Asymptotic representations of augmented q-Onsager algebra and boundary K-operators related to Baxter Q-operators

We consider intertwining relations of the augmented $q$-Onsager algebra introduced by Ito and Terwilliger, and obtain generic (diagonal) boundary $K$-operators in terms of the Cartan element of $U_{q}(sl_2)$. These $K$-operators solve reflection equations. Taking appropriate limits of these $K$-operators in Verma modules, we derive $K$-operators for Baxter Q-operators and corresponding reflection equations.


Introduction
In the context of quantum integrable systems with periodic boundary conditions, the Baxter Q-operator [1] is an important object. It contains the information about the eigenfunctions and Bethe roots of the Hamiltonian and transfer matrix that are studied within a Bethe ansatz approach. Importantly, a key ingredient in the construction of Baxter Q-operators are L-operators. In the context of the representation theory, the Loperators that are suitable for Baxter Q-operators are obtained as certain homomorphic tum algebra U q ( sl 2 ), U q (sl 2 ) and two q-oscillator algebras that will be needed for our purpose. L-operators and their limits are recalled in Section 3. In Section 4, we introduce the augmented q-Onsager algebra though generators and relations. Two different types of realizations of the augmented q-Onsager algebra are considered, namely as a right or left coideal subalgebra U q ( sl 2 ). The two corresponding intertwining relations are considered and solved, giving an explicit expression for K-operators. The reflection and dual reflection equations they satisfy are displayed. Certain limits of those K-operators are considered, that are required for the construction of Q-operators. The rational limit (q → 1) of these K-operators for Q-operators correspond to the K-operators found in [19]. In Appendices, we give some material that is needed for the main discussion. In Appendix A, to make the text self-contained we give a brief review on the universal R-matrix. In Appendix B, the contracted algebras associated with U q ( sl 2 ) and corresponding L-operators are reviewed. A universal form of the intertwining relations among L-operators for Q-operators is presented. In Appendix C, contractions of the augmented q-Onsager algebra are introduced and corresponding K-operators are described. In Appendices D,E,F, miscellaneous results are collected. In Appendix G, definitions of universal T-and Q-operators in terms K-operators text are explained. Throughout this paper, we will work on the general gradation of U q ( sl 2 ). This does not produce particularly new results since the L-operators in the general gradation can easily be obtained from the ones in a particular gradation by similarity transformation and rescaling of the spectral parameter. However, we expect that this may clarify some relations rather ambiguously treated in literatures.

Quantum algebras
In this section, basic definitions that will be used in the next sections are introduced. Successively, we recall the definitions of the quantum affine algebra U q ( sl 2 ), the quantum algebra U q (sl 2 ) and two q-oscillator algebras through generators and relations. Coproduct, automorphisms and certain finite dimensional representations are also displayed. We will follow the style of presentation in [9].

2.1
The quantum affine algebra U q ( sl 2 ) The quantum affine algebra U q ( sl 2 ) is a Hopf algebra generated by the generators e i , f i , h i , d, where i ∈ {0, 1}. For i, j ∈ {0, 1}, the defining relations of the algebra U q ( sl 2 ) are given by where (a ij ) 0≤i,j≤1 is the Cartan matrix The algebra has automorphisms σ and τ defined by We use the following co-product ∆ : U q ( sl 2 ) → U q ( sl 2 ) ⊗ U q ( sl 2 ): We will also utilize an opposite co-product defined by The automorphisms (2.4) and (2.5) are related to the co-product as We always assume that the central element h 0 +h 1 is zero. Anti-pode, co-unit and grading element d are not explicitly used in this paper. The Borel subalgebra B + (resp. B − ) is generated by e i , h i (resp. f i , h i ), where i ∈ {0, 1}. For complex numbers c i ∈ C which obey the relation There exists a unique element [23,24] R in a completion of B + ⊗B − called the universal R-matrix which satisfies the following relations for ∀ a ∈ U q ( sl 2 ) , (∆ ⊗ 1) R = R 13 R 23 , (2.10) (1 ⊗ ∆) R = R 13 R 12 where R 12 = R ⊗ 1, R 23 = 1 ⊗ R, R 13 = (p ⊗ 1) R 23 . We will use the relation (2.11) which follows from the first relation in (2.10). The Yang-Baxter equation is a corollary of these relations (2.10). For R = R 21 = (p ⊗ 1)R 12 , the relations (2.10) become ∆(a) R = R ∆ ′ (a) for ∀ a ∈ U q ( sl 2 ) , One can also check that R −1 21 (resp. R −1 ) satisfies (2.10) (resp. (2.13)). The universal R-matrix can be written in the form (2.14) HereR is the reduced universal R-matrix, which is a series in e j ⊗ 1 and 1 ⊗ f j and does not contain Cartan elements. Thus the reduced universal R-matrix is unchanged under the shift automorphism τ c 1 of B + , see (2.9), while the prefactor K is shifted as The universal R-matrix is invariant under σ ⊗ σ: (σ ⊗ σ)R = R (2.16) Then we will use the following relation, which follows from this: The following elements are central in U q (sl 2 ): Note that the following map gives an automorphism of the algebra.
ν : E → F, F → E, H → −H. (2.20) We will also use an anti-automorphism defined by where (ab) t = b t a t holds for any a, b ∈ U q (sl 2 ). There is an evaluation map ev x : U q ( sl 2 ) → U q (sl 2 ): where x ∈ C is a spectral parameter and s 0 , s 1 ∈ C. We set s = s 0 +s 1 . If we apply the sim- ) for a ∈ U q ( sl 2 ) and the rescaling of the spectral parameter x → x 2 s (resp. x → x 1 s ), we will obtain the principal gradation s 0 = s 1 = 1 (resp. the homogeneous gradation s 0 = 1, s 1 = 0). Let π + µ be the Verma module over U q (sl 2 ) with the highest weight µ. In a basis {v n |n ∈ Z ≥0 }, we have (2.23) For µ ∈ Z ≥0 , the finite dimensional irreducible module π µ with the highest weight µ is given as quotient of Verma modules: In particular, π 1 (E) = E 12 , π 1 (F ) = E 21 and π 1 (H) = E 11 − E 22 gives the fundamental representation of U q (sl 2 ), where E ij is a 2 × 2 matrix unit whose (k, l)-element is δ i,k δ j,l . In this case, (2.21) coincides with transposition of matrices. Then the compositions π + µ (x) = π + µ • ev x and π µ (x) = π µ • ev x give evaluation representations of U q ( sl 2 ).

The q-oscillator algebras
We introduce two kinds of oscillator algebras Osc i (i = 1, 2). They are generated by the elements h i , e i , f i which obey the following relations: Note that Osc 1 and Osc 2 can be swapped by the transformation q → q −1 . One can prove the following corollaries of (2.25) and (2.26): (2.28) We will use anti-automorphisms of Osc i , which are analogues of (2.21), defined by

L-operators and limits
In this section, we first recall the definition of the Lax operators which follows from the universal R−matrix. In the context of quantum integrable systems, it is known that certain limits (cf. [2,3]) of L−operators provide the basic ingredient for the construction of Baxter Q-operators associated with the Yang-Baxter algebra (see [5,6,7,8] for examples of L-operators for Q-operators, and [10] for examples of the rational case). By analogy, here we consider different limits of L−operators that will be useful in the construction of 5 Contractions of a quantum algebra and its relation to a q-oscillator algebra was discussed in [31]. 6 for (2.30): on the renormalized basis v ′ n = q − µs 0 n s v n ; for (2.31): on the renormalized basis v ′ n = q µs 0 n s v n Q−operators associated with the reflection equation algebra. Technically, we follow the presentation of [9]. From now on we denote λ = q − q −1 . We set These are images of the universal R-matrix (see Appendix A) where the overall factor is defined by , and t 2 is the transposition in the second component of the tensor product. Evaluating the first space of these L-operators for the fundamental representation, we obtain R-matrices of the 6-vertex model.

L-operators for Q-operators
Applying the limits (2.30) and (2.31) to (3.1) and (3.2), we define four type of L-operators as 7 8 (3.13) As mentioned above, the L-operators (3.10) and (3.11) are essential ingredients in the construction of Baxter Q-operators associated with the Yang-Baxter algebra. For instance, for a spin chain with periodic boundary conditions, the corresponding Q-operators are defined as trace over product of these L-operators: where ξ 1 , . . . , ξ L ∈ C \ {0} are inhomogeneities on the spectral parameter in the quantum space; α ∈ C; the trace is taken over the auxiliary space (a Fock space W a for Osc a denoted as 0). Here the normalization operator Z (a) is defined by The factor q − µ⊗π 1 (1)(h 1 ) 2 came from (2.15) for c 1 = −µ. 8 We could also use automorphisms of U q ( sl 2 ) or U q (sl 2 ) to derive various L-operators: for example, This could be a substitute of (3.11) (cf. [7]). Instead of using automorphisms, we will use Chevalley like generators of the q-oscillator algebras and take various different limits of the L-operators (as already demonstrated in [9]). A merit for this is that the resultant L-operators become just reductions of the original L-operators (for example, compare (3.1) with (3.10), (3.16) and (3.18)). This is also the case with the intertwining relations and the K-operators.
In Appendix G, we will propose Q-operators Q-operators associated with different types of reflection equation algebra (cf. eqs. (4.53), (4.72)). Such operators are useful in the analysis of spin chains with open diagonal boundary conditions. For this purpose, we need additional L-operators (3.12) and (3.13) that are introduced as follows. Observe that the pair of L-operators (3.10) and (3.12) (or (3.11) and (3.13)) no longer satisfies relations corresponding to (3.5) and (3.6). For this reason, consider the following L-operators: Then the limits of (3.5)-(3.6) are given by where the following relations are used: These relations are among the conditions that are necessary to establish the commutativity of T-and Q-operators. The intertwining relations for these L-operators have unusual form (cf. [8]). For example, (3.10) satisfies

The augmented q−Onsager algebra
In this section, we first recall the definition of the augmented q−Onsager algebra [21,22] through generators and relations. Realizations of the augmented q−Onsager algebra as either right or left coideal subalgebras of U q ( sl 2 ) are then introduced, and co-actions maps are given. Correspondingly, two different intertwiners of the augmented q−Onsager algebra are constructed explicitly. They solve a reflection equation and dual reflection in U q (sl 2 ) ⊗ U q (sl 2 ). Under the specialization π 1 , known results are recovered. The augmented q−Onsager algebra -denoted below O aug q -is generated by four generators K 0 , K 1 , Z 1 ,Z 1 subject to the defining relations [22]: This algebra can be embedded into U q ( sl 2 ). Below, we will introduce two different realizations of the algebra O aug q . They are related each other via the automorphism (2.5) of U q ( sl 2 ).

The first realization
In this subsection, the augmented q-Onsager algebra is realized as a right coidal subalgebra of U q ( sl 2 ). According to the coaction map, an intertwiner K(x) is explicitly constructed.
4.1.1 Right coideal subalgebra of U q ( sl 2 ) and the intertwiner K(x) A realization of the augmented q−Onsager algebra O aug q , as a right coideal subalgebra of U q ( sl 2 ) is known [22]. Let ǫ ± be non-zero scalars. It is given by 9 : Note that the automorphism (2.4) of U q ( sl 2 ) also gives the automorphism of O aug q , that is compatible with the relations (4.1) corresponds to the restriction of the co-product (2.6) of U q ( sl 2 ) to O aug q under the realization (4.3).It is such that: On the other hand, the restriction of the opposite co-product Let us now consider the following intertwining relations associated with the first realization of the augmented q−Onsager algebra. They read: The equations for a ∈ {K 0 , K 1 } imply that [K(x), q H ] = 0. The equations for a ∈ {Z 1 ,Z 1 } give:

Solutions of the intertwining relations
According to the intertwining relations (4.7) and (4.8), solutions are defined up to a We find various different solutions with different non-trivial prefactors. Here we present two typical examples of them: 10 and These solutions (4.9)-(4.12) satisfy Note that other expressions are given in Appendix E. 10 After we obtained these solutions, we were informed by S. Belliard that he obtained a solution for the rational case Y (sl 2 ).

Reflection equations
Let us define the R-operators in U q (sl 2 ) ⊗ U q (sl 2 ) by R 12 (x, y) = (ev x ⊗ ev y )R and R 21 (x, y) = (ev x ⊗ ev y )R 21 . Then the first relations in (2.10) and (2.13) produce the following intertwining relations (4.14) The intertwining relations (4.6) and (4.14) imply the following reflection equation in U q (sl 2 ) ⊗ U q (sl 2 ): In fact, the intertwining relations )r i for any a ∈ O aug q follow 11 from (4.6) and (4.14), where the right hand side and the left hand side of (4.15) are denoted as r 1 and r 2 , respectively. Evaluating (4.15) for 1 ⊗ π 1 , we obtain the following reflection equation for the L-operators where we set K(x) = π 1 (K(x)) and used the difference property with respect to the spectral parameters. Expanding (4.16) with respect to the spectral parameter y, one recognizes the intertwining relations (4.6).
Specialization to π 1 : Evaluating (4.16) further for π 1 ⊗ 1, we obtain the following reflection equation for the R-matrices.
The solution of (4.17) is given by Here κ(x) is an overall factor. In case one uses (4.9) for |q| > 1, it reads 11 This is not a substitute of a proof of (4.15). One will be able to prove this on the level of irreducible representations of O aug q by using the Schur's lemma (which fixes r 1 = scalar × r 2 ) and an assumption on the behavior of r i with respect to the spectral parameters x, y (which determines scalar = 1). We do not have a universal K-matrix relevant to our discussion. Thus we do not have a proof of (4.15) on the level of the algebra. On the other hand, we have a proof of the reflection equation for the L-operators (4.16), which follows from this generic reflection equation (4.15).
Note that the solution (4.18) is a special case of the most general scalar solution 12 (F.1) of the reflection equation (4.17) [25,26]. In the context of quantum integrable systems, it characterizes systems with arbitrary diagonal boundary conditions.

The second realization
Next, the augmented q-Onsager algebra is realized as a left coidal subalgebra of U q ( sl 2 ). An intertwiner K(x) is explicitly constructed.

Left coideal subalgebra of U q ( sl 2 ) and the intertwiner K(x)
Using the automorphism (2.5) of U q ( sl 2 ), a second realization of the augmented q−Onsager algebra O aug q , now as a left-coideal subalgebra of U q ( sl 2 ). Let ǫ ± be non-zero scalars. It is given by 13 : under the condition σ(ǫ ± ) = ǫ ∓ . The co-action map ∆ : O aug q → U q ( sl 2 ) ⊗ O aug q , that is compatible with the relations (4.1) corresponds to the restriction of the co-product (2.6) of U q ( sl 2 ) to O aug q under the realization (4.21). It is such that: (4.23) 12 Here the word 'scalar' means the matrix elements of the solution are not operators but scalar quantities. 13 [22] under the transformations q → q −1 and ǫ ± → ǫ ∓ . In addition, instead of using the automorphism (2.5), one can also use an anti-automorphismτ defined bȳ Recall the evaluation map (2.22). Define 14 ev x = ev x | (s 0 ,s 1 )→(−s 1 ,−s 0 ) . It follows: We now consider the following intertwining relations associated with the second realization of the augmented q−Onsager algebra: where g = q H(s 0 −s 1 )/s . Here t is the transposition. One may drop it from (4.24) since the K-operator here is a diagonal operator. The equations for a ∈ {K 0 ,

Solutions of the intertwining relations
The solutions of the intertwining relations (4.24) follow from the ones for the first realization (4.6) under the identification
Specialization to π 1 : Specializing to the two-dimensional representation of U q (sl 2 ), the solution of the intertwining relations is unique (up to an overall factor). It reads: Here κ(x) is an overall factor. In case one uses (4.9) for |q| > 1 and (4.27), it reads By construction, it solves the specialization of the dual reflection equation 15 (4.29): The solution (4.30) is a special case of the general scalar solution (F.2) of the dual reflection equation. Note that the solutions of the reflection and dual reflection equations are related by the following transformation (see [16]):

Limit of intertwining relations and their solutions: Koperators for Q-operators
In this subsection, we consider the limit q ∓µ → 0 of the intertwining relations and their solutions in the Verma module π + µ . In order to avoid divergences, we have to renormalize the generators of the augmented q-Onsager algebra and the K-operators. The resulting K-operators serve as building blocks of Q-operators in that they solve reflection equations for the L-operators for Q-operators. Similar K-operators for the rational case can be found in [19].

Rational limit q → 1
One can take the rational limit q → 1 of the formulas in this paper easily. The q-gamma function (see for example, [27]) is defined by This reduces to the normal gamma function in the rational limit. Then these generators satisfy [a, a † ] = 1, aa † = n + 1 2 , a † a = n − 1 2 . Let q 2p = −ǫ − /ǫ + . Then one can take the rational limit of renormalized versions of (4.9)-(4.12), (3.10), (3.12), (4.18) and (4.37) as for (4.9), (4.77) for (4.10), (4.78) for (4.12), (4.80) It is important to note that the above limits keep the reflection equation (4.53) unchanged. This is because of the relation (2.11) for ξ = (1 − q 2 ) h 1 2 and where ρ (1) x is defined in (B.12). The rational limit of all the other K-and L-operators can be taken in the same way. In this way, we have recovered K-operators for Q-operators for rational (XXX-) models which are similar to the ones in [19].

Concluding remarks
In the context of quantum groups and related coideal subalgebras, finding a universal product formula for the K-matrix by analogy with the known product formula (A.1) for the universal R-matrix proposed in [29] is an interesting problem. In this direction, a universal formulation of the reflection equation algebra and related intertwining relations associated with a given coideal subalgebra are highly desirable (see recent progress in [28]). In the present paper, we have focused on homomorphic images of two different coideal subalgebras of U q ( sl 2 ) (onto U q (sl 2 )), that are related with the augmented q-Onsager algebra first introduced by Ito and Terwilliger in [21] (see also [22]). Based on the intertwining relations, product formulae for the K-matrix solutions in terms of the generators of U q (sl 2 ) are derived. They solve certain reflection and dual reflection equations associated with L-operators. In the second part of the paper, certain limits of these K-operators are studied. For these limits, contracted versions of the augmented q-Onsager algebra are considered and q-oscillator representations are constructed. Importantly, these K-operators are the basic ingredient for the construction of Q-operators that are relevant in the analysis of quantum integrable models with non-periodic diagonal boundary conditions. An interesting problem would be to extend the analysis presented here to the case of the q-Onsager algebra [35,36], which is isomorphic to the fixed point subalgebra of U q ( sl 2 ) under the action of the Chevalley involution [37]. A product formula in this case is an open problem, that should find applications to the analysis of integrable models with non-diagonal boundary conditions.
It is known that L-operators for Verma modules of the quantum affine algebra (or the Yangian) factorize with respect to L-operators for the Q-operators. Examples for such factorization formulas appeared in a number of papers (see for example, [12,10] and references therein). In [9], such factorization formulas were reconsidered in relation to properties of the universal R-matrix, and a universal factorization formula, which is independent of the quantum space, was proposed. One of our motivations was to generalize the universal factorization formula [9] to the case of open boundary conditions in the light of the augmented q-Onsager algebra [22]. The main obstacle for this is that we do not have the defining relations of the universal K-matrix corresponding to the relations (2.10) for the universal R-matrix of U q ( sl 2 ). This is a reason why we focused our discussions only on one of the most essential objects, the K-matrices, without application to Q-operators and their properties. Still, in appendix G we suggest universal T-and Q-operators. For a class of representations, the commutativity is proven. It is desirable to reconsider the problem after formulation of the universal K-matrix for the universal R-matrix of U q ( sl 2 ) in the future.

A The universal R-matrix
In this section, we briefly review the product expression of the universal R-matrix given by Khoroshkin and Tolstoy in [29]. Their universal R-matrix was already reviewed by several authors [30,7]. Here we basically follow these. Let be a positive root system of sl 2 in the notation of [29]. We choose the root ordering as α + (k − 1)δ ≺ α + kδ ≺ lδ ≺ (l + 1)δ ≺ δ − α + mδ ≺ δ − α + (m − 1)δ for any k, l, m ∈ Z ≥1 . In this case, the universal R-matrix has the following expression: where we use notations Let e α = e 1 , e δ−α = e 0 , f α = f 1 , f δ−α = f 0 . Then the other root vectors are defined by the following recursion relations: where the root vectors with prime are given by the following generating functions.
In general, root vectors contain many commutators. However, simplification occurs under the evaluation map.
where the central elements C k are defined by Inserting these 18 into (A.1), we obtain R(x, y). In order to obtain R 21 (x, y), one has to swap the first and the second components of the tensor product in (A.1) beforehand. Based on these product formulas, one can check

B Contraction of the quantum affine algebra
A systematic study of the asymptotic representation theory of the Borel subalgebras of quantum affine algebras was given in [13]. How to evaluate the universal R-matrix for the purpose of Q-operators was explained in detail in [7]. We nevertheless review the subject in the spirit of [8], which is inspired by earlier discussions [2,3,32,33,6]. We are interested in considering limits of representations of the whole quantum affine algebra rather than those of its Borel subalgebras. In particular, we will present the universal form of the intertwining relations for the L-operators for Q-operators (B.23)-(B.26). 18 For the second component of the tensor product, one has to replace x with y beforehand.
The Borel subalgebras of U q ( sl(2; I)) and U q ( sl 2 ) share the same defining relations. U q ( sl(2; I)) is a subalgebra of U q ( sl(2; I)). Taking note of this fact, we purposely use the same symbols for the generators of these algebras.
where the relation The intertwining relations for these operators are given 28 by Evaluating this for π + µ (xq − µ s ) ⊗ 1, we obtain where (2.15) for c 1 = −µ is used. Then the limit q −µ → 0 produces where (B.14) for B + is applied to the first component of the tensor product in R ∈ B + ⊗ B -is generated by four generators K 0 , K 1 , Z 1 ,Z 1 subject to the defining relations: Note that the last two relations in (C.1) automatically hold 29 true under (C.3). This algebra can be embedded into U q ( sl(2; I)). Below, we will introduce four different realizations of the algebra O(I) aug q .

C.1 The first realization
In this subsection, the contracted augmented q-Onsager algebra is embedded into U q ( sl(2; I)). We consider two types of realization of the contracted augmented q−Onsager algebra O aug q (I), as a subalgebra of U q ( sl(2; I)). The realizations of O aug q ({0}) in terms of U q ( sl(2; {0})) are given by and the realizations of O aug q ({1}) in terms of U q ( sl(2; {1})) are given by Here we attach symbols (1, +), (1, −), (2, +), (2, −) on the generators to distinguish different realizations of the algebras. We remark that O aug q (I) is realized by (C.5)-(C.8) even if {e 0 , e 1 , f 0 , f 1 , h 0 , h 1 } are generators of U q ( sl(2)), while O aug q (I) is realized only by the generators of U q ( sl(2; I)).
Limits of the renormalized generators of the augmented q-Onsager algebra in Verma modules are related to the images of the contracted augmented q-Onsager algebra under the maps (B.12)-(B.13) as follows: , a = 0, 1. (C.14) Based on these relations and the commutation relations (4.1) , one can show 30 that the contracted commutation relations (C.1) hold under the maps (B.12)-(B.13). The limit of the intertwining relations associated with the first realization of the augmented q-Onsager algebra given in the main text can now be compactly summarized, in terms of the contracted augmented q-Onsager algebra, as

C.2 The second realization
In this subsection, the contracted augmented q-Onsager algebra is also embedded into U q ( sl(2; I)). We consider two types of realization of the contracted augmented q−Onsager algebra O aug q (I) with I = {0, 1} \ I, as subalgebra of U q ( sl(2; I)). The realizations of O aug q ({1}) in terms of U q ( sl(2; {0})) are given by where τ (ǫ ± ) = ǫ ∓ is assumed. The realizations of O aug q ({0}) in terms of U q ( sl(2; {1})) are given by 30 For example, multiplying the equation in the 4-th line in (4.1) by q −2µ− 4s 0 µ s , one obtains Then take the limit q −µ → 0 in the representation π + µ (xq  x . The other relations in (C.1) can be checked in the same way.

G Universal T-and Q-operators
In this appendix, we propose several version of operators in terms of L-and K-operators in the main text, which are candidate of universal T-and Q-operators for integrable systems with open boundary conditions associated with U q ( sl 2 ), and mention merit and demerit of them. We only sketch our idea on the definition of them and do not discuss convergence of the trace, explicit rational limit, functional relations among T-and Qoperators, Bethe equations, etc, which we prefer to consider in a separate publication (if there is an opportunity). We define universal L-operators as and define universal dressed K-operators as 32 Note that (G.2) is an element of U q (sl 2 ) ⊗ B − and (G.3) is an element of U q (sl 2 ) ⊗ U q ( sl 2 ) . One can show 33 that (G.2) satisfies the following dressed reflection equation under (4.15). In contrast, we have no proof (or disproof) that (G.3) satisfies even if we assume (4.15) since we do not have an analogue of (3.5) for the universal Loperators. Evaluating 34 (G.4) and (G.5) for 1 ⊗ π 1 ⊗ 1, we obtain the following dressed 32 As remarked in subsection 2.1, R −1 = R −1 21 also satisfies the relation (2.10). Thus one may also define a universal dressed K-operator as K(x) = L −1 (x −1 )(K(x) ⊗ 1)L(x), where L(x) = (ev x ⊗ 1)R. In this case, K(x) is an element of U q (sl 2 ) ⊗ B + . 33 One has to use R 12 (x, y)L 13 (x)L 23 (y) = L 23 (y)L 13 (x)R 12 (x, y) and R 21 (x, y)L 23 (y)L 13 (x) = L 13 (x)L 23 (y)R 21 (x, y), which follow from (2.12), and (4.15). 34 Here we abuse notation and use the same expression for both the image of K 23 (y) for 1 ⊗ π 1 ⊗ 1 and the original one. reflection equations for the L-operators L 12 x −1 y K 13 (x)L 12 (xy) K 23 (y) = K 23 (y)L 12 x −1 y −1 K 13 (x)L 12 xy −1 . (G.6) L 12 x −1 y K 13 (x)L 12 (xy)K 23 (y) =K 23 (y)L 12 x −1 y −1 K 13 (x)L 12 xy −1 . (G.7) One can also prove (G.6) independent of (G.4) since (G.6) is a dressed version of (4.16).
T π 1 (x)T π 1 (y) =T π 1 (y)T π 1 (x),Q (a) (x)T π 1 (y) =T π 1 (y)Q (a) (x), a = 1, 2, x, y ∈ C. Any object we consider under the trace is a linear combination of elements of the form J m,n = E m F n q ξH ∈ U q (sl 2 ) (or J m,n = e m a f n a q ξha ∈ Osc a , a = 1, 2) for m, n ∈ Z ≥0 , ξ ∈ C. In particular, only the terms for m = n (J := J m,m ) contribute to the trace. Then TrJ t = TrJ holds if the trace converges and the cyclicity of the trace holds. In the following, we assume 40 this. One can check the following relations for the L-operators 41 : The first relation in (G.29) can be proven similarly (easier than the second relation). 40 This is not a trivial issue in particular for infinite dimensional representations. 41 The anti-automorphism t defined by (2.21) or (2.29) might not be enough for more general Roperators. A relation corresponding to (G.32) (similar to (A.22)) may not hold true in general situation. However, it is enough for the L-operators discussed here. = tr 2 K ′ 2 (y −1 )K 23 (y) tr 1 Ǩ (a)′ 1 (x −1 ) t 1 K (a) 13 (x) t 1 = T π 1 (y)Q (a) (x). (G. 33)