The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
Assume that is a field with char ≠ 2. The Racah algebra ℜ is a unital associative -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 ℜ. The Bannai-Ito...
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| Cite this: | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra. Hau-Wen Huang. SIGMA 16 (2020), 075, 15 pages |
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| description | Assume that is a field with char ≠ 2. The Racah algebra ℜ is a unital associative -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 ℜ. The Bannai-Ito algebra is a unital associative -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 I. It was discovered that there exists an -algebra homomorphism ζ: ℜ → 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 ℜ can be considered as an -subalgebra of . Moreover, we show that any Casimir element of ℜ can be uniquely expressed as a polynomial in {X, Y} − Z, {Y, Z} − X, {Z, X} − Y, and X + Y + Z with coefficients in .
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| first_indexed | 2026-03-20T06:42:49Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 075, 15 pages
The Racah Algebra as a Subalgebra
of the Bannai–Ito Algebra
Hau-Wen HUANG
Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
E-mail: hauwenh@math.ncu.edu.tw
Received May 22, 2020, in final form July 31, 2020; Published online August 10, 2020
https://doi.org/10.3842/SIGMA.2020.075
Abstract. Assume that F is a field with charF 6= 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 7→ (2X−3)(2X+1)
16 , B 7→ (2Y−3)(2Y+1)
16 ,
C 7→ (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.
Key words: Bannai–Ito algebra; Racah algebra; Casimir elements
2020 Mathematics Subject Classification: 81R10; 81R12
1 Introduction
Throughout this paper we adopt the following conventions: Assume that F is a field with
charF 6= 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 [19, 23] and the Bannai–Ito algebra [26] 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 [7, 9, 16, 18] 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
mailto:hauwenh@math.ncu.edu.tw
https://doi.org/10.3842/SIGMA.2020.075
2 H.-W. Huang
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 ex-
plored in [3, 6, 8, 11, 12, 13, 14, 15, 17, 19, 20, 23]. For more information and recent progress,
see [2, 4, 5, 7, 9, 10, 22].
A result of [16] made the following link between the Racah algebra < and the Bannai–Ito
algebra BI. The standard realization for BI is a representation π : BI → End(F[x]) given
in [26, Section 4]. Inspired by π, a representation τ : < → End(F[x]) was constructed in [16,
Section 2] as well as an F-algebra homomorphism ζ : < → BI that sends
A 7→ (2X − 3)(2X + 1)
16
, B 7→ (2Y − 3)(2Y + 1)
16
, C 7→ (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
AiBjCkD`αrβs for all i, j, k, `, r, s ∈ N (1.1)
are an F-basis for < and the monomials
XiY jZkκrλsµt for all i, j, k, r, s, t ∈ N (1.2)
are an F-basis for BI. We consider the following F-subspaces of BI induced from the basis (1.2)
for BI: Let wX , wY , wZ , wκ, wλ, wµ ∈ N be given. For each n ∈ N let BIn denote the F-subspace
of BI spanned by XiY jZkκrλsµt for all i, j, k, r, s, t ∈ N with
wXi+ wY j + wZk + wκr + wλs+ wµt ≤ n.
We show that the sequence {BIn}n∈N is an N-filtration of BI if and only if
max{wZ , wκ} ≤ wX + wY , max{wX , wλ} ≤ wY + wZ , max{wY , wµ} ≤ wZ + wX .
We apply the basis (1.1) for < and the N-filtration {BIn}n∈N of BI associated with
(wX , wY , wZ , wκ, wλ, wµ) = (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 [15, Section 2], each element
of
D2 +A2 +B2 +
(δ + 2){A,B} −
{
A2, B
}
−
{
A,B2
}
2
+A(β − δ) +B(δ − α) + C
is called a Casimir element of < [22]. Each Casimir element of < is central in <. We locate the
expressions for the D6-symmetric Casimir elements [22, Section 5] 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 {BIn}n∈N of BI associated with
(wX , wY , wZ , wκ, wλ, wµ) = (1, 1, 2, 0, 0, 0)
to prove that for any Casimir element Ω of < there exists a unique four-variable polynomial
P (x1, x2, x3, x4) over F such that
Ω = P (ι, κ, λ, µ).
The outline of this paper is as follows: In Sections 2 and 3 we present the required backgrounds
on < and BI, especially the basis (1.1) for < and the criterion for {BIn}n∈N as an N-filtration
of BI. In Section 4 we review the homomorphism ζ : < → BI and evaluate the image of D
under ζ. In Section 5 we give the proof for the injectivity of ζ. In Section 6 we show that each
Casimir element of < can be uniquely expressed as a polynomial in ι, κ, λ, µ over F.
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 3
2 The Racah algebra <
Definition 2.1 ([7, 16, 19, 23]). 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 (2.1)
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, (2.2)
β = [B,D] +BA− CB, (2.3)
γ = [C,D] + CB −AC, (2.4)
δ = A+B + C. (2.5)
Lemma 2.2 ([22, Lemma 3.2]). The following (i)–(iii) hold:
(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.
Proposition 2.3. The F-algebra < has a presentation with generators A, B, C, D, α, β and
relations
BA = AB − 2D, (2.6)
CB = BC − 2D, (2.7)
CA = AC + 2D, (2.8)
DA = AD −AB +AC + 2D − α, (2.9)
DB = BD −BC +AB − β, (2.10)
DC = CD −AC +BC − 2D + α+ β, (2.11)
αA = Aα, αB = Bα, αC = Cα, αD = Dα, (2.12)
βA = Aβ, βB = Bβ, βC = Cβ, βD = Dβ, βα = αβ. (2.13)
Proof. Relations (2.6)–(2.8) are immediate from (2.1). Relation (2.9) follows from (2.2), (2.6).
Relation (2.10) follows from (2.3), (2.6) and (2.7). Relation (2.11) follows from (2.4), (2.7) and
Lemma 2.2(iii). Relations (2.12) and (2.13) follow from Lemma 2.2(ii). �
Theorem 2.4. The elements
AiBjCkD`αrβs for all i, j, k, `, r, s ∈ N (2.14)
are an F-basis <.
Proof. To prove the result we invoke the diamond lemma [1, Theorem 1.2]. The relations
(2.6)–(2.13) are regarded as a reduction system. The F-linear combinations of (2.14) are exactly
the irreducible elements under the reduction system. There are no inclusion ambiguities in the
4 H.-W. Huang
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 − 2D2 +A2B −AB2 − 2AD + 2BD + 2AB − 2BC −Aβ −Bα− 2β,
DCB = BCD − 2D2 +B2C −BC2 − 2BD + 2CD − 2AC + 2BC +Bα+Bβ − Cβ
− 4D + 2α+ 2β,
DCA = ACD + 2D2 −A2C +AC2 − 2AD − 2AC + 2BC + 2CD +Aα+Aβ − Cα
− 4D + 2α+ 2β.
Hence each of the overlap ambiguities is resolvable.
LetM denote the free monoid with the alphabet set S = {A,B,C,D, α, β}. Let ` : M → N de-
note the length function ofM . Consider an element w = s1s2 · · · sn ∈M where s1, s2, . . . , sn ∈ S.
An operation on w is called an elementary operation if it is one of the following actions on w:
• We interchange si and sj where 1 ≤ i < j ≤ n and the position of sj is left to the position
of si in the list
A, B, C, D, α, β.
• Choose si ∈ {B,C,D} and replace si by the left neighbor of si 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
`(u) < `(w) or u is obtained from w by an elementary operation. For any u,w ∈ M we define
u � w if there exist u0, u1, . . . , uk ∈M with k ∈ N such that
u = u0 → u1 → · · · → uk−1 → uk = 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 (2.6)–(2.13).
Therefore, by diamond lemma the monomials (2.14) form an F-basis for <. �
Recall that the dihedral group D6 has a presentation with generators σ, τ and relations
σ2 = 1, τ6 = 1, (στ)2 = 1. (2.15)
Proposition 2.5 ([22, Propositions 4.1 and 4.3]). There exists a unique D6-action on < such
that (i), (ii) hold:
(i) σ acts on < as an F-algebra antiautomorphism of < given in the following way:
u A B C D α β γ δ
σ(u) B A C D −β −α −γ δ
(ii) τ acts on < as an F-algebra antiautomorphism of < given in the following way:
u A B C D α β γ δ
τ(u) B C A −D β γ α δ
Moreover the D6-action on < is faithful.
Let C denote the F-subalgebra of < generated by α, β, γ, δ. It follows from Lemma 2.2(ii)
that C is commutative.
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 5
Definition 2.6 ([22, Definition 5.2]). The coset
D2 +A2 +B2 +
(δ + 2){A,B} −
{
A2, B
}
−
{
A,B2
}
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
ΩA = D2 +
BAC + CAB
2
+A2 +Bγ − Cβ −Aδ, (2.16)
ΩB = D2 +
CBA+ABC
2
+B2 + Cα−Aγ −Bδ, (2.17)
ΩC = D2 +
ACB +BCA
2
+ C2 +Aβ −Bα− Cδ. (2.18)
Note that ΩA, ΩB, ΩC are mutually distinct [22, Corollary 6.5].
Lemma 2.7 ([22, Proposition 3.7]). Each of ΩA, ΩB, ΩC is a Casimir element.
Lemma 2.8 ([22, Lemma 3.6]). The set {ΩA,ΩB,ΩC} is invariant under the D6-action on <.
Moreover the restrictions of σ and τ to {ΩA,ΩB,ΩC} are as follows:
u ΩA ΩB ΩC
σ(u) ΩB ΩA ΩC
τ(u) ΩB ΩC ΩA
Definition 2.9 ([22, Section 5]). The elements ΩA, ΩB, ΩC are called theD6-symmetric Casimir
elements of <.
3 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 {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, (3.1)
κ = {X,Y } − Z, (3.2)
λ = {Y, Z} −X, (3.3)
µ = {Z,X} − Y. (3.4)
Proposition 3.2. There exists a unique D6-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:
u X Y Z ι κ λ µ
σ(u) Y X Z ι κ µ λ
(ii) τ acts on BI as an F-algebra antiautomorphism of BI given in the following way:
6 H.-W. Huang
u X Y Z ι κ λ µ
τ(u) Y Z X ι λ µ κ
Moreover the D6-action on BI is faithful.
Proof. It is straightforward to verify the existence of the D6-action on BI by using (2.15)
and Definition 3.1. Since D6 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 [22, Lemma 4.2] that the
D6-action on BI is faithful. �
Proposition 3.3. The F-algebra BI has a presentation with generators X, Y , Z, κ, λ, µ and
relations
Y X = −XY + Z + κ, ZY = −Y Z +X + λ, ZX = −XZ + Y + µ,
κX = Xκ, κY = Y κ, κZ = Zκ,
λX = Xλ, λY = Y λ, λZ = Zλ, λκ = κλ,
µX = Xµ, µY = Y µ, µZ = Zµ, µκ = κµ, µλ = λµ.
Proof. Immediate from Definition 3.1. �
Applying the diamond lemma to Proposition 3.3, we obtain the following Poincaré–Birkhoff–
Witt basis for BI. Since the argument is similar to the proof of Theorem 2.4, we omit the proof
here.
Theorem 3.4. The elements
XiY jZkκrλsµt for all i, j, k, r, s, t ∈ N (3.5)
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 {An}n∈N of F-subspaces of A satisfies the following conditions:
(N1)
⋃
n∈NAn = A.
(N2) An ⊆ An+1 for all n ∈ N.
(N3) Am · An ⊆ Am+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 3.4: Let wX , wY , wZ ,
wκ, wλ, wµ denote the nonnegative integers. For each n ∈ N let BIn denote the F-subspace
of BI spanned by XiY jZkκrλsµt for all i, j, k, r, s, t ∈ N with
wXi+ wY j + wZk + wκr + wλs+ wµt ≤ n.
We call {BIn}n∈N the F-subspaces of BI associated with (wX , wY , wZ , wκ, wλ, wµ). In what
follows we give a simple criterion for the above F-subspaces of BI to be an N-filtration of BI.
Theorem 3.5. Let wX , wY , wZ , wκ, wλ, wµ ∈ N. Let {BIn}n∈N denote the F-subspaces of BI
associated with (wX , wY , wZ , wκ, wλ, wµ). Then {BIn}n∈N is an N-filtration of BI if and only
if
max{wZ , wκ} ≤ wX + wY , (3.6)
max{wX , wλ} ≤ wY + wZ , (3.7)
max{wY , wµ} ≤ wZ + wX . (3.8)
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 7
Proof. (⇒) By the construction of {BIn}n∈N and Theorem 3.4 the element
Z + κ /∈ BImax{wZ ,wκ}−1.
On the other hand, by (N3) we have {X,Y } ∈ BIwX+wY . The equation (3.2) implies
Z + κ = {X,Y }.
By the above comments we see that BIwX+wY contains Z + κ which is not in BImax{wZ ,wκ}−1.
Combined with (N2) the inequality (3.6) follows. The inequalities (3.7) and (3.8) follow by
similar arguments.
(⇐) Condition (N1) is immediate from Theorem 3.4. Condition (N2) is immediate from the
construction of {BIn}n∈N. Set S = {X,Y, Z, κ, λ, µ}. For all n ∈ N, let In denote the set of
all (i, j, k, r, s, t) ∈ N6 with wXi + wY j + wZk + wκr + wλs + wµt ≤ 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) = wu for all u ∈ S.
By (3.6)–(3.8), for each relation of Proposition 3.3, 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′) ∈ Im and (i′′, j′′, k′′, r′′, s′′, t′′) ∈ In the product
Xi′Y j′Zk
′
κr
′
λs
′
µt
′ ·Xi′′Y j′′Zk
′′
κr
′′
λs
′′
µt
′′
is equal to an F-linear combination of XiY jZkκrλsµt for all (i, j, k, r, s, t) ∈ Im+n. In other
words (N3) holds. The theorem follows. �
4 The homomorphism ζ : < → BI
According to [16, Section 2] there exists an F-algebra homomorphism ζ : < → BI and the images
of A, B, C, α, β, γ, δ under ζ are as follows:
Theorem 4.1 ([16]). There exists a unique F-algebra homomorphism ζ : < → BI that sends
A 7→ (2X − 3)(2X + 1)
16
, B 7→ (2Y − 3)(2Y + 1)
16
, C 7→ (2Z − 3)(2Z + 1)
16
,
α 7→ (2ι− κ− µ− 3)(κ− µ)
64
, β 7→ (2ι− λ− κ− 3)(λ− κ)
64
,
γ 7→ (2ι− µ− λ− 3)(µ− λ)
64
, δ 7→ ι2 − 2ι− κ− λ− µ
4
− 9
16
.
We are now going to evaluate the image of D under ζ.
Lemma 4.2.
(i) The following equations hold in BI:[
X2, Y
]
= [X,Z],
[
Y 2, Z
]
= [Y,X],
[
Z2, X
]
= [Z, Y ],[
Y 2, X
]
= [Y, Z],
[
Z2, Y
]
= [Z,X],
[
X2, Z
]
= [X,Y ].
(ii) The following elements of BI are equal:
{X, [Z, Y ]}, {Y, [X,Z]}, {Z, [Y,X]},[
X2, Y 2
]
,
[
Y 2, Z2
]
,
[
Z2, X2
]
.
8 H.-W. Huang
Proof. (i) Since κ is central in BI and by (3.2) it follows that
[X,Z] = [X, {X,Y }].
Observe that [X, {X,Y }] =
[
X2, Y
]
. Therefore[
X2, Y
]
= [X,Z]. (4.1)
Applying Proposition 3.2 to (4.1) yields the remaining equations in (i).
(ii) By Proposition 3.2(ii) it suffices to show that
{X, [Z, Y ]} = {Y, [X,Z]}, (4.2)
{X, [Z, Y ]} =
[
X2, Y 2
]
. (4.3)
With trivial cancellations we obtain
{X, [Z, Y ]} − {Y, [X,Z]} = [Z, {Y,X}]. (4.4)
Since κ is central in BI and by (3.2) the element Z commutes with {Y,X}. Hence the right-hand
side of (4.4) is zero. Therefore (4.2) follows. Using (3.2) twice we find that
X2Y 2 = XY 2X +XZY −XY Z. (4.5)
By Proposition 3.2(ii), τ3 is an F-algebra antiautomorphism of BI that fixes X, Y , Z. Thus,
applying τ3 to (4.5) yields that
Y 2X2 = XY 2X + Y ZX − ZY X. (4.6)
Subtracting (4.6) from (4.5) yields (4.3). Hence (ii) follows. �
For convenience we let L denote the common element of BI from Lemma 4.2(ii).
Proposition 4.3. The image of D under ζ is equal to
[X,Y ] + [Y, Z] + [Z,X] + L
32
.
Proof. By (2.1) we have 2Dζ =
[
Aζ , Bζ
]
. A direct calculation yields that [Aζ , Bζ ] is equal to[
X2, Y 2
]
+ [X,Y ] +
[
Y 2, X
]
+
[
Y,X2
]
16
.
By Lemma 4.2(i),
[
Y 2, X
]
= [Y, Z] and
[
Y,X2
]
= [Z,X]. By Lemma 4.2(ii),
[
X2, Y 2
]
= L.
The proposition follows. �
Corollary 4.4. For each g ∈ D6 the following diagram commutes:
< BI
< BI.
ζ
g
ζ
g
Proof. It is routine to verify the corollary by using Propositions 2.5, 3.2 and Theorem 4.1. �
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 9
We end this section with a comment: Recall from [18, 21] that a universal analogue of the
additive DAHA (double affine Hecke algebra) of type
(
C∨1 , C1
)
, denoted by H here, is an F-
algebra generated by t0, t1, t
∨
0 , t∨1 and the relations assert that
t0 + t1 + t∨0 + t∨1 = −1
and each of t20, t
2
1, t
∨2
0 , t∨21 is central in H. By [18, Proposition 2] there exists an F-algebra
isomorphism \ : BI→ H that sends
X 7→ t0 + t1 +
1
2
, Y 7→ t0 + t∨0 +
1
2
, Z 7→ t0 + t∨1 +
1
2
, ι 7→ 2t0 +
1
2
,
κ 7→ t20 − t21 − t∨20 + t∨21 , λ 7→ t20 − t∨20 − t∨21 + t21, µ 7→ t20 − t∨21 − t21 + t∨20 .
The universal Askey–Wilson algebra [24] and the universal DAHA of type
(
C∨1 , C1
)
[25] are
the q-analogues of < and H, respectively. Therefore [25, Theorem 4.1] is a q-analogue of the
homomorphism \ ◦ ζ : < → H. Note that \ ◦ ζ sends
A 7→
(
t∨1 + t∨0
)(
t∨1 + t∨0 + 2
)
4
, B 7→
(
t1 + t∨1
)(
t1 + t∨1 + 2
)
4
,
C 7→
(
t∨0 + t1
)(
t∨0 + t1 + 2
)
4
,
α 7→
(
t∨21 − t∨20
)(
t21 − t20 + 2t0 − 1
)
16
, β 7→
(
t21 − t∨21
)(
t∨20 − t20 + 2t0 − 1
)
16
,
γ 7→
(
t∨20 − t21
)(
t∨21 − t20 + 2t0 − 1
)
16
, δ 7→ t20 + t21 + t∨20 + t∨21
4
− t0
2
− 3
4
.
5 The injectivity of ζ
Throughout this section, we let {BIn}n∈N denote the F-subspaces of BI associated with
(wX , wY , wZ , wκ, wλ, wµ) = (4, 4, 6, 8, 9, 9). (5.1)
Since the number sequence (5.1) satisfies (3.6)–(3.8), it follows from Theorem 3.5 that {BIn}n∈N
is an N-filtration of BI.
Lemma 5.1.
(i) For any even integer n ≥ 0 the following equations hold:
Y nX = XY n (mod BI4n+3),
XnY = Y Xn (mod BI4n+3),
ZnY = Y Zn (mod BI6n+3),
Y nZ = ZY n (mod BI4n+5),
XnZ = ZXn (mod BI4n+5),
ZnX = XZn (mod BI6n+3).
(ii) For any odd integer n ≥ 1 the following equations hold:
Y nX = −XY n + κY n−1 (mod BI4n+3),
XnY = −Y Xn + κXn−1 (mod BI4n+3),
Y nZ = −ZY n (mod BI4n+5),
ZnY = −Y Zn (mod BI6n+3),
XnZ = −ZXn (mod BI4n+5),
ZnX = −XZn (mod BI6n+3).
10 H.-W. Huang
Proof. All equations are established by routine inductions and using (3.2)–(3.4). �
Lemma 5.2.
(i) For any integer n ≥ 0 the following equations hold:
(
Aζ
)n
=
(
X
2
)2n
(mod BI8n−1),
(
Bζ
)n
=
(
Y
2
)2n
(mod BI8n−1),
(
Cζ
)n
=
(
Z
2
)2n
(mod BI12n−1),(
αζ
)n
=
(µ
8
)2n
(mod BI18n−1),(
βζ
)n
= (−1)n
(
λ
8
)2n
(mod BI18n−1).
(ii) For any even integer n ≥ 0 the following equation holds:
(
Dζ
)n
=
1
16n
n
2∑
i=0
(−4)i
(n
2
i
)
X2iY 2iκn−2iZn (mod BI14n−1).
(iii) For any odd integer n ≥ 1 the following equation holds:
(
Dζ
)n
=
1
16n
n−1
2∑
i=0
(−4)i
(n−1
2
i
)(
X2iY 2iκn−2i − 2X2i+1Y 2i+1κn−2i−1
)
Zn
(mod BI14n−1).
Proof. (i) Immediate from Theorem 4.1 and the construction of {BIn}n∈N.
(ii) It follows from Proposition 4.3 that
Dζ =
L
32
(mod BI13).
Evaluating L mod BI13 by using Lemma 4.2(ii) and Lemma 5.1(ii) yields that
Dζ =
Zκ
16
− XY Z
8
(mod BI13). (5.2)
Squaring the equation (5.2) a direct calculation shows that
(
Dζ
)2
=
Z2κ2
256
− X2Y 2Z2
64
(mod BI27). (5.3)
It follows from Lemma 5.1(i) that
Z2 ·X2Y 2Z2 = X2Y 2Z2 · Z2 (mod BI39). (5.4)
Now it is routine to derive (ii) by using (5.3) and (5.4).
(iii) To get (iii), one may multiply (5.2) by the equation from (ii) and simplify the resulting
equation by using Lemma 5.1(i). �
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 11
Lemma 5.3. Let i, j, k, `, n, r, s ∈ N with 8i+8j+12k+14`+18r+18s = n. Then the following
(i)–(iii) hold:
(i) For all i′, j′, k′, r′, s′, t′ ∈ N with 4i′+4j′+6k′+8r′+9s′+9t′ = n and r′ > `, the coefficient
of
Xi′Y j′Zk
′
κr
′
λs
′
µt
′
in
(
Aζ
)i(
Bζ
)j(
Cζ
)k(
Dζ
)`(
αζ
)r(
βζ
)s
with respect to the F-basis (3.5) for BI is zero.
(ii) For all i′, j′, k′, r′, s′, t′ ∈ N with 4i′+4j′+6k′+8r′+9s′+9t′ = n and r′ = `, the coefficient
of
Xi′Y j′Zk
′
κr
′
λs
′
µt
′
in
(
Aζ
)i(
Bζ
)j(
Cζ
)k(
Dζ
)`(
αζ
)r(
βζ
)s
with respect to the F-basis (3.5) for BI is nonzero if
and only if
(i′, j′, k′, s′, t′) = (2i, 2j, 2k + `, 2s, 2r).
(iii) The coefficient of
X2iY 2jZ2k+`κ`λ2sµ2r
in
(
Aζ
)i(
Bζ
)j(
Cζ
)k(
Dζ
)`(
αζ
)r(
βζ
)s
with respect to the F-basis (3.5) for BI is
(−1)s4−i−j−k−2`−3r−3s.
Proof. Using Lemmas 5.1(i) and 5.2 one may express(
Aζ
)i(
Bζ
)j(
Cζ
)k(
Dζ
)`(
αζ
)r(
βζ
)s
+ BIn−1
as an F-linear combination of Xi′Y j′Zk
′
κr
′
λs
′
µt
′
+ BIn−1 for all i′, j′, k′, r′, s′, t′ ∈ N with 4i′ +
4j′ + 6k′ + 8r′ + 9s′ + 9t′ = n. The lemma follows from the expression. �
Theorem 5.4. The homomorphism ζ : < → BI is injective.
Proof. Suppose on the contrary that there exists a nonzero element I in the kernel of ζ. For
all i, j, k, `, r, s ∈ N let c(i, j, k, `, r, s) denote the coefficient of
AiBjCkD`αrβs
in I with respect to the F-basis (2.14) for <. Let S denote the set of all (i, j, k, `, r, s) ∈ N6 with
c(i, j, k, `, r, s) 6= 0. For each n ∈ N we let S(n) denote the set of all (i, j, k, `, r, s) ∈ S with
8i+ 8j + 12k + 14`+ 18r + 18s = n. We may write
I =
∑
n∈N
∑
(i,j,k,`,r,s)∈S(n)
c(i, j, k, `, r, s)AiBjCkD`αrβs. (5.5)
Applying ζ to (5.5) we have
0 =
∑
n∈N
∑
(i,j,k,`,r,s)∈S(n)
c(i, j, k, `, r, s)
(
Aζ
)i(
Bζ
)j(
Cζ
)k(
Dζ
)`(
αζ
)r(
βζ
)s
. (5.6)
12 H.-W. Huang
Since I 6= 0 there exists at least one n ∈ N with S(n) 6= ∅. Set
N = max{n |S(n) 6= ∅}.
Among the elements in S(N) we choose a 6-tuple (i, j, k, `, r, s) that has the maximum value
at `. In what follows we evaluate the coefficient of
X2iY 2jZ2k+`κ`λ2sµ2r (5.7)
in the right-hand side of (5.6) with respect to the F-basis (3.5) for BI. Denote by c the coefficient.
Suppose that (i′, j′, k′, `′, r′, s′) is a 6-tuple in S(n) for some n ∈ N such that(
Aζ
)i′(
Bζ
)j′(
Cζ
)k′(
Dζ
)`′(
αζ
)r′(
βζ
)s′
(5.8)
contributes to the coefficient c. By Theorem 3.4 the monomial (5.7) lies in BIN not in BIN−1.
By Lemma 5.2 the term (5.8) lies in BIn. It follows from (N2) that n ≥ N and the maximality
of N implies n = N . By Lemma 5.3(i) we have `′ ≥ ` and the maximality of ` forces that `′ = `.
Combined with Lemma 5.3(ii) this yields that (i′, j′, k′, r′, s′) = (i, j, k, r, s). Therefore(
Aζ
)i(
Bζ
)j(
Cζ
)k(
Dζ
)`(
αζ
)r(
βζ
)s
is the only summand in the right-hand side of (5.6) contributes to the coefficient c. By Lem-
ma 5.3(iii) the coefficient c is the nonzero scalar
(−1)s · 4−i−j−k−2`−3r−3s · c(i, j, k, `, r, s).
It follows from Theorem 3.4 that the right-hand side of (5.6) is nonzero, a contradiction. The
theorem follows. �
As a consequence of Theorem 5.4 the F-algebra homomorphism \ ◦ ζ : < → H described in
Section 4 is injective. Note that [25, Theorem 4.5] is a q-analogue of the injectivity for \ ◦ ζ.
6 The images of the Casimir elments of < under ζ
In light of Theorem 5.4 the Racah algebra < 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 < in BI.
Proof. By Theorem 4.1 and (3.1) the commutator [ι, A] is equal to 1
4 times[
Y + Z,X2
]
− [Y + Z,X]. (6.1)
Simplifying (6.1) by using Lemma 4.2(i) yields that (6.1) 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 < 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 {BIn}n∈N denote the F-subspaces of BI associated with
(wX , wY , wZ , wκ, wλ, wµ) = (1, 1, 2, 0, 0, 0). (6.2)
Since the sequence (6.2) satisfies (3.6)–(3.8), it follows from Theorem 3.5 that {BIn}n∈N is an
N-filtration of BI.
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 13
Lemma 6.2. Zn = ιn mod BI2n−1 for all n ∈ N.
Proof. Proceed by induction on n. It is trivial for n = 0. By (3.1) we have
ι− Z = X + Y ∈ BI1. (6.3)
Hence the lemma holds for n = 1. Suppose that n ≥ 2. We divide ιn − Zn into
Z
(
ιn−1 − Zn−1
)
+ (ι− Z)ιn−1. (6.4)
Since Z ∈ BI2 and by induction hypothesis, the first summand of (6.4) is in BI2n−1. By (3.1)
the element ι ∈ BI2 and hence ιn−1 ∈ BI2n−2. Combined with (6.3) the second summand
of (6.4) is in BI2n−1. The lemma follows. �
Lemma 6.3.
(i) For all n ∈ N the elements
XiY jZkκrλsµt + BIn−1 for all i, j, k, r, s, t ∈ N with i+ j + 2k = n
are an F-basis for BIn/BIn−1.
(ii) For all n ∈ N the elements
XiY jιkκrλsµt + BIn−1 for all i, j, k, r, s, t ∈ N with i+ j + 2k = n
are an F-basis for BIn/BIn−1.
(iii) For all n ∈ N the elements
XiY jιkκrλsµt for all i, j, k, r, s, t ∈ N with i+ j + 2k ≤ n
are an F-basis for BIn.
Proof. (i) Immediate from Theorem 3.4 and the construction of {BIn}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
XiY jιkκrλsµt for all i, j, k, r, s, t ∈ N (6.5)
are an F-basis for BI.
Proof. Immediate from (N1) and Lemma 6.3(iii). �
Corollary 6.5. The elements ι, κ, λ, µ of BI are algebraically independent over F.
Proof. Immediate from Theorem 6.4. �
Lemma 6.6. The F-algebra BI has a presentation with generators X, Y , ι, κ, λ, µ and relations
Y X = −XY −X − Y + ι+ κ,
ιY = 2Y 2 − Y ι− Y + ι+ κ+ λ,
ιX = 2X2 −Xι−X + ι+ κ+ µ,
κX = Xκ, κY = Y κ, κι = ικ,
λX = Xλ, λY = Y λ, λι = ιλ, λκ = κλ,
µX = Xµ, µY = Y µ, µι = ιµ, µκ = κµ, µλ = λµ.
14 H.-W. Huang
Proof. This is a reformulation of Proposition 3.3 by using (3.1). �
Recall the D6-symmetric Casimir elements ΩA, ΩB, ΩC of < from (2.16)–(2.18).
Proposition 6.7. The D6-symmetric Casimir elements ΩA, ΩB, ΩC of < have the following
expressions:
ΩA =
λ(λ− 2ι+ 3)
(
4ι2 − 8ι− 4κ− 4λ− 4µ+ 7
)
1024
− Γ, (6.6)
ΩB =
µ(µ− 2ι+ 3)
(
4ι2 − 8ι− 4κ− 4λ− 4µ+ 7
)
1024
− Γ, (6.7)
ΩC =
κ(κ− 2ι+ 3)
(
4ι2 − 8ι− 4κ− 4λ− 4µ+ 7
)
1024
− Γ, (6.8)
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
.
Proof. Applying Theorem 4.1 and Proposition 4.3 to (2.16) and replacing Z by ι − X − Y ,
we may express ΩA in terms of X, Y , ι, κ, λ, µ. To get (6.6) we apply Lemma 6.6 to express
the resulting expression as an F-linear combination of (6.5). Combined with Lemma 2.8 and
Proposition 3.2 we obtain (6.7) and (6.8). �
Theorem 6.8. For each Casimir element Ω of < there exists a unique four-variable polynomial
P (x1, x2, x3, x4) over F such that
Ω = P (ι, κ, λ, µ).
Proof. By Definition 2.6 there exists a four-variable polynomial Q(y1, y2, y3, y4) over F such
that Ω = ΩA +Q(α, β, γ, δ). Set Q̂(x1, x2, x3, x4) = Q(y1, y2, y3, y4) by substituting
y1 =
(2x1 − x2 − x4 − 3)(x2 − x4)
64
, y2 =
(2x1 − x3 − x2 − 3)(x3 − x2)
64
,
y3 =
(2x1 − x4 − x3 − 3)(x4 − x3)
64
, y4 =
x21 − 2x1 − x2 − x3 − x4
4
− 9
16
.
It follows from Theorem 4.1 that Ω = ΩA + Q̂(ι, κ, λ, µ). Combined with Proposition 6.7 the
existence follows. The uniqueness is immediate from Corollary 6.5. �
Acknowledgements
The research is supported by the Ministry of Science and Technology of Taiwan under the project
MOST 106-2628-M-008-001-MY4.
References
[1] Bergman G.M., The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218.
[2] Crampé N., Frappat L., Vinet L., Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient
of the Bannai–Ito algebra, J. Phys. A: Math. Theor. 52 (2019), 424001, 11 pages, arXiv:1906.03936.
[3] De Bie H., De Clercq H., The q-Bannai–Ito algebra and multivariate (−q)-Racah and Bannai–Ito polyno-
mials, J. London Math. Soc., to appear, arXiv:1902.07883.
https://doi.org/10.1016/0001-8708(78)90010-5
https://doi.org/10.1088/1751-8121/ab433f
https://arxiv.org/abs/1906.03936
https://doi.org/10.1112/jlms.12367
https://arxiv.org/abs/1902.07883
The Racah Algebra as a Subalgebra of the Bannai–Ito Algebra 15
[4] De Bie H., De Clercq H., van de Vijver W., The higher rank q-deformed Bannai–Ito and Askey–Wilson
algebra, Comm. Math. Phys. 374 (2020), 277–316, arXiv:1805.06642.
[5] De Bie H., Genest V.X., Tsujimoto S., Vinet L., Zhedanov A., The Bannai–Ito algebra and some applications,
J. Phys. Conf. Ser. 597 (2015), 012001, 16 pages, arXiv:1411.3913.
[6] De Bie H., Genest V.X., van de Vijver W., Vinet L., Bannai–Ito algebras and the osp(1; 2) superalgebra,
in Physical and Mathematical Aspects of Symmetries, Proceedings of the 31st International Colloquium in
Group Theoretical Methods in Physics (Rio de Janeiro, June 19–25, 2016), Editors Duarte S., Gazeau J.-P.,
Faci S., Micklitz T., Scherer R., Toppan F., Springer, Cham, 2017, 349–354, arXiv:1610.04797.
[7] De Bie H., Genest V.X., van de Vijver W., Vinet L., A higher rank Racah algebra and the Zn2 Laplace–Dunkl
operator, J. Phys. A: Math. Theor. 51 (2018), 025203, 20 pages, arXiv:1610.02638.
[8] De Bie H., Genest V.X., Vinet L., A Dirac–Dunkl equation on S2 and the Bannai–Ito algebra, Comm. Math.
Phys. 344 (2016), 447–464, arXiv:1501.03108.
[9] De Bie H., Genest V.X., Vinet L., The Zn2 Dirac–Dunkl operator and a higher rank Bannai–Ito algebra,
Adv. Math. 303 (2016), 390–414, arXiv:1511.02177.
[10] Genest V.X., Lapointe L., Vinet L., osp(1, 2) and generalized Bannai–Ito algebras, Trans. Amer. Math. Soc.
372 (2019), 4127–4148, arXiv:1705.03761.
[11] Genest V.X., Vinet L., Zhedanov A., The Bannai–Ito algebra and a superintegrable system with reflections
on the two-sphere, J. Phys. A: Math. Theor. 47 (2014), 205202, 13 pages, arXiv:1401.1525.
[12] Genest V.X., Vinet L., Zhedanov A., The Bannai–Ito polynomials as Racah coefficients of the sl−1(2)
algebra, Proc. Amer. Math. Soc. 142 (2014), 1545–1560, arXiv:1205.4215.
[13] Genest V.X., Vinet L., Zhedanov A., The equitable Racah algebra from three su(1, 1) algebras, J. Phys. A:
Math. Theor. 47 (2014), 025203, 12 pages, arXiv:1309.3540.
[14] Genest V.X., Vinet L., Zhedanov A., The Racah algebra and superintegrable models, J. Phys. Conf. Ser.
512 (2014), 012011, 15 pages, arXiv:1312.3874.
[15] Genest V.X., Vinet L., Zhedanov A., Superintegrability in two dimensions and the Racah–Wilson algebra,
Lett. Math. Phys. 104 (2014), 931–952, arXiv:1307.5539.
[16] Genest V.X., Vinet L., Zhedanov A., Embeddings of the Racah algebra into the Bannai–Ito algebra, SIGMA
11 (2015), 050, 11 pages, arXiv:1504.00558.
[17] Genest V.X., Vinet L., Zhedanov A., A Laplace–Dunkl equation on S2 and the Bannai–Ito algebra, Comm.
Math. Phys. 336 (2015), 243–259, arXiv:1312.6604.
[18] Genest V.X., Vinet L., Zhedanov A., The non-symmetric Wilson polynomials are the Bannai–Ito polyno-
mials, Proc. Amer. Math. Soc. 144 (2016), 5217–5226, arXiv:1507.02995.
[19] Granovskĭı Y.A., Zhedanov A.S., Nature of the symmetry group of the 6j-symbol, Soviet Phys. JETP 94
(1988), 1982–1985.
[20] Granovskĭı Y.I., Zhedanov A.S., Lutsenko I.M., Quadratic algebras and dynamical symmetry of the
Schrödinger equation, Soviet Phys. JETP 99 (1991), 205–209.
[21] Huang H.W., Finite-dimensional modules of the Racah algebra and the additive DAHA of type (C∨1 , C1),
arXiv:1906.09160.
[22] Huang H.W., Bockting-Conrad S., The Casimir elements of the Racah algebra, J. Algebra Appl., to appear,
arXiv:1711.09574.
[23] Lévy-Leblond J.M., Lévy-Nahas M., Symmetrical coupling of three angular momenta, J. Math. Phys. 6
(1965), 1372–1380.
[24] Terwilliger P., The universal Askey–Wilson algebra, SIGMA 7 (2011), 069, 24 pages, arXiv:1104.2813.
[25] Terwilliger P., The universal Askey–Wilson algebra and DAHA of type
(
C∨1 , C1
)
, SIGMA 9 (2013), 047,
40 pages, arXiv:1202.4673.
[26] Tsujimoto S., Vinet L., Zhedanov A., Dunkl shift operators and Bannai–Ito polynomials, Adv. Math. 229
(2012), 2123–2158, arXiv:1106.3512.
https://doi.org/10.1007/s00220-019-03562-w
https://arxiv.org/abs/1805.06642
https://doi.org/10.1088/1742-6596/597/1/012001
https://arxiv.org/abs/1411.3913
https://doi.org/10.1007/978-3-319-69164-0_52
https://arxiv.org/abs/1610.04797
https://doi.org/10.1088/1751-8121/aa9756
https://arxiv.org/abs/1610.02638
https://doi.org/10.1007/s00220-016-2648-1
https://doi.org/10.1007/s00220-016-2648-1
https://arxiv.org/abs/1501.03108
https://doi.org/10.1016/j.aim.2016.08.007
https://arxiv.org/abs/1511.02177
https://doi.org/10.1090/tran/7733
https://arxiv.org/abs/1705.03761
https://doi.org/10.1088/1751-8113/47/20/205202
https://arxiv.org/abs/1401.1525
https://doi.org/10.1090/S0002-9939-2014-11970-8
https://arxiv.org/abs/1205.4215
https://doi.org/10.1088/1751-8113/47/2/025203
https://doi.org/10.1088/1751-8113/47/2/025203
https://arxiv.org/abs/1309.3540
https://doi.org/10.1088/1742-6596/512/1/012011
https://arxiv.org/abs/1312.3874
https://doi.org/10.1007/s11005-014-0697-y
https://arxiv.org/abs/1307.5539
https://doi.org/10.3842/SIGMA.2015.050
https://arxiv.org/abs/1504.00558
https://doi.org/10.1007/s00220-014-2241-4
https://doi.org/10.1007/s00220-014-2241-4
https://arxiv.org/abs/1312.6604
https://doi.org/10.1090/proc/13141
https://arxiv.org/abs/1507.02995
https://arxiv.org/abs/1906.09160
https://doi.org/10.1142/S0219498821501358
https://arxiv.org/abs/1711.09574
https://doi.org/10.1063/1.1704786
https://doi.org/10.3842/SIGMA.2011.069
https://arxiv.org/abs/1104.2813
https://doi.org/10.3842/SIGMA.2013.047
https://arxiv.org/abs/1202.4673
https://doi.org/10.1016/j.aim.2011.12.020
https://arxiv.org/abs/1106.3512
1 Introduction
2 The Racah algebra
3 The Bannai–Ito algebra BI
4 The homomorphism 2mu-:6muplus1muBI
5 The injectivity of
6 The images of the Casimir elments of under
References
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| id | nasplib_isofts_kiev_ua-123456789-210773 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T06:42:49Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Huang, Hau-Wen 2025-12-17T14:35:43Z 2020 The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra. Hau-Wen Huang. SIGMA 16 (2020), 075, 15 pages 1815-0659 2020 Mathematics Subject Classification: 81R10; 81R12 arXiv:1906.11745 https://nasplib.isofts.kiev.ua/handle/123456789/210773 https://doi.org/10.3842/SIGMA.2020.075 Assume that is a field with char ≠ 2. The Racah algebra ℜ is a unital associative -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 ℜ. The Bannai-Ito algebra is a unital associative -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 I. It was discovered that there exists an -algebra homomorphism ζ: ℜ → 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 ℜ can be considered as an -subalgebra of . Moreover, we show that any Casimir element of ℜ can be uniquely expressed as a polynomial in {X, Y} − Z, {Y, Z} − X, {Z, X} − Y, and X + Y + Z with coefficients in . The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 106-2628-M-008-001-MY4. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra Article published earlier |
| spellingShingle | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra Huang, Hau-Wen |
| title | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra |
| title_full | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra |
| title_fullStr | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra |
| title_full_unstemmed | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra |
| title_short | The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra |
| title_sort | racah algebra as a subalgebra of the bannai-ito algebra |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210773 |
| work_keys_str_mv | AT huanghauwen theracahalgebraasasubalgebraofthebannaiitoalgebra AT huanghauwen racahalgebraasasubalgebraofthebannaiitoalgebra |