On a Certain Subalgebra of $U_q(\widehat{\mathfrak{sl}}_2)$ Related to the Degenerate $q$-Onsager Algebra

In [Kyushu J. Math. 64 (2010), 81-144, arXiv:0904.2889], it is discussed that a certain subalgebra of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}}_2)$ controls the second kind TD-algebra of type I (the degenerate $q$-Onsager algebra). The subalgebra, which we denote by $U'_q(\widehat{\mathfrak{sl}}_2)$, is generated by $e_0^+$, $e_1^\pm$, $k_i^{\pm1}$ $(i=0,1)$ with $e^-_0$ missing from the Chevalley generators $e_i^\pm$, $k_i^{\pm1}$ $(i=0,1)$ of $U_q(\widehat{\mathfrak{sl}}_2)$. In this paper, we determine the finite-dimensional irreducible representations of $U'_q(\widehat{\mathfrak{sl}}_2)$. Intertwiners are also determined.


Introduction
Throughout this paper, the ground field is C and q stands for a nonzero scalar that is not a root of unity. The symbols ε, ε * stand for an integer chosen from {0, 1}. Let A q = A (ε,ε * ) q denote the associative algebra with 1 generated by z, z * subject to the defining relations [4] (TD) where [X, Y ] = XY − Y X, [X, Y ] q = qXY − q −1 Y X. This paper deals with a subalgebra of the quantum affine algebra U q ( sl 2 ) that is closely related to A q in the case of (ε, ε * ) = (1, 0). If (ε, ε * ) = (0, 0), A q is isomorphic to the positive part of U q ( sl 2 ). If (ε, ε * ) = (1, 1), A q is called the q-Onsager algebra. If (ε, ε * ) = (1, 0), A q may well be called the degenerate q-Onsager algebra.
The algebra A q arises in the course of the classification of TD-pairs of type I, which is a critically important step in the study of representations of Terwilliger algebras for P-and Q-polynomial association schemes [3]. For this reason, A q is called the TD-algebra of type I. Precisely speaking, the TD-algebra of type I is standardized to be the algebra A q , where q is the main parameter for TD-pairs of type I; so q = ±1 and q is allowed to be a root of unity. In our case where we assume q is not a root of unity, to classify the TD-pairs of type I is to determine the finite-dimensional irreducible representations ρ : A q −→ End (V ) with the property that ρ(z), ρ(z * ) are both diagonalizable, and vice versa. Such irreducible representations of A q are determined in [4] via embeddings of A q into the augmented TD-algebra T q . (In the case of (ε, ε * ) = (1, 1), the diagonalizability condition of ρ(z), ρ(z * ) can be dropped, because it turns out that this condition always holds for every finite-dimensional irreducible representation ρ of the q-Onsager algebra A q .) T q is easier than A q to study representations for, and each finite-dimensional irreducible representation ρ : A q −→ End (V ) with ρ(z), ρ(z * ) diagonalizable can be extended to a finite-dimensional irreducible representation of T q via a certain embedding of A q into T q .
The augmented TD-algebra T q = T (ε,ε * ) q is the associative algebra with 1 generated by x, y, k ±1 subject to the defining relations and where δ = −(q−q −1 )(q 2 −q −2 )(q 3 −q −3 )q 4 . The finite-dimensional irreducible representations of T q are determined in [4] via embeddings of T q into the U q (sl 2 )-loop algebra U q (L(sl 2 )).
Let e ± i , k ±1 i (i = 0, 1) be the Chevalley generators of U q (L(sl 2 )). So the defining relations of U q (L(sl 2 )) are Note that if k 0 k 1 = k 1 k 0 = 1 is replaced by k 0 k 1 = k 1 k 0 in (4), we have the quantum affine algebra U q ( sl 2 ): U q (L(sl 2 )) is the quotient algebra of U q ( sl 2 ) by the two-sided ideal generated by k 0 k 1 − 1. For a nonzero scalar s, define the elements Then the mapping gives an injective algebra homomorphism. If (ε, ε * ) = (0, 0), the image ϕ s (T q ) coincides with the Borel subalgebra generated by e + i , k ±1 i (i = 0, 1). If (ε, ε * ) = (1, 0), the image ϕ s (T q ) is properly contained in the subalgebra generated by e + 0 , e ± 1 , k ±1 i (i = 0, 1), e − 0 missing from the generators; we denote this subalgebra by U ′ q (L(sl 2 )). Through the natural homomorphism U q ( sl 2 ) −→ U q (L(sl 2 )), pull back the subalgebra U ′ q (L(sl 2 )) and denote the pre-image by U ′ q ( sl 2 ): In [4], it is shown that in the case of (ε, ε * ) = (1, 0), all the finitedimensional irreducible representations of T q are produced by tensor products of evaluation modules for U ′ q (L(sl 2 )) via the embedding ϕ s of T q into U ′ q (L(sl 2 )). Using this fact and the Drinfel'd polynomials, we show in this paper that there are no finite-dimensional irreducible representations of U ′ q (L(sl 2 )) and hence of U ′ q ( sl 2 ) other than those afforded by tensor products of evaluation modules, if we apply suitable automorphisms of U ′ q (L(sl 2 )), U ′ q ( sl 2 ) to adjust the types of the representations to be (1, 1). Here we note that the evaluation parameters are allowed to be zero for U ′ q (L(sl 2 )), U ′ q ( sl 2 ). Details will be discussed in Section 2, where the isomorphism classes of finitedimensional irreducible representations of U ′ q ( sl 2 ) are also determined. In Section 3, intertwiners will be determined for finite-dimensional irreducible U ′ q ( sl 2 )-modules. Drinfel'd polynomials are not the main subject of this paper but the key tool for the classification of finite-dimensional irreducible representations of U q ( sl 2 ), U ′ q ( sl 2 ). They are defined in [4], directly attached to T q -modules, not to U q ( sl 2 )-or U ′ q ( sl 2 )-modules. (In the case of (ε, ε * ) = (0, 0), they turn out to coincide with the original ones up to the reciprocal of the variable.) So if our approach is applied to the case of (ε, ε * ) = (0, 0), finite-dimensional irreducible representations are naturally classified in the first place for the Borel subalgebra of U q ( sl 2 ) and then for U q ( sl 2 ) itself. This reverses the process adopted in [1] and will be briefly demonstrated in Section 2 as a warm-up for the case of (ε, ε * ) = (1, 0), thus giving another proof to the classical result of Chari-Pressley [2].
The subalgebra U ′ q ( sl 2 ) of the quantum affine algebra U q ( sl 2 ) is generated by e + 0 , e ± 1 , k ±1 i (i = 0, 1), e − 0 missing from the generators, and has by the triangular decomposition of U q ( sl 2 ) the defining relations Note that if k 0 k 1 = k 1 k 0 is replaced by k 0 k 1 = k 1 k 0 = 1 in (8), we have the defining relations for U ′ q (L(sl 2 )). Let V be a finite-dimensional irreducible U ′ q ( sl 2 )-module. Let us first observe that the U ′ q ( sl 2 )-module V is obtained from a U ′ q (L(sl 2 ))-module by applying an automorphism of U ′ q ( sl 2 ). Since the element k 0 k 1 belongs to the centre of U ′ q ( sl 2 ), k 0 k 1 acts on V as a scalar s by Schur's lemma. Since k 0 k 1 is invertible, the scalar s is nonzero: k 0 k 1 | V = s ∈ C × . Observe that U ′ q ( sl 2 ) admits an automorphism that sends k 0 to s −1 k 0 and preserves k 1 . Hence we may assume k 0 k 1 | V = 1. Then we can regard V as a U ′ q (L(sl 2 ))-module.
is the corresponding eigenspace of k 0 . For an eigenvalue θ and an eigenvector v ∈ V (θ), it holds that e + 0 v ∈ V (q 2 θ) by the relation Choose an eigenvalue θ of k 0 on V . Then i∈Z V (q ±2i θ) is invariant under the actions of the generators by (9), and so we have V = i∈Z V (q ±2i θ) by the irreducibility of the U ′ q (L(sl 2 ))module V . Since V is finite-dimensional, there exists a positive integer d and a nonzero scalar s 0 such that the eigenspace decomposition of k 0 is We want to show that s 0 = ±1 holds in (10). Consider the subalgebra of U ′ q (L(sl 2 )) generated by e ± 1 , k ±1 1 and denote it by U : In particular, if θ is an eigenvalue of k 0 , so is θ −1 . The collection of such eigenvalues gives rise to the eigenspace decomposition of (10). We obtain s 0 = ±1. Observe that U ′ q (L(sl 2 )) admits an automorphism that sends k i to −k i (i = 0, 1) and e + 1 to −e + 1 . Hence we may assume s 0 = 1 in (10). Note that in this case, k 1 has the eigenvalues {s 1 q 2i−ℓ | 0 ≤ i ≤ ℓ} with s 1 = 1. Such an irreducible module or the irreducible representation afforded by such is said to be of type (1, 1), indicating (s 0 , s 1 ) = (1, 1). We conclude that the determination of finite-dimensional irreducible representations for U ′ q ( sl 2 ) is, via automorphisms, reduced to that of type (1, 1) for U ′ q (L(sl 2 )). In the rest of this section, we shall introduce evaluation modules for U ′ q (L(sl 2 )) and show that every finite-dimensional irreducible representation of type (1, 1) of U ′ q (L(sl 2 )) is afforded by a tensor product of them. For a ∈ C and ℓ ∈ Z ≥0 , let V (ℓ, a) denote the (ℓ where is called a standard basis. The vector v 0 is called the highest weight vector. Note that the evaluation parameter a is allowed to be zero. Also note that if ℓ = 0, V (ℓ, a) is the trivial module. We denote the evaluation module V (ℓ, 0) by V (ℓ), allowing ℓ = 0, and use the notation V (ℓ, a) only for an evaluation module with a = 0 and ℓ ≥ 1.
The U q (sl 2 )-loop algebra U q (L(sl 2 )) has the coproduct ∆ : The subalgebra U ′ q (L(sl 2 )) is closed under ∆. Thus given a set of evaluation becomes a U ′ q (L(sl 2 ))-module via ∆. Note that by the coassociativity of ∆, the way of putting parentheses in the tensor product of (13) does not affect the isomorphism class as a U ′ q (L(sl 2 ))-module. Also note that if ℓ = 0 in (13), then V (0) is the trivial module and the tensor product of (13) is isomorphic to V (ℓ 1 , a 1 ) ⊗ · · · ⊗ V (ℓ n , a n ) as U ′ q (L(sl 2 ))-modules. Finally we allow n = 0, in which case we understand that the tensor product of (13) means V (ℓ).
With the evaluation module V (ℓ, a), we associate the set S(ℓ, a) of scalars aq −ℓ+1 , aq −ℓ+3 , . . . , aq ℓ−1 : The set S(ℓ, a) is called a q-string of length ℓ. Two q-strings S(ℓ, a), S(ℓ ′ , a ′ ) are said to be in general position if either Below is the main theorem of this paper. It classifies the isomorphism classes of the finite-dimensional irreducible U ′ q (L(sl 2 ))-modules of type (1, 1).
Discard the evaluation module V (ℓ) from the statement of Theorem 1 and replace U ′ q (L(sl 2 )) by U q (L(sl 2 )) or B, where B is the Borel subalgebra of U q (L(sl 2 )) generated by e + i , k ±1 i (i = 0, 1). Then it precisely describes the classification of the isomorphism classes of finite-dimensional irreducible modules of type (1, 1) for U q (L(sl 2 )) [2] or B [1]. There is a one-to-one correspondence of finite-dimensional irreducible modules of type (1, 1) between U q (L(sl 2 )) and B: every finite-dimensional irreducible U q (L(sl 2 ))-module of type (1, 1) is irreducible as a B-module and every finite-dimensional irreducible B-module of type (1, 1) is uniquely extended to a U q (L(sl 2 ))-module. This sort of correspondence of finite-dimensional irreducible modules exists between U ′ q (L(sl 2 )) and T q via the embedding ϕ s of (6), where T q is the augmented TD-algebra with (ε, ε * ) = (1, 0), and this gives a proof of Theorem 1. The key to our understanding of the correspondence is the following two lemmas. Let U denote the quantum algebra U q (sl 2 ): U is the associative algebra with 1 generated by X ± , K ±1 subject to the defining relations Lemma 1. [4, Lemma 7.5] Let V be a finite-dimensional U-module that has the following weight-space (K-eigenspace) decomposition: Let W be a subspace of V as a vector space. Assume that W is invariant under the actions of X + and K: If it holds that then X − W ⊆ W , i.e., W is a U-submodule.

Lemma 2.
If V is a finite-dimensional U-module, the action of X − on V is uniquely determined by those of X + , K ±1 on V .
Proof. The claim holds if V is irreducible as a U-module. By the semisimplicity of U, it holds generally.
As a warm-up for the proof of Theorem 1, we shall demonstrate how to use these lemmas in the case of the corresponding theorem [2] for U q (L(sl 2 )). We want, and it is enough, to show part (iii) of the theorem for U q (L(sl 2 )) by using the classification of finite-dimensional irreducible B-modules, since the parts (i), (ii) are well-known in advance of [2] and the finite-dimensional irreducible B-modules are classified in [4] without using the part (iii) in question.
Let V be a finite-dimensional irreducible U q (L(sl 2 ))-module of type (1, 1). Then V has the weight-space decomposition Consider the algebra homomorphism from U to U q (L(sl 2 )) that sends X + , X − , K ±1 to e + 0 , e − 0 , k ±1 0 , respectively. Regard V as a U-module via this algebra homomorphism. Then Similarly, Lemma 1 can be applied to the U-module V via the algebra homomorphism from U to U q (L(sl 2 )) that sends X + , X − , K ±1 to e + 1 , e − 1 , k ±1 1 , respectively, in which case the weight-space decomposition of )-invariant and we have W = V by the irreducibility of the U q (L(sl 2 ))-module V . We conclude that every finite-dimensional irreducible U q (L(sl 2 ))-module of type (1, 1) is irreducible as a B-module. Now consider the class of finite-dimensional irreducible B-modules V , where V runs through all tensor products of evaluation modules that are irreducible as a U q (L(sl 2 ))-module: V = V (ℓ 1 , a 1 ) ⊗ · · · ⊗ V (ℓ n , a n ).
Then it turns out that the Drinfel'd polynomials P V (λ) of the irreducible B-modules V exhaust all that are possible for finite-dimensional irreducible B-modules of type (1, 1), as shown in [4] by the product formula Since the Drinfel'd polynomial P V (λ) determines the isomorphism class of the B-module V of type (1, 1) [4, the injectivity part of Theorem 1.9'], there are no other finite-dimensional irreducible B-modules of type (1, 1). This means that every finite-dimensional irreducible B-module of type (1, 1) comes from some tensor product of evaluation modules for U q (L(sl 2 )).
Let V be a finite-dimensional irreducible U q (L(sl 2 ))-module of type (1, 1). Then V is irreducible as a B-module and so there exists an irreducible U q (L(sl 2 ))-module V ′ = V (ℓ 1 , a 1 )⊗· · ·⊗V (ℓ n , a n ) such that V, V ′ are isomorphic as B-modules. By Lemma 2, V, V ′ are isomorphic as U q (L(sl 2 ))-modules. This completes the proof of part (iii) of the theorem for U q (L(sl 2 )).
The proof of Theorem 1 can be given by an argument very similar to the one we have seen above for the case of U q (L(sl 2 )). We prepare two more lemmas for the case of U ′ q (L(sl 2 )) to make the point clearer. Set (ε, ε * ) = (1, 0) and let T q be the augmented TD-algebra defined by (TD) 0 , (TD) 1 in (2), (3). For s ∈ C × , let ϕ s be the embedding of T q into U ′ q (L(sl 2 )) given by (5), (6).
is isomorphic to the quantum algebra U q (sl 2 ), the action of e − 1 on V i (i = 1, 2) is uniquely determined by those of e + 1 , k ±1 1 ∈ ϕ s (T q ) by Lemma 2. Apparently the action of e + 0 on V i (i = 1, 2) is uniquely determined by those of se + 0 + s −1 e − 1 k 1 , e − 1 , k 1 , and hence by that of ϕ s (T q ). So the action of U ′ q (L(sl 2 )) on V i (i = 1, 2) is uniquely determined by that of ϕ s (T q ).

Lemma 4.
Let V be a finite-dimensional irreducible U ′ q (L(sl 2 ))-module of type (1,1). Then there exists a finite set Λ of nonzero scalars such that V is irreducible as a ϕ s (T q )-module for each s ∈ C × − Λ.
Proof. For s ∈ C × , regard V be a ϕ s (T q )-module. Let W be a minimal ϕ s (T q )-submodule of V . It is enough to show that W = V holds if s avoids finitely many scalars. By (10) with s 0 = 1, the eigenspace decomposition of . We want to show it is a bijection if s avoids finitely many scalars. Identify U d−i with U i as vector spaces by the bijection (e + 1 ) d−2i between them. Then it makes sense to consider the determinant of a linear map from U i to U d−i . Set t = s −2 and expand (e + 0 + te − 1 k 1 ) d−2i as Each term of the expansion gives a linear map from U i to U d−i . So the determinant of (e + 0 + te − and this is a polynomial in t of degree . Therefore by Lemma 1, we have e − 1 W ⊆ W . Since (e + 0 + s −2 e − 1 k 1 )W ⊆ W , the inclusion e + 0 W ⊆ W follows from e − 1 W ⊆ W and so W is U ′ q (L(sl 2 ))-invariant. Thus W = V holds by the irreducibility of V as a U ′ q (L(sl 2 ))-module.
(ii) Consider two tensor products V = V (ℓ)⊗V (ℓ 1 , a 1 )⊗· · ·⊗V (ℓ n , a n ), of evaluation modules and assume that they are both irreducible as a ϕ s (T q )-module. Then V, V ′ are isomorphic as ϕ s (T q )-modules if and only if ℓ = ℓ ′ , n = m and (ℓ i , a i ) = (ℓ ′ i , a ′ i ) for all i ∈ {1, . . . , n} with a suitable reordering of the evaluation modules V (ℓ 1 , a 1 ), . . . , V (ℓ n , a n ). a 1 )⊗· · ·⊗V (ℓ n , a n ) on which T q acts via some embedding ϕ s : T q −→ U ′ q (L(sl 2 )). Part (i) of Theorem 1 follows immediately from the part (i) above, due to Lemma 4. Part (ii) of Theorem 2 follows immediately from the part (ii) above, the 'if' part due to Lemma 3 (and Lemma 4) and the 'only if' part due to Lemma 4.
We want to show part (iii) of Theorem 1. Let V be a finite-dimensional irreducible U ′ q (L(sl 2 ))-module of type (1, 1). By Lemma 4, there exists a nonzero scalar s such that V is irreducible as a ϕ s (T q )-module. By the part (iii) above, the T q -module V via ϕ s is isomorphic to some T q -module V ′ = V (ℓ) ⊗ V (ℓ 1 , a 1 ) ⊗ · · · ⊗ V (ℓ n , a n ) via some embedding ϕ s ′ of T q into U ′ q (L(sl 2 )). Since k 0 has the same eigenvalues on V, V ′ , we have s = s ′ and so V, V ′ are isomorphic as ϕ s (T q )-modules. By Lemma 3, V, V ′ are isomorphic as U ′ q (L(sl 2 ))-modules. This completes the proof of Theorem 1.

Intertwiners
In this section, we show that for ℓ, m ∈ Z >0 , a ∈ C × , there exists an intertwiner between the U ′ q (L(sl 2 ))-modules V (ℓ, a) ⊗ V (m), V (m) ⊗ V (ℓ, a), i.e., If such an intertwiner R exists, then it is routinely concluded that V (ℓ, a) ⊗ V (m) is isomorphic to V (m) ⊗ V (ℓ, a) as U ′ q (L(sl 2 ))-modules and any other intertwiner is a scalar multiple of R, since V (m) ⊗ V (ℓ, a) is irreducible as a U ′ q (L(sl 2 ))-module by Theorem 1.
Using the theory of Drinfel'd polynomials [4] for the augmented TD- )-modules. We shall then construct an intertwiner explicitly.
Let us denote the U ′ q (L(sl 2 ))-modules Recall we assume that the integers ℓ, m and the scalar a are nonzero. Let us denote a standard basis of the in the sense of (11). Recall V (m) is an abbreviation of V (m, 0) and the action of U ′ q (L(sl 2 )) on V, V ′ are via the coproduct ∆ of (12).
Let U denote the subalgebra of U ′ q (L(sl 2 )) generated by e ± 1 , K ± 1 . The subalgebra U is isomorphic to the quantum algebra U q (sl 2 ). Let V (n) denote the irreducible U-module of dimension n + 1: V (n) has a standard basis x 0 , x 1 , . . . , x n on which U acts as We call x n (resp. x 0 ) the lowest (highest) weight vector : Let x n denote a lowest weight vector of the U-module V (n). So the lowest weight vector x n of V (n) can be expressed as Solving ∆(e − 1 ) x n = 0 for the coefficients c j , we obtain and so with a suitable choice of c 0 where n = ℓ + m − 2ν and [t]! = [t][t − 1] · · · [1].
Proof. Let T q = T (ε,ε * ) q be the augmented TD-algebra with (ε, ε * ) = (1, 0). Let ϕ s : T q −→ U ′ q (L(sl 2 )) denote the embedding of T q into U ′ q (L(sl 2 )) given in (6). By Theorem 5.2 of [4], the Drinfel'd polynomial P V (λ) of the ϕ s (T q )module V = V (ℓ, a) ⊗ V (m) turns out to be (Note that the parameter s of the embedding ϕ s does not appear in P V (λ). So the polynomial P V (λ) can be called the Drinfel'd polynomial attached to the U ′ q (L(sl 2 ))-module V .) Let W be a minimal U ′ q (L(sl 2 ))-submodule of V . By Corollary 1, W contains the lowest and hence highest weight vectors of V . In particular, the irreducible U ′ q (L(sl 2 ))-module W is of type (1, 1). By Lemma 4, there exists a finite set Λ of nonzero scalars such that W is irreducible as a ϕ s (T q )-module for any s ∈ C × − Λ. By the definition [4, (25)], the Drinfel'd polynomial P W (λ) of the irreducible ϕ s (T q )-module W coincides with P V (λ): By Theorem 1, V ′ = V (m)⊗V (ℓ, a) is irreducible as a U ′ q (L(sl 2 ))-module. So by Lemma 4, there exists a finite set Λ ′ of nonzero scalars such that V ′ is irreducible as a ϕ s (T q )-module for any s ∈ C × − Λ ′ . By Theorem 5.2 of [4], the Drinfel'd polynomial P V ′ (λ) of the irreducible ϕ s (T q )-module V ′ coincides with P V (λ) : Both of the irreducible ϕ s (T q )-modules W, V ′ have type s, diameter d = ℓ + m and the Drinfel'd polynomial P V (λ). By Theorem 1.9' of [4], W and V ′ are isomorphic as ϕ s (T q )-modules. By Lemma 3, W and V ′ are isomorphic as U ′ q (L(sl 2 ))-modules. In particular, dim W = dim V ′ . Since dim V ′ = dim V , we have W = V , i.e., V and V ′ are isomorphic as U ′ q (L(sl 2 ))-modules.
Finally we want to construct an intertwiner R between the irreducible U ′ q (L(sl 2 ))-modules V, V ′ . Regard V ′ = V (m) ⊗ V (ℓ, a) as a U-module. By the Clebsch-Gordan formula, we have the direct sum decomposition where V ′ (n) is the unique irreducible U-submodule of V ′ isomorphic to V (n) (n = ℓ + m − 2ν). Let x ′ n be a lowest weight vector of the U-module V ′ (n). By (22), we have up to a scalar multiple, where n = ℓ + m − 2ν. It can be easily checked as in Lemma 5 that the lowest weight vectors x ′ n (n = ℓ + m − 2ν, 0 ≤ ν ≤ min{ℓ, m}) are related by where V ′ = V (m) ⊗ V (ℓ, a) is regarded as a (U ⊗ U)-module in the natural way.
There exists a unique linear map that commutes with the action of U and sends x n to x ′ n . The linear map R n vanishes on V (t) for t = n and affords an isomorphism between V (n) and V ′ (n) as U-modules. If R is an intertwiner in the sense of (17), R can be expressed as regarding R as an intertwiner for the U-modules V, V ′ . By (17), we have R ∆(e + 0 ) = ∆ (e + 0 )R.