An embedding of the universal Askey-Wilson algebra into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$

The Askey--Wilson algebras were used to interpret the algebraic structure hidden in the Racah--Wigner coefficients of the quantum algebra $U_q(\mathfrak{sl}_2)$. In this paper, we display an injection of a universal analog $\triangle_q$ of Askey--Wilson algebras into $U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)\otimes U_q(\mathfrak{sl}_2)$ behind the application. Moreover we formulate the decomposition rules for $3$-fold tensor products of irreducible Verma $U_q(\mathfrak{sl}_2)$-modules and of finite-dimensional irreducible $U_q(\mathfrak{sl}_2)$-modules into the direct sums of finite-dimensional irreducible $\triangle_q$-modules.

Consider a Hopf * -algebra of U q (sl 2 ), for instance U q (su 2 ) with q a real number. Let V denote a 3-fold tensor product of irreducible unitary U q (su 2 )-modules. The inner products between two coupled bases of V are called the Racah-Wigner coefficients of U q (sl 2 ). In fact, the two coupled bases of V are the orthonormal eigenbases of on V . Granovskiȋ and Zhedanov [2] realized that when restricting to an eigenspace of ∆ 2 (Λ) on V , there exist complex scalars ̺, ̺ * , η, η * , ω such that the operators K 0 , K 1 and their q-commutator satisfy the relations The research is supported by the Ministry of Science and Technology of Taiwan under the project MOST 105-2115-M-008-013. 1 The unital associative algebra over C generated by three generators, abusively denoted by K 0 , K 1 , K 2 , subject to the relations of the forms (1)-(3) is called the Askey-Wilson algebra [12]. A universal analog △ q of Askey-Wilson algebras was recently proposed by Terwilliger [10] with q 4 = 1. The defining generators of △ q are usually denoted by A, B, C and the relations assert that each of Let α, β, γ denote the central elements of △ q given by The Casimir element of △ q has the expression Inspired by the work [2], we discover a homomorphism ♭ : △ q → U q (sl 2 ) ⊗ U q (sl 2 ) ⊗ U q (sl 2 ) that sends The image of Ω under ♭ is equal to Moreover the homomorphism ♭ is shown to be injective. As a consequence, given any Hopf * -algebra of U q (sl 2 ) we show that there exists a unique algebra involution † of △ q such that the following diagram commutes: △ q -modules are completely reducible. Furthermore the work [5] allows us to give the decomposition rules for these △ q -modules into the direct sums of finite-dimensional irreducible △ q -modules. As a consequence, on each irreducible component A and B form a Leonard pair [9,11]. The preliminaries are contained in §2 and §3.
It is worthwhile to remark that the paper [3] produces a similar work on an algebraic structure behind the Clebsch-Gordan coefficients of U q (sl 2 ).
2. The quantum algebra U q (sl 2 ) and its modules The conventions for this paper are as follows. An algebra is meant to be a unital associative algebra. Let Z denote the ring of integers. Let N denote the set of nonnegative integers and N * = N \ {0}.
Let F denote a field and fix a nonzero scalar q ∈ F with q 4 = 1. Recall that the q-brackets are defined as [n] = q n − q −n q − q −1 for all n ∈ N and the q-binomial coefficients are defined as for all i, n ∈ N.
By [6,Lemma 2.7] the element is central in U q (sl 2 ). The central element is called the Casimir element of U q (sl 2 ). Multiplying the Casimir element by (q − q −1 ) 2 we obtain the element Throughout this paper we use the normalized Casimir element Λ instead of the original Casimir element of U q (sl 2 ). Lemma 2.2. For each i ∈ N the following relations (i), (ii) hold in U q (sl 2 ): Proof. Proceed by inductions on i and apply (4).
Lemma 2.3. For each n ∈ N the following (i), (ii) hold: (i) U n has the F-basis (ii) U −n has the F-basis Proof. Fix an n ∈ N. By Lemma 2.1 the elements for all h ∈ Z and i ∈ N are an F-basis of U n . For each ℓ ∈ N let U n,ℓ denote the F-subspace of U n spanned by F i K h E i+n for all h ∈ Z and 0 ≤ i ≤ ℓ. Clearly We claim that the elements form an F-basis of U n,ℓ . To see this we proceed by induction on ℓ. It is nothing to prove for ℓ = 0. Suppose that ℓ ≥ 1. By construction U n,ℓ /U n,ℓ−1 has the F-basis For each h ∈ Z we have The first equality follows from the defining relation KF = q −2 F K and the second equality follows from Lemma 2.2(ii) and the induction hypothesis. Therefore the elements are an F-basis of U n,ℓ /U n,ℓ−1 . Combined with the induction hypothesis the claim follows. Now (i) follows from the claim and (5). Statement (ii) follows by a similar argument.

2.2.
The Hopf algebra structure of U q (sl 2 ). Recall that the algebras and algebra homomorphisms can be defined in the following ways: An F-vector space A endowed with two F-linear maps and ∇ • (1 ⊗ ι) and ∇ • (ι ⊗ 1) are equal to the canonical maps A ⊗ F → A and F ⊗ A → A, respectively. Here ∇ and ι are called the multiplication and the unit of an F-algebra A respectively. Suppose that A and A ′ are two F-algebras. Let ∇, ∇ ′ denote the multiplications of A, A ′ and let ι, ι ′ denote the units of A, A ′ respectively. An F-linear map φ : The coalgebras and coalgebra homomorphisms are defined in dual ways: An F-vector space A endowed with two F-linear maps ∆ : A → A ⊗ A and ǫ : A → F is called an F-coalgebra if and (1 ⊗ ǫ) • ∆ and (ǫ ⊗ 1) • ∆ are equal to the canonical maps A → A ⊗ F and A → F ⊗ A, respectively. Here ∆ and ǫ are called the comultiplication and the counit of an F-coalgebra A respectively. Suppose that A and A ′ are two F-coalgebras. Let ∆, ∆ ′ denote the comultiplications of A, A ′ and let ǫ, ǫ ′ denote the counits of A, A ′ respectively. An F-linear map φ : An F-vector space A is called an F-bialgebra if A is an F-algebra and an F-coalgebra such that the multiplication and the unit of A are F-coalgebra homomorphisms or equivalently the comultiplication and the counit of A are F-algebra homomorphisms. Suppose that A and A ′ are two F-bialgebras. An F-linear map φ : A → A ′ is called an F-bialgebra homomorphism if φ is an F-algebra homomorphism and an F-coalgebra homomorphim. An F-bialgebra A endowed with an F-linear map S : A → A is called an F-Hopf algebra if the multiplication ∇, the unit ι, the comultiplication ∆ and the counit ǫ of A satisfy Here S is called the antipode of an F-Hopf algebra A. Suppose that A and A ′ are two F-Hopf algebras. Let S, S ′ denote the antipodes of A, A ′ respectively. An F-linear map φ : A → A ′ is called an F-Hopf algebra homomorphism if φ is an F-bialgebra homomorphism satisfying The quantum algebra U q (sl 2 ) admits the Hopf algebra structure: The comultiplication ∆ : The counit ǫ : U q (sl 2 ) → F is an F-algebra homomorphism defined by The antipode S : U q (sl 2 ) → U q (sl 2 ) is an F-algebra antiautomorphism defined by Let ∆ 0 denote the identity map on U q (sl 2 ). We recurrently define The map ∆ n (n ∈ N) is called the n-fold comultiplication of U q (sl 2 ).
Lemma 2.5. For all n ∈ N * and 1 ≤ i ≤ n the following equation holds: Proof. Proceed by induction on n and apply (6).
Note that each U q (sl 2 ) ⊗n -module can be regarded as a U q (sl 2 )-module by pulling back via ∆ n−1 for n ∈ N * . 2.3. Verma U q (sl 2 )-modules and finite-dimensional U q (sl 2 )-modules. For notational convenience we set [λ; n] = λq n − λ −1 q −n q − q −1 for any nonzero λ ∈ F and n ∈ Z.
For any nonzero scalar λ ∈ F there exists a unique U q (sl 2 )-module M (λ) that has an F-basis {m [6, §2.4] the Verma U q (sl 2 )module M (λ) satisfies the following universal property: then there exists a unique U q (sl 2 )-module homomorphism M (λ) → V that sends m (λ) 0 to v. Recall the normalized Casimir element Λ from (4). A direct calculation yields that Lemma 2.7. Let λ denote a nonzero scalar in F. Then the element Λ acts on the Verma U q (sl 2 )module M (λ) as scalar multiplication by λq + λ −1 q −1 .
A necessary and sufficient condition for the Verma U q (sl 2 )-module M (λ) to be irreducible is given below. For a proof please see [6,Proposition 2.5].
A necessary and sufficient condition for V (n, ε) to be irreducible is given below.
Lemma 2.10. For all n ∈ N and ε ∈ {±1} the following (i), (ii) are equivalent: Proof. (i) ⇒ (ii): Suppose on the contrary that there exists 1 ≤ i ≤ n such that q 2i = 1. Clearly To see the irreducibility of V (n, ε), we let V denote any nonzero U q (sl 2 )-submodule of V (n, ε) and show that V = V (n, ε). Choose a nonzero vector v ∈ V . For 0 ≤ i ≤ n let Since V is a U q (sl 2 )-module the vector E k v ∈ V . By (ii) the scalars [i] = 0 for all 1 ≤ i ≤ n. Hence v (n,ε) 0 ∈ V . Observe that Hence v (n,ε) i ∈ V for all 1 ≤ i ≤ n. Therefore V = V (n, ε). For all n ∈ N it is known that any (n+1)-dimensional irreducible U q (sl 2 )-module is isomorphic to V (n, 1) or V (n, −1) provided that F is algebraically closed and q is not a root of unity [6, Theorem 2.6].
3. The universal Askey-Wilson algebra △ q and its modules 3.1. The universal Askey-Wilson algebra. The universal Askey-Wilson algebra △ q is an Falgebra generated by A, B, C and the relations assert that each of Recall from §1 that the Askey-Wilson algebra is an F-algebra generated by K 0 , K 1 , K 2 subject to the relations of the forms (1)- (3). Recall that (1)-(3) involve five additional parameters ̺, ̺ * , η, η * , ω. It is straightforward to verify that △ q satisfies the following property: Under the mild constraints ̺ = 0 and ̺ * = 0 there exists a unique surjective homomorphism from △ q into the Askey-Wilson algebra that sends Let α, β, γ denote the elements of △ q defined by By the definition of △ q the elements α, β, γ are central in △ q . Recall from [10, Theorem 4.1] that Lemma 3.1. The elements Observe that C is an F-linear combination of AB, BA, γ. Therefore A, B, γ form a set of generators of △ q . A presentation for △ q with respect to the generators is given in [10, Theorem 2.2]: By [10, Theorem 6.2] the element is central in △ q . The central element is called the Casimir element of △ q [10,12].
, c) satisfies the following universal property: The action of A, B, C on V d (a, b, c) with respect to the F-basis {v i } d i=0 is as follows: The central elements α, β, γ act on V d (a, b, c) as scalar multiplications by ω, ω * , ω ε respectively. The △ q -module V d (a, b, c) satisfies the following universal property: By the Cayley-Hamilton theorem N ν (a, b, c) is contained in the kernel of φ. The lemma follows.
A necessary and sufficient condition for V d (a, b, c) to be irreducible is given below. For a proof please see [5,Theorem 4.4].
By [5,Theorem 4.7], for any (d + 1)-dimensional irreducible △ q -module V , there exist nonzero scalars a, b, c ∈ F satisfying Lemma 3.5(ii) such that V is isomorphic to V d (a, b, c), provided that F is algebraically closed and q is not a root of unity.

3.3.
Finite-dimensional irreducible △ q -modules and Leonard pairs. In this subsection we give a brief introduction to two famous families of finite-dimensional irreducible △ q -modules studied in [4,5].
To describe these △ q -modules, we begin with some terms. A matrix is said to be tridiagonal if all nonzero entries lie on either the diagonal, the subdiagonal, or the superdiagonal. A tridiagonal matrix is said to be irreducible if each entry on the subdiagonal and superdiagonal is nonzero. Recall from [9] that two linear transformations on a finite-dimensional vector space V are called a Leonard pair whenever for each of the two transformations, there exists a basis of V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. [5, Theorem 5.2] gave the following necessary and sufficient conditions for any two of A, B, C as a Leonard pair on a finite-dimensional irreducible △ q -module. Lemma 3.6. Assume that the △ q -module V d (a, b, c) is irreducible. Then the following (i)-(iii) are equivalent: Recall from [1] that three linear transformations on a finite-dimensional vector space V are called a Leonard triple [1] whenever for each of the three transformations, there exists a basis of V with respect to which the matrix representing that transformation is diagonal and the matrices representing the other transformations are irreducible tridiagonal. [5,Theorem 5.3] gave the following necessary and sufficient conditions for A, B, C as a Leonard triple on a finite-dimensional irreducible △ q -module.
The outline of this section is as follows. In §4.1 we prove the existence and uniqueness of the Moreover we display the image of the Casimir element Ω of △ q under ♭ in terms of the normalized Casimir element Λ of U q (sl 2 ). In §4.2 we prove that the homomorphism ♭ is injective. As an application of the injectivity of ♭, in §4.3 we show that for any Hopf can be uniquely pulled back to an algebra involution of △ q via ♭.
. In this subsection we prove the principal result of this paper: To verify the existence of ♭ we make some preparation.
Lemma 4.3. The homogeneous components of ∆(Λ) and ∆ 2 (Λ) are given in the tables below. Any homogeneous component not displayed is zero.
Lemma 4.4. The homogeneous components of A ♭ , B ♭ , C ♭ , α ♭ , β ♭ , γ ♭ are given in the tables below. Any homogeneous component not displayed is zero. We are now in the position to prove Theorem 4.1.
Proof of Theorem 4.1. By Lemma 4.2 with n = 1 the element ∆ 2 (Λ) commutes with 1 ⊗ ∆(Λ) and To see the existence of ♭ it remains to verify that Equation (15) is immediate from the construction of A ♭ , B ♭ , C ♭ , γ ♭ . To verify (13) and (14), one may utilize Lemmas 2.2 and 4.4 to express each homogeneous component of the right-hand sides of (13), (14) as a linear combination of the corresponding basis given in Lemma 2.4. Then it is able to check that the corresponding homogeneous components of both sides of (13), (14) are equal to each other. After the tedious verification (13), (14) follow. This shows the existence of ♭. The homomorphism ♭ is unique since the F-algebra △ q is generated by A, B, C.
Recall the formula (11) for the Casimir element Ω of △ q .
Theorem 4.5. The image of the Casimir element Ω of △ q under ♭ is equal to Proof. By (11) the image of Ω under ♭ is Similar to the idea of the verification for (13) and (14), we express each homogeneous component of (16), (17) as a linear combination of the corresponding basis given in Lemma 2.4. After then we can check that the corresponding homogeneous components of (16), (17) are equal to each other. The theorem follows.
We end this subsection with a corollary of Theorem 4.1.
Proof. By Lemma 4.2 with n = 1 the elements A ♭ = ∆(Λ) ⊗ 1 and B ♭ = 1 ⊗ ∆(Λ) are in the centralizer of ∆ 2 (U q (sl 2 )) in U q (sl 2 ) ⊗3 . Since Λ is in the center of U q (sl 2 ) the element C ♭ is in the centralizer of ∆ 2 (U q (sl 2 )) in U q (sl 2 ) ⊗3 as well. Since the F-algebra △ q is generated by A, B, C the corollary follows.

4.2.
The injectivity of the homomorphism ♭. To see the injectivity of ♭, we equip the ring Z 3 with the lexicographical order. Let u denote a nonzero element of U q (sl 2 ) ⊗ U q (sl 2 ) ⊗ U q (sl 2 ). We call a nonzero homogeneous component of u the leading component of u if each homogeneous component of u of higher degree is zero. The degree of u is meant to be the degree of the leading component of u.
Lemma 4.7. For all i, j, k, r, s, t ∈ N the following (i), (ii) hold: We are ready to prove the injectivity of ♭.

Proof. Observe from Lemma 4.4 that the degrees of
Theorem 4.8. The homomorphism ♭ is injective.
Proof. Suppose on the contrary that there exists a nonzero element I in the kernel of ♭. For any i, j, k, r, s, t ∈ N let c(i, j, k, r, s, t) denote the coefficient of A i B j C k α r β s γ t in I with respect to the basis of △ q given in Lemma 3.1. Let S denote the set consisting of all 6-tuples (i, j, k, r, s, t) of nonnegative integers with c(i, j, k, r, s, t) = 0. Thus (i,j,k,r,s,t)∈S c(i, j, k, r, s, t)A i B j C k α r β s γ t = I.
Applying ♭ to either side of the equality, we obtain that Furthermore, for all m, n, p ∈ Z let S(m, n, p) denote the set consisting of those (i, j, k, r, s, t) ∈ S with (m, n, p) = (i + k + r + s + t, j − i, −j − k − r − s − t).
By Lemma 4.7(i) each summand in the inner summation of (19) is of degree (m, n, p). Since I = 0 the finite set S is nonempty. Hence we may define (M, N, P ) = max{(m, n, p) ∈ Z 3 | S(m, n, p) = ∅}.
We end this subsection with a corollary of Theorem 4.8. Corollary 4.9. Let φ denote an F-Hopf algebra automorphism of U q (sl 2 ). For any F-algebra automorphism ϕ of △ q the following (i), (ii) are equivalent: Proof. By [8, §3.1.2, Proposition 6] there exists a nonzero scalar α ∈ F satisfying
(i) ⇒ (ii): By Theorem 4.8 any automorphism ϕ of △ q with (i) must sends A, B, C to A, B, C respectively. Therefore ϕ = 1.

4.3.
Hopf * -algebras of U q (sl 2 ) and algebra involutions of △ q . Throughout this subsection, we assume that the underlying field is the complex number field C and let − : C → C denote the complex conjugation.
Recall that an involution * is a function with * • * = 1. A vector space V over C with an involution * : V → V is called a * -vector space if (λu + µv) * = λu * + µv * for all u, v ∈ V and λ, µ ∈ C. An algebra A over C with an involution * : A → A is called a * -algebra if A is a * -vector space and (xy) * = y * x * for all x, y ∈ A. Here * is called the algebra involution of a * -algebra A. A coalgebra A over C with an involution * is called a * -coalgebra if the coalgebra A is a * -vector space and the comultiplication ∆ and the counit ǫ of A satisfy A bialgebra over C with an involution * is called a * -bialgebra if the bialgebra is a * -algebra and a * -coalgebra. A Hopf algebra over C with an involution * is called a Hopf * -algebra if the Hopf algebra is a * -bialgebra. A Hopf * -algebra A is said to be equivalent to a Hopf * ′ -algebra A ′ if there exists a Hopf algebra isomorphism φ : The equivalence relations on * -algebras, * -coalgebras and * -bialgebras are defined in similar ways. Let R denote the real number field. Recall from [8, §3. 1.4] that the Hopf algebra of U q (sl 2 ) has the following Hopf * -algebra structures up to equivalence: The cases (R1)-(R3) correspond to the real forms su 2 , su 1,1 , sl 2 (R) of sl 2 (C) respectively. Hence these Hopf * -algebras of U q (sl 2 ) are denoted by U q (su 2 ), U q (su 1,1 ), U q (sl 2 (R)) respectively. The cases (R4), (R5) have no classical counterparts. Proof. To see (i), (ii) one may apply each algebra involution * of U q (sl 2 ) given in (R1)-(R5) to (4) and simplify the resulting equation by using the defining relations of U q (sl 2 ).
Theorem 4.11. For any Hopf * -algebra of U q (sl 2 ) there exists a unique algebra involution † : Proof. By Theorem 4.8, if the algebra involution † of △ q exists then it is unique. In what follows we prove the existence of †.

5.
The U q (sl 2 ) ⊗ U q (sl 2 ) ⊗ U q (sl 2 )-modules as △ q -modules By Theorem 4.1 each U q (sl 2 ) ⊗ U q (sl 2 ) ⊗ U q (sl 2 )-module can be considered as a △ q -module via pulling back the homomorphism ♭. The aim of this section is to study the 3-fold tensor products of irreducible Verma U q (sl 2 )-modules and of finite-dimensional U q (sl 2 )-modules from the viewpoint of △ q -modules.
Proof. Since q is not a root of unity the scalars {κλq −2h } h∈N are mutually distinct. Therefore j } i,j∈N are an F-basis of M (κ) ⊗ M (λ), the lemma follows.
A decomposition rule for the U q (sl 2 )-module M (κ) ⊗ M (λ) is given below.
Proposition 5.2. Assume that q is not a root of unity. Let κ and λ denote nonzero scalars in F with κ, λ, κλ ∈ {±q n | n ∈ N}.
Then the following (i), (ii) hold: Proof. Fix an h ∈ N. Recall the F-subspace N (h) of M (κ) ⊗ M (λ) from Lemma 5.1. Observe that We show that the (h + 1) vectors (25) are linearly independent over F. Suppose on the contrary that there exist scalars Let j = max{j | c j = 0}. Applying ∆(E) j to either side of (26) it follows that Since κλ = ±q n for all n ∈ N the coefficient in the left-hand side of (27) is nonzero. Therefore the left-hand side of (27) is a nonzero vector, a contradiction. This shows that (25) are linearly independent over F. By . By the construction of M the homomorphism φ is surjective. Since κλ = ±q n for all n ∈ N and by Lemma 2.8, the Verma U q (sl 2 )-module M (κλq −2h ) is irreducible. Since M = 0 it follows that φ is injective. Therefore φ is a U q (sl 2 )-module isomorphism. By (24) the vector m Combined with (i) the statement (ii) follows.
The F-basis of M (κ) ⊗ M (λ) given in Proposition 5.2(i) is called the coupled F-basis of M (κ) ⊗ M (λ). On the other hand the U q (sl 2 )-module M (κ) ⊗ M (λ) has the F-basis For all h, i, j, k ∈ N let Proposition 5.3. Assume that q is not a root of unity. Let κ and λ denote nonzero scalars in F with κ, λ, κλ ∈ {±q n | n ∈ N}. For all h, i, j, k ∈ N the following (i), (ii) hold: Proof. By (23), for all h, i, j ∈ N we have Comparing the coefficients of m in both sides of (24) yields that for all h ∈ N and i, j, k ∈ N * . The lemma follows by solving the recurrence relation (30) with initial values (29).

5.2.
Three-fold tensor products of irreducible Verma U q (sl 2 )-modules. In this subsection, we study the 3-fold tensor products of irreducible Verma U q (sl 2 )-modules from the viewpoint of △ q -modules.
(iv) For all d, k ∈ N the following are equivalent: (a) A, B, C act on M k (d) as a Leonard triple.
Proof. Since q is not a root of unity and κ, λ, κλ, κλµ = ±q n for all n ∈ N we may apply Proposition 5.2(i) to see that Since q is not a root of unity and κλµ = ±q n for all n ∈ N the sum in the right-hand side is a direct sum. Therefore (i) follows. Moreover we have Lemma 5.5. For all d, k ∈ N the following (i), (ii) hold: is an eigenvector of A associated with eigenvalue is an eigenvector of B associated with eigenvalue (iii) The elements α, β, γ act on M k (d) as scalar multiplications by respectively.

By Proposition 5.2(ii) we have
By Lemma 5.5(i) and since q is not a root of unity the above equation induces an F-linear isomorphism from M 0 (d) onto M k (d). By Corollary 4.6 each element of △ ♭ q commutes with ∆ 2 (F ). Hence the induced F-linear isomorphism M 0 (d) → M k (d) is a △ q -module isomorphism. By the above comment, we may consider the △ q -module Let a, b, c be as in the statement (ii). Let {θ i } i∈Z , {θ * i } i∈Z , {ϕ i } i∈Z and ω, ω * , ω ε denote the corresponding scalars defined in §3.2 with ν = q d . It is straightforward to check that θ h , θ * h , ω, ω * , ω ε coincide with (31)-(35) respectively. By Lemmas 5.5(i) and 5.6(i) the element A is diagonalizable on V with eigenvalues {θ i } d i=0 . Hence the characteristic polynomial of A on V is d i=0 (X − θ i ). By Lemma 5.6(ii) we have Also, it follows from Lemma 5.6(iii) that To apply Proposition 3.4 it remains to verify that To check (37) we consider the F-basis with respect to (38). By the construction of w with respect to (38) is equal to a 0 · c 0 (d, 0, 0) + a 1 · c 0 (d − 1, 1, 0) where Here c 0 (d − 1, 1, 0) is interpreted as an indeterminate if d = 0. By Proposition 5.3 the scalars Comparing the coefficient of m On the other hand, we apply w (0,d) 0 to either side of the second relation shown in Lemma 3.2. Simplifying the resulting equation by using (36) we obtain that Since q is not a root of unity and λµ = ±q n for all n ∈ N, the scalars {θ * i } d i=0 are mutually distinct and θ * −1 = θ * i for all 2 ≤ i ≤ d. Combined with Lemmas 5.5(ii) and 5.6(ii) the element B is diagonalizable on V with simple eigenvalues {θ * i } d i=0 . Therefore B − θ * −1 can be dropped from (40) even if θ * −1 = θ * 0 or θ * −1 = θ * 1 . It follows from (36) that Hence the left-hand side of (37) is a scalar multiple of w (0,d) 0 by the scalar (39) times θ * 0 − θ * 1 . To see (37) it is now routine to check that ϕ 1 coincides with the aforementioned scalar.
Thanks to Proposition 3.4 there exists a unique △ q -module homomorphism V d (a, b, c) → V that sends v 0 to w (0,d) 0 . By the assumptions on κ, λ, µ, q the scalars a, b, c, q satisfy Lemma 3.5(i), (ii). Therefore the △ q -module V d (a, b, c) is irreducible and the above △ q -module homomorphism We have seen that A and B are diagonalizable on V . Therefore (iii) is immediate from Lemma 3.6. Using Lemma 3.7 it is routine to verify (iv). The theorem follows.
For notational convenience, we write v Then the following (i), (ii) hold: that sends v (m,n;h) k to v (m+n−2h) k for all 0 ≤ h ≤ min{m, n} and 0 ≤ k ≤ m + n − 2h.
The F-basis of V (m) ⊗ V (n) given in Proposition 5.7(i) is called the coupled F-basis of V (m) ⊗ V (n). On the other hand, the U q (sl 2 )-module V (m) ⊗ V (n) has the F-basis in v (m,n;h) k with respect to the F-basis (41) of V (m) ⊗ V (n). By a similar argument to Proposition 5.2 these coefficients can be expressed as below.

5.4.
Three-fold tensor products of finite-dimensional irreducible U q (sl 2 )-modules. In the final subsection we show an analogue of Theorem 5.4 for the finite-dimensional U q (sl 2 )-modules V (n). Although the proof idea is similar to that of Theorem 5.4, some places need nontrivial adjustments and hence a complete proof is included below. Lemma 5.9. Let m, n, p denote any three real numbers. For any real numbers h, ℓ the following (i)-(iii) are equivalent: Proof. Rewriting the second inequality in (i) as ℓ − p ≤ h ≤ min{ℓ, m + n − ℓ}. Combining the inequality with the first inequality in (i) yields the equivalence of (i) and (ii). Since max{0, ℓ − p} = ℓ − min{p, ℓ} and min{m, n, ℓ, m + n − ℓ} = min{m, ℓ} + min{n, ℓ} − ℓ the inequalities (ii) and (iii) are equivalent.
The latter is equivalent to (ii). Therefore (i) and (ii) are equivalent. Rewriting (ii) as an inequality about ℓ we obtain (iii). Therefore (ii) and (iii) are equivalent.
After all this preparation let us prove the last result of this paper.
Theorem 5.11. Assume that q is not a root of unity. Let m, n, p denote any nonnegative integers. Let Σ denote the set consisting of all pairs (ℓ, k) of integers satisfying 0 ≤ ℓ ≤ min m + n, n + p, p + m, m+n+p 2 , 0 ≤ k ≤ m + n + p − 2ℓ.
(iii) For all (ℓ, k) ∈ Σ the action of A, B, C on V k (ℓ) is as a Leonard triple.