Y-3)(2 Y+1)/16,     
    C ↦(2 Z-3)(2 Z+1)/16,
    
    α↦(2ι-κ-μ-3)(κ-μ)/64,
        β↦(2ι-λ-κ-3)(λ-κ)/64,
    
    γ↦(2ι-μ-λ-3)(μ-λ)/64,
        δ↦ι^2-2ι-κ-λ-μ/4-9/16.



We are now going to evaluate the image of D under ζ.

  =0pt

  (i) The following equations hold in 𝔅ℑ:

    [X^2,Y]=[X,Z],    [Y^2,Z]=[Y,X],    [Z^2,X]=[Z,Y],
    
    [Y^2,X]=[Y,Z],     [Z^2,Y]=[Z,X],    [X^2,Z]=[X,Y].



  (ii) The following elements of 𝔅ℑ are equal:

    {X,[Z,Y]},    {Y,[X,Z]},    {Z,[Y,X]},
    [X^2,Y^2],    [Y^2,Z^2],    [Z^2,X^2].




(i) Since κ is central in 𝔅ℑ and by (<ref>) it follows that

    [X,Z]=[X,{X,Y}].

Observe that [X,{X,Y}]=[X^2,Y]. Therefore

    [X^2,Y]=[X,Z].

Applying Proposition <ref> to (<ref>) yields the remaining equations in (i).

(ii) By Proposition <ref>(ii) it suffices to show that

    {X,[Z,Y]}={Y,[X,Z]},
    {X,[Z,Y]}=[X^2,Y^2].

With trivial cancellations we obtain

    {X,[Z,Y]}-{Y,[X,Z]}=[Z,{Y,X}].

Since κ is central in 𝔅ℑ and by (<ref>) the element Z commutes with {Y,X}. Hence the right-hand side of (<ref>) is zero. Therefore (<ref>) follows.
Using (<ref>) twice we find that

    X^2 Y^2 =XY^2 X+XZY-XYZ.

By Proposition <ref>(ii), τ^3 is an 𝔽-algebra antiautomorphism of 𝔅ℑ that fixes X, Y, Z.
Thus, applying τ^3 to (<ref>) yields that

    Y^2X^2 =XY^2X+YZX-ZYX.

Subtracting (<ref>) from (<ref>) yields (<ref>). Hence (ii) follows.


For convenience we let L denote the common element of 𝔅ℑ from Lemma <ref>(ii).


The image of D under ζ is equal to

    [X,Y]+[Y,Z]+[Z,X]+L/32.



By (<ref>) we have 2D^ζ=[A^ζ,B^ζ].
A direct calculation yields that [A^ζ, B^ζ] is equal to

    [X^2,Y^2]+[X,Y]+[Y^2,X]+[Y,X^2]/16.

By Lemma <ref>(i), [Y^2,X]=[Y,Z] and [Y,X^2]=[Z,X]. By Lemma <ref>(ii), [X^2,Y^2]=L. The proposition follows.



For each g∈ D_6 the following diagram commutes:




It is routine to verify the corollary by using Propositions <ref>, <ref> and Theorem <ref>.


We end this section with a comment:
Recall from <cit.> 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

    t_0+t_1+t_0^∨+t_1^∨=-1

and each of t_0^2, t_1^2, t_0^∨ 2, t_1^∨ 2 is central in ℌ.
By <cit.> there exists an 𝔽-algebra isomorphism ♮𝔅ℑ→ℌ that sends

    X ↦ t_0+t_1+1/2,
        
    Y ↦ t_0+t_0^∨+1/2,
        
    Z ↦ t_0+t_1^∨+1/2,
        ι↦ 2t_0+1/2,
    κ↦ t_0^2-t_1^2-t_0^∨ 2+t_1^∨ 2,
        λ↦ t_0^2- t_0^∨ 2-t_1^∨ 2+t_1^2,
        μ↦ t_0^2-t_1^∨ 2-t_1^2+t_0^∨ 2.

The universal Askey–Wilson algebra <cit.> and the universal DAHA of type (C_1^∨,C_1) <cit.> are the q-analogues of  and ℌ, respectively. Therefore
<cit.> is a q-analogue of the homomorphism ♮∘ζ→ℌ. Note that ♮∘ζ sends

    A ↦(t_1^∨+t_0^∨)(t_1^∨+t_0^∨+2)/4,
        
    B ↦(t_1+t_1^∨)(t_1+t_1^∨+2)/4,
    
    
    C ↦(t_0^∨+t_1)(t_0^∨+t_1+2)/4,
    
    α↦(t_1^∨ 2-t_0^∨ 2)(t_1^2-t_0^2+2t_0-1)/16,
        β↦(t_1^2-t_1^∨ 2)(t_0^∨ 2-t_0^2+2t_0-1)/16,
    
    γ↦(t_0^∨ 2-t_1^2)(t_1^∨ 2-t_0^2+2t_0-1)/16,
        δ↦t_0^2+t_1^2+t_0^∨ 2+t_1^∨ 2/4-t_0/2-3/4.




§ THE INJECTIVITY OF Ζ


Throughout this section, we let {𝔅ℑ_n}_n∈ℕ denote the 𝔽-subspaces of 𝔅ℑ associated with

    (w_X,w_Y,w_Z,w_κ,w_λ,w_μ)=(4,4,6,8,9,9).

Since the number sequence (<ref>) satisfies (<ref>)–(<ref>), it follows from Theorem <ref> that {𝔅ℑ_n}_n∈ℕ is an ℕ-filtration of 𝔅ℑ.

  =0pt

  (i) For any even integer n≥ 0 the following equations hold:

    Y^n X =X Y^n𝔅ℑ_4n+3,
    
    X^n Y = Y X^n𝔅ℑ_4n+3,
    
    Z^n Y = Y Z^n𝔅ℑ_6n+3,
    
    Y^n Z = Z Y^n𝔅ℑ_4n+5,
    
    X^n Z = Z X^n𝔅ℑ_4n+5,
    
    Z^n X = X Z^n𝔅ℑ_6n+3.



  (ii) For any odd integer n≥ 1 the following equations hold:

    Y^n X = -X Y^n+κ Y^n-1𝔅ℑ_4n+3,
    
    X^n Y = -Y X^n+κ X^n-1𝔅ℑ_4n+3,
    
    Y^n Z = -Z Y^n𝔅ℑ_4n+5,
    
    Z^n Y = -Y Z^n𝔅ℑ_6n+3,
    
    X^n Z = -Z X^n𝔅ℑ_4n+5,
    
    Z^n X = -X Z^n𝔅ℑ_6n+3.




All equations are established by routine inductions and using (<ref>)–(<ref>).


  =0pt

  (i) For any integer n≥ 0 the following equations hold:

    (A^ζ)^n = (X/2)^2n𝔅ℑ_8n-1,
    (B^ζ)^n =(Y/2)^2n𝔅ℑ_8n-1,
    (C^ζ)^n =(Z/2)^2n𝔅ℑ_12n-1,
    (α^ζ)^n =(μ/8)^2n𝔅ℑ_18n-1,
    (β^ζ)^n = (-1)^n(λ/8)^2n𝔅ℑ_18n-1.



  (ii) For any even integer n≥ 0 the following equation holds:

    (D^ζ)^n=1/16^n∑_i=0^n/2(-4)^in/2iX^2i Y^2iκ^n-2iZ^n𝔅ℑ_14n-1.



  (iii) For any odd integer n≥ 1 the following equation holds:

    (D^ζ)^n=1/16^n∑_i=0^n-1/2(-4)^i
    n-1/2i(X^2i Y^2iκ^n-2i-2 X^2i+1 Y^2i+1κ^n-2i-1)Z^n 
    𝔅ℑ_14n-1.




(i) Immediate from Theorem <ref> and the construction of {𝔅ℑ_n}_n∈ℕ.

(ii) It follows from Proposition <ref> that

    D^ζ=L/32𝔅ℑ_13.

Evaluating L 𝔅ℑ_13 by using Lemma <ref>(ii) and Lemma <ref>(ii) yields that

    D^ζ=Zκ/16-XYZ/8𝔅ℑ_13.

Squaring the equation (<ref>) a direct calculation shows that

    (D^ζ)^2=Z^2 κ^2/256-X^2Y^2Z^2/64𝔅ℑ_27.

It follows from Lemma <ref>(i) that

    Z^2 · X^2Y^2Z^2=X^2Y^2Z^2· Z^2
    𝔅ℑ_39.

Now it is routine to derive (ii) by using (<ref>) and (<ref>).

(iii) To get (iii), one may multiply (<ref>) by the equation from (ii)
and simplify the resulting equation by using Lemma <ref>(i).



Let i,j,k,ℓ,n,r,s∈ℕ with 8i+8j+12 k+14ℓ+18r+18s=n.
Then the following (i)–(iii) hold:
=0pt

  (i) For all i',j',k',r',s',t'∈ℕ with 4i'+4j'+6k'+8r'+9s'+9t'=n and r'> ℓ, the coefficient of

    X^i' Y^j' Z^k'κ^r'λ^s'μ^t'

in (A^ζ)^i(B^ζ)^j(C^ζ)^k(D^ζ)^ℓ(α^ζ)^r(β^ζ)^s
with respect to the 𝔽-basis (<ref>) for 𝔅ℑ is zero.


  (ii) For all i',j',k',r',s',t'∈ℕ with 4i'+4j'+6k'+8r'+9s'+9t'=n and r'= ℓ, the coefficient of

    X^i' Y^j' Z^k'κ^r'λ^s'μ^t'

in (A^ζ)^i(B^ζ)^j(C^ζ)^k(D^ζ)^ℓ(α^ζ)^r(β^ζ)^s
with respect to the 𝔽-basis (<ref>) for 𝔅ℑ is nonzero if and only if

    (i',j',k',s',t')=(2i, 2j, 2k+ℓ, 2s, 2r).



  (iii) The coefficient of

    X^2i Y^2j Z^2k+ℓκ^ℓλ^2sμ^2r

in
(A^ζ)^i(B^ζ)^j(C^ζ)^k(D^ζ)^ℓ(α^ζ)^r(β^ζ)^s
with respect to the 𝔽-basis (<ref>) for 𝔅ℑ is

    (-1)^s 4^-i-j-k-2ℓ-3r-3s.



Using Lemmas <ref>(i) and <ref> one may express

    (A^ζ)^i
    (B^ζ)^j
    (C^ζ)^k
    (D^ζ)^ℓ(α^ζ)^r
    (β^ζ)^s
    +𝔅ℑ_n-1

as an 𝔽-linear combination of X^i' Y^j' Z^k'κ^r'λ^s'μ^t'+𝔅ℑ_n-1 for all i',j',k',r',s',t'∈ℕ with 4i'+4j'+6k'+8r'+9s'+9t'=n. The lemma follows from the expression.



The homomorphism ζ→𝔅ℑ is injective.


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

    A^iB^jC^kD^ℓα^rβ^s

in I with respect to the 𝔽-basis (<ref>) for . 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+12 k+14ℓ+18r+18s=n. We may write

    I=∑_n∈ℕ∑_(i,j,k,ℓ,r,s)∈ S(n) c(i,j,k,ℓ,r,s) A^i B^j C^k D^ℓα^rβ^s.


Applying ζ to (<ref>) we have

    0=∑_n∈ℕ∑_(i,j,k,ℓ,r,s)∈ S(n)c(i,j,k,ℓ,r,s)(A^ζ)^i (B^ζ)^j (C^ζ)^k(D^ζ)^ℓ(α^ζ)^r (β^ζ)^s.

Since I≠0 there exists at least one n∈ℕ with S(n)≠∅.
Set

    N=max{n |  S(n)≠∅}.

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

    X^2i Y^2j Z^2k+ℓκ^ℓλ^2sμ^2r

in the right-hand side of (<ref>) with respect to the 𝔽-basis (<ref>) 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

    (A^ζ)^i'(B^ζ)^j'(C^ζ)^k'(D^ζ)^ℓ'(α^ζ)^r'(β^ζ)^s'

contributes to the coefficient c. By Theorem <ref> the monomial (<ref>) lies in 𝔅ℑ_N not in 𝔅ℑ_N-1. By Lemma <ref> the term (<ref>) lies in 𝔅ℑ_n. It follows from (N2) that n≥ N and the maximality of N implies n=N.
By Lemma <ref>(i) we have ℓ'≥ℓ and the maximality of ℓ forces that ℓ'=ℓ. Combined with Lemma <ref>(ii) this yields that (i',j',k',r',s')=(i,j,k,r,s). Therefore

    (A^ζ)^i (B^ζ)^j (C^ζ)^k (D^ζ)^ℓ(α^ζ)^r (β^ζ)^s

is the only summand in the right-hand side of (<ref>) contributes to the coefficient c. By Lem­ma <ref>(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 <ref> that the right-hand side of (<ref>) is nonzero, a contradiction. The theorem follows.


As a consequence of Theorem <ref> the 𝔽-algebra homomorphism ♮∘ζ→ℌ described in Section <ref> is injective. Note that <cit.> is a q-analogue of the injectivity for ♮∘ζ.



§ THE IMAGES OF THE CASIMIR ELMENTS OF  UNDER Ζ


In light of Theorem <ref> the Racah algebra  can be viewed as an 𝔽-subalgebra of the Bannai–Ito algebra 𝔅ℑ via ζ.

The element ι is in the centralizer of  in 𝔅ℑ.

By Theorem <ref> and (<ref>) the commutator [ι,A] is equal to 1/4 times

    [Y+Z,X^2]-[Y+Z,X].

Simplifying (<ref>) by using Lemma <ref>(i) yields that (<ref>) is zero. Therefore ι commutes with A. Similarly ι commutes with B and C. Combined with Lemma <ref>(i) the lemma follows.


By Lemma <ref> each of ι, κ, λ, μ lies in the centralizer of  in 𝔅ℑ. The intention of the final section is to show that each Casimir element of  can be uniquely expressed as a polynomial in ι, κ, λ, μ with coefficients in 𝔽.

Throughout this section, let {𝔅ℑ_n}_n∈ℕ denote the 𝔽-subspaces of 𝔅ℑ associated with

    (w_X,w_Y,w_Z,w_κ,w_λ,w_μ)=(1,1,2,0,0,0).

Since the sequence (<ref>) satisfies (<ref>)–(<ref>), it follows from Theorem <ref> that {𝔅ℑ_n}_n∈ℕ is an ℕ-filtration of 𝔅ℑ.


Z^n=ι^n 𝔅ℑ_2n-1 for all n∈ℕ.


Proceed by induction on n. It is trivial for n=0. By (<ref>) we have

    ι-Z=X+Y∈𝔅ℑ_1.

Hence the lemma holds for n=1.
 Suppose that n≥ 2. We divide ι^n-Z^n into

    Z(ι^n-1-Z^n-1)+(ι-Z)ι^n-1.

Since Z∈𝔅ℑ_2 and by induction hypothesis, the first summand of (<ref>) is in 𝔅ℑ_2n-1. By (<ref>) the element ι∈𝔅ℑ_2 and hence ι^n-1∈𝔅ℑ_2n-2. Combined with (<ref>) the second summand of (<ref>) is in 𝔅ℑ_2n-1. The lemma follows.


  =0pt

  (i) For all n∈ℕ the elements

    X^i Y^j Z^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.



  (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.



(i) Immediate from Theorem <ref> and the construction of {𝔅ℑ_n}_n∈ℕ.

(ii) Immediate from Lemma <ref> and (i).

(iii) Using (ii) the statement (iii) follows by a routine induction on n.



The elements

    X^i Y^j ι^k κ^r λ^s μ^t
        for all i,j,k,r,s,t∈ℕ

are an 𝔽-basis for 𝔅ℑ.


Immediate from (N1) and Lemma <ref>(iii).



The elements ι, κ, λ, μ of 𝔅ℑ are algebraically independent over 𝔽.

Immediate from Theorem <ref>.



The 𝔽-algebra 𝔅ℑ has a presentation with generators X, Y, ι, κ, λ, μ and relations

    YX=-XY-X-Y+ι+κ,
    ι Y =2 Y^2 -Yι- Y+ι+κ+λ,
    ι X =2 X^2-Xι- X+ι+κ+μ,
    κ X=Xκ,    κ Y=Yκ,    κι=ικ,
    λ X=Xλ,    λ Y=Yλ,    λι=ιλ,    λκ=κλ,
    μ X=Xμ,    μ Y=Yμ,    μι=ιμ,    μκ=κμ,    μλ=λμ.



This is a reformulation of Proposition <ref> by using (<ref>).


Recall the D_6-symmetric Casimir elements Ω_A, Ω_B, Ω_C of  from (<ref>)–(<ref>).


The D_6-symmetric Casimir elements Ω_A, Ω_B, Ω_C of  have the following expressions:

    Ω_A=λ(λ-2ι+3)(4ι^2-8ι-4κ-4λ-4μ+7)/1024-Γ,
    Ω_B=μ(μ-2ι+3)(4ι^2-8ι-4κ-4λ-4μ+7)/1024-Γ,
    Ω_C=κ(κ-2ι+3)(4ι^2-8ι-4κ-4λ-4μ+7)/1024-Γ,

where

    Γ=3(2ι+3)(2ι+1)(2ι-5)(2ι-7)/4096-(2ι+1)(6ι-13)(κ+λ+μ)/512
     + (κ+λ+μ)(κ+λ+μ+4)/64
    -(2ι-3)(κλ+λμ+μκ)/512
    +κλμ/256.


Applying Theorem <ref> and Proposition <ref> to (<ref>) and replacing Z by ι-X-Y, we may express Ω_A in terms of X, Y, ι, κ, λ, μ. To get (<ref>) we apply Lemma <ref> to express the resulting expression as an 𝔽-linear combination of (<ref>). Combined with Lemma <ref> and Proposition <ref> we obtain (<ref>) and (<ref>).


For each Casimir element Ω of  there exists a unique four-variable polynomial P(x_1,x_2,x_3,x_4) over 𝔽 such that

    Ω=P(ι,κ,λ,μ).



By Definition <ref> there exists a four-variable polynomial Q(y_1,y_2,y_3,y_4) over 𝔽 such that
Ω=Ω_A+Q(α,β,γ,δ).
Set
Q(x_1,x_2,x_3,x_4) = Q(y_1,y_2,y_3,y_4)
by substituting

    y_1 =(2 x_1- x_2 -x_4-3)(x_2-x_4)/64,     y_2 =(2 x_1-x_3-x_2-3)(x_3-x_2)/64,
    
    y_3 =(2 x_1-x_4-x_3-3)(x_4-x_3)/64,     y_4 =x_1^2-2x_1-x_2-x_3-x_4/4-9/16.

It follows from Theorem <ref> that Ω=Ω_A+Q(ι,κ,λ,μ). Combined with Proposition <ref> the existence follows.
The uniqueness is immediate from Corollary <ref>.




 §.§ Acknowledgements


The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4.

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