Universal Baxter TQ-relations for open boundary quantum integrable systems

Based on properties of the universal R-matrix, we derive universal Baxter TQ-relations for quantum integrable systems with (diagonal) open boundaries associated with $U_{q}(\widehat{sl_{2}})$. The Baxter TQ-relations for the open XXZ-spin chain are images of these universal Baxter TQ-relations.


Introduction
Baxter Q-operators [1] are fundamental objects in quantum integrable systems. They give information about the eigenfunctions and Bethe roots. In particular, Bazhanov, Lukyanov and Zamolodchikov [2] defined Baxter Q-operators as traces of the universal R-matrix over q-oscillator representations of one of the Borel subalgebras of the quantum affine algebra U q ( sl 2 ). The universal R-matrix is an element of the tensor product of two Borel subalgebras. Hence, the Baxter Q-operators are universal in the sense that they are elements of a Borel subalgebra and thus do not depend on the concrete quantum space of states on which the operators act. Baxter Q-operators for concrete physical models can be obtained by specifying representations of the Borel subalgebra. Much work has been done related to this 'q-oscillator construction' of Baxter Q-operators (see, for example, the following papers and references therein: [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] for the trigonometric case; [18,19,20,21] for the rational case; [22,23,24,25,26] for some other methods). Moreover, systematic studies related to this from the point of view of the asymptotic representation theory of quantum affine algebras were done in [27,28,29].
All these works are for quantum integrable systems with periodic boundary condition. In contrast with the case for models with periodic boundary condition, there are only a few works [30,31,32] on the q-oscillator construction 1 of Baxter Q-operators for models with open boundary conditions. The first breakthrough on this topic was brought by Frassek and Szecsenyi [30] who proposed Baxter Q-operators for the (diagonal) open XXX-spin chain (q = 1 case). In [31], we proposed universal Baxter Q-operators for (diagonal) open boundary quantum integrable systems associated with U q ( sl 2 ) and gave Baxter Q-operators for open XXZ-spin chains as holomorphic images (for a tensor product of the fundamental representation) of them. One of the fundamental equations for Baxter Q-operators are the so-called Baxter TQ-relations [1]. Vlaar and Weston [32] proved an operator Baxter TQ-relation 2 for the (diagonal) open XXZ-spin chain based on a representation theoretical method. Thus a Baxter Q-operator for the open XXZ-spin chain proposed in [31] indeed satisfies the TQ-relation. The purpose of this paper is to supplement [31] and give universal analogues of Baxter TQ-relations for (diagonal) open boundary quantum integrable systems. Our universal TQ-relations are equations in U q ( sl 2 ). The Baxter TQ-relation in [32] follows from our universal Baxter TQ-relation as a holomorphic image.
The layout of this paper is as follows. In Section 2, we summarize the definitions of quantum algebras. In Section 3, we review L-operators, which are building blocks of T-and Q-operators. In particular, the L-operators are defined as holomorphic images of the universal R-matrix. In Section 4, we recall solutions (K-operators) [31] of the reflection equation, which are asymptotic limits of solutions of the intertwining relations for the augmented q-Onsager algebra [37,38]. The solutions are expressed in terms of Cartan elements of q-oscillator algebras. In Section 5, we present the universal Baxter TQ-relations, which are our main results. In Appendix A, we review Khoroshkin and Tolstoy's explicit formula [41,42] of the universal R-matrix. In Appendix B, we explain the derivation of the universal TQ-relations. In Appendix C, we reconsider the dressed reflection equation in relation to the universal R-matrix. In Appendix D, we discuss unitarity relations of R-operators in general situation. Throughout the paper we assume that the deformation parameter q is not a root of unity and use the following notation.

Notation:
• For any elements X, Y of the quantum algebras, the q-commutator is defined by [X, Y ] q = XY − qY X. In particular, we set [X, Y ] 1 = [X, Y ].
• A q-analogue of the exponential function is defined by . This has infinite product expressions.
• We introduce free parameters s 0 , s 1 ∈ Z. In particular, we set s = s 0 + s 1 .

Quantum algebras
In this section, we review quantum algebras. There is overlap among this section and the corresponding sections in [31,43]. We also refer to [45,44,46] for review on this subject.

The quantum affine algebra
The quantum affine algebra U q ( sl 2 ) (at level 0, i.e. the quantum loop algebra) is a Hopf algebra generated by the elements e i , f i , q ξh i for i ∈ {0, 1} and ξ ∈ C obeying the following relations: where (a ij ) 0≤i,j≤1 is the Cartan matrix The algebra has an automorphism σ defined by The algebra also has an anti-automorphism t defined by 3 Note that this means σ(ab) = σ(a)σ(b) and (ab) t = b t a t for a, b ∈ U q ( sl 2 ). We use the following co-multiplication ∆ : We will also use the opposite co-multiplication defined by Co-unit, anti-pode and grading element d are not used in the present paper. The Borel subalgebras B + and B − are generated by the elements e i , q ξh i and f i , q ξh i , respectively, where i ∈ {0, 1}, ξ ∈ C. There exists a unique element [47,48] R in a completion of B + ⊗B − called the universal R-matrix which satisfies the following relations for ∀ a ∈ U q ( sl 2 ), , whereR is called the reduced universal Rmatrix, which is a series on {e 0 ⊗ 1, e 1 ⊗ 1, 1 ⊗ f 0 , 1 ⊗ f 1 }. We symbolically write this as R =R({e 0 ⊗ 1, e 1 ⊗ 1, 1 ⊗ f 0 , 1 ⊗ f 1 }). The Yang-Baxter equation follows from the relations (2.8). We will also use which is an element of a completion of B − ⊗ B + . Taking note on the first relation in (2.8), one can show that the universal R-matrices commute with the co-multiplication of any Cartan elements The following relations follow from the uniqueness of the universal R-matrix [48] and the fact that (2.4) is an automorphism of each Borel subalgebra.
(2.12) 4 We will use similar notation for the L-operators to indicate the space on which they non-trivially act.

The quantum algebra
The quantum algebra U q (sl 2 ) is generated by the elements E, F, q ξH for ξ ∈ C obeying the following relations: The upper (resp. lower) Borel subalgebra is generated by the elements E, q ξH (resp. F, q ξH ). The Casimir element is central in U q (sl 2 ). We have an automorphism and an anti-automorphism of the algebra. These are U q (sl 2 ) analogues of (2.4) and (2.5), respectively. There is an algebra homomorphism called evaluation map 5 ev x : where x ∈ C × is the spectral parameter. We introduce an operation to permute the parameters s 0 and s 1 : One can verify consistency of these: where • is composition of maps and ζ · ev x is the map (2.17) with the replacement of the parameters (s 0 , s 1 ) → (s 1 , s 0 ). In case the objects on which (2.19) is acting do not depend on the parameters s 0 and s 1 , (2.19) reduces to σ • ev x = ζ • ev x • σ. The fundamental representation π of U q (sl 2 ) is given by π(E) = E 12 , π(F ) = E 21 and π(q ξH ) = q ξ E 11 + q −ξ E 22 , where E ij is the 2 × 2 matrix unit whose (k, l)-element is δ i,k δ j,l . The composition π x = π • ev x gives the (fundamental) evaluation representation of U q ( sl 2 ). 5 We follow [10] and consider the general gradation of the algebra.
For the fundamental representation, we define an algebra automorphism σ and an algebra anti-automorphism t of the algebra of 2 × 2 matrices over C by In this case, the anti-homomorphism t coincides with transposition of 2 × 2 matrices. We have an identity of algebra homomorphisms and an identity of algebra anti-homomorphisms π(a t ) = (π(a)) t for a ∈ U q (sl 2 ), (2.24) which justify our use of the same symbol for different maps.

q-oscillator algebras
We introduce two kinds of oscillator algebras 6 Osc i (i = 1, 2). They are generated by the elements e i , f i , q ξh i (for ξ ∈ C) obeying the following relations: (2.28) 6 Osc 1 is same as the one defined by eq.(2.25) in [31]; while Osc 2 is slightly different from the one defined by eq.(2.26) in [31]. Let h ′ 2 , e ′ 2 , f ′ 2 be h 2 , e 2 , f 2 in eq.(2.26) in [31]. They are related to h 2 , e 2 , f 2 in (2.28) in this paper as h ′ 2 = h 2 + 2, e ′ 2 = q β e 2 , f ′ 2 = q −β−2 f 2 , β ∈ C. Moreover, one can define Osc 1 as a contraction of U q (sl 2 ) (cf. [49]). We define a homomorphism ρ µ : U q (sl 2 ) → Osc 1 (see for example [44]) by the relations (2.25) This realizes the Verma module of U q (sl 2 ) with the highest weight µ on the Fock space. The generators of the q-oscillator algebra Osc 1 can be given by contraction of (2.25) (see [49], and eq. (2.30) in [31]. (The similarity transformation by the factor q − µs 0 h 1 2s is related to the change of basis mentioned in footnote 7 in [31].)) where the limit is taken with respect to µ (q is constant).
Note that Osc 2 can be realized in terms of Osc 1 : The following relations follow from (2.27) and (2.28): (2.31) We will use anti-involutions of Osc i (analogues of (2.16)) defined by where (ab) t = b t a t holds for any a, b ∈ Osc i , i = 1, 2. We define the homomorphism ρ (i) x : B + → Osc i , i = 1, 2 by the relations These maps are related each other as (cf. (2.29)) x is the map (2.33) or (2.34) with the replacement of the parameters (s 0 , s 1 ) → (s 1 , s 0 ). In case the objects on which (2.35) is acting do not depend on the parameters s 0 and s 1 , (2.35) reduces to ρ

L-operators
In this section, we review various L-operators, which are building blocks of T-and Qoperators. They are holomorphic images of the universal R-matrix in various representations of Borel subalgebras of U q ( sl 2 ). We will make use of the product expression of the universal R-matrix given by Khoroshkin and Tolstoy [41,42], which is reviewed in Appendix A. Their universal R-matrix was already reviewed by several authors (see for example, [50,51,10,12,15,31,26]). In particular, a pedagogical account on how to evaluate it in the context of Baxter Q-operators can be found in [10,12].
Evaluating the second components of these universal L-operators in the fundamental evaluation representation π 1 = π x | x=1 , we obtain the R-matrices of the 6-vertex model.
where the overall factor is defined by

L-operators for Q-operators
We define the universal L-operators for Q-operators by One can calculate these based on the explicit expression of the universal R-matrix in Appendix A. In particular, L (1) (x) and L (2) (x) have simple expressions: where the root vectors e kδ and f kδ , which can be expressed in terms of the basic generators e 1 , e 0 , f 1 , f 0 , come from (A16). Now we evaluate the second component of the universal L-operators in the fundamental evaluation representation π 1 = π x | x=1 . We normalize the L-operators as These are L-operators for Q-operators for XXZ-spin chain. Explicitly, one obtains 8 In addition to the above L-operators, we need L-operators proportional to the inverse of them. 9Ľ The L-operators with the superscript ' (2) ' can be obtained from the ones with ' (1) '.
(3.25) 9 We could interpret these as follows (cf. [31]). Consider universal L-operators of the form: The L-operators (3.18)-(3.20) are images of these: One can check that these L-operators satisfy , g 2 = 1 ⊗ g, and t 2 is the transposition in the second component of the tensor product. Note that the matrix g is invariant under the map ζ • σ: One can also check the following relations for the L-operators:

Reflection equation and K-operators
A systematic approach for construction of quantum integrable systems with open boundaries was developed by Sklyanin [39]. The key equation for this is the reflection equation (boundary Yang-Baxter equation) [52]. We start from the following form of the reflection equation and the dual reflection equation for the R-matrices (3.4) and (3.5): The most general non-diagonal 2 × 2 matrix solutions of the reflection equations are known in [53,54,55]. The diagonal solutions of (4.1) and (4.2) are specialization of them: where ǫ ± and ǫ ± are scalar parameters 10 . We assume ǫ + ǫ − ǫ + ǫ − = 0 since we will deal with solutions which contain ǫ −1 + , ǫ −1 − , ǫ −1 + or ǫ −1 − . We remark that these solutions (4.3) and (4.4) are related each other by the following transformation [55]: (4.5) In addition to (2.18), we assume The K-matrices (4.3) and (4.4) are invariant under the operation ζ • σ: Next, we consider the reflection equations and the dual reflection equations for the Loperators for Q-operators (3.12), (3.14), (3.18) and (3.19): x y , a = 1, 2. (4.9) We have solutions (K-operators) [31] 11 of these equations for a = 1.
where the normalization functions are defined by We remark that the same type of normalization is used in [32]. The solutions (4.10) and (4.11) are related each other by the following transformation: (4.14) 10 Up to an overall factor, K(xq Solutions for a = 2 follow from the first ones (4.10) and (4.11): One can check this by applying ζ • (1 ⊗ σ) to (4.8) and (4.9) for a = 1 (with the help of (3.22)-(3.25), (3.30) and (4.7)).

Universal Baxter TQ-relation
In this section, we apply a universal version of Sklyanin's dressing method [39] to the K-operators in the previous section and obtain more general solutions of the reflection equation. Then we define universal T-and Q-operators for open boundaries integrable systems [31], which are elements in U q (sl 2 ). We will present the universal TQ-relations among them. We define the universal dressed K-operator for a T-operator by One can show that (5.1) satisfies the following universal dressed reflection equation for a T-operator (see Appendix C).
We define the universal T-operator by where the trace is taken over the space End(C 2 ). This is invariant under the operation ζ • σ: One can show this by using the relations: L(x) (these follow from (2.12), (2.19), (2.23)), (4.7) and tr σ(A) = trA for any 2 × 2 matrix A. We define the universal dressed K-operators for Q-operators by One can prove that (5.5) satisfy the following universal dressed reflection equations for Q-operators (see Appendix C). We define the universal Q-operators [31] by where g (a) = q (s 0 −s 1 )ha s and W a are Fock spaces generated by Osc a . Note that the second Q-operator follows from the first one One can show this by using the relations (3.9), (3.10) and (4.15). We find that the universal T-and Q-operators satisfy the following universal TQ-relations (see Appendix B for derivation).

Concluding remarks
In this paper, we gave universal Baxter TQ-relations for diagonal open boundary integrable systems associated with U q ( sl 2 ). This supplements and expands the discussions in [31]. By fixing the representation on the quantum space, we recovered the Baxter TQ-relation for the open XXZ spin chain [32]. One of the unsolved problems related to this paper is generalization to quantum integrable systems with non-diagonal open boundaries. The key objects for construction of the Baxter Q-operators for open boundary integrable systems are K-operators. The Koperators for Q-operators can be obtained as asymptotic limits (or contraction) of generic K-operators which are expressed in terms of generators of symmetry algebras. In the case of the Yangian Y (sl 2 ), generic K-operators for general non-diagonal boundaries were constructed in [56], and in the case of U q ( sl 2 ), generic K-operators for general triangular boundaries were constructed in [43].
The generalization to the higher rank case is also not fully understood yet. In [57], diagonal K-operators for U q ( gl n ) were expressed in terms of Cartan elements of a quotient of U q (gl n ). Non diagonal K-matrices for the symmetric tensor representations of U q ( sl n ) were constructed in [58] (see also [59] for some aspect on n = 2 case). By taking limits of these, one will be able to obtain a subset of the K-operators for Q-operators for the higher rank case.
14 Set s 0 = s 1 = 1, s = 2. In this case, R(x) = R(x −1 ) and L (1) Another way to construct Baxter TQ-relations for open boundaries would be to use a generating function of the T-operators for the anti-symmetric representations. In the case of the periodic boundary condition, it is a column-ordered determinant over a function of the monodromy matrix for a transfer matrix [60]. For models with open boundaries, one may have to use a dressed K-matrix (in our case, the universal dressed K-operator (5.1)) instead of the usual monodromy matrix.
It is known that T-operators can be expressed as concise Wronskian-like determinants (Casoratian) in terms of Q-operators (in addition to the references for Baxter Q-operators referred in Introduction, see also [61,62,63,64,65] and references therein). In contrast with integrable systems with periodic boundary, not much is known about this for integrable systems with open boundaries (cf. [66]).

Appendix 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 [41,42]. Their universal R-matrix was already reviewed by several authors (see for example, [50,51,10,12,15,31,26]). Here we basically follow these in the convention in Appendix A in [31].
be a positive root system of sl 2 in the notation of [41]. We fix 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 explicit expression: where each element is defined by 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: and the following generating functions: In general, root vectors contain many (q-deformed) commutators. However, simplification occurs under the evaluation map. For k ∈ Z ≥0 , we have and for k ∈ Z ≥1 , where the central elements C k are defined by In addition to the root vectors defined above, we will also use another set of root vectors. Let e α = e α , e δ−α = e δ−α , f α = f α , f δ−α = f δ−α . Then the other root vectors are defined by the following recursion relations: and the following generating functions: One can prove the following relations by induction.

Appendix B: Derivation of the universal Baxter TQrelations
In this section, we derive the universal Baxter TQ-relations (5.9) under the assumption 15 that convergence and cyclicity of the traces in the T-and Q-operators hold. Baxter TQrelations with fixed quantum spaces were already derived for the open XXX-spin chains [30] and for the open XXZ-spin chains [32]. Here we consider the problem on the level of the universal T-and Q-operators.
We introduce two kinds of elements G, G ∈ Osc 1 ⊗ End(C 2 ) and their inverse: One can check the following relations by direct calculations.
where x ∈ C. Let us apply (B5)-(B12) to the second equation in (2.8). Taking note on the fact that the co-multiplication of the universal R-matrix has the form 16 1 2 , and the relations E ij E kl = δ ik δ jl , we obtain where F 13 (x) and F 13 (x) are elements in Osc 1 ⊗ End(C 2 ) ⊗ B − and Osc 1 ⊗ End(C 2 ) ⊗ B + , respectively. 17 One can also show the following relations by direct calculations. 16 See also similar discussions in section 4 in [13]. 17 Although explicit expressions of them are not necessary for the proof of the universal Baxter TQrelation, one can calculate them based on the explicit expression of the universal R-matrix. For example, we obtain Evaluating (B13) and (B14) in the fundamental evaluation representation in the third component of the tensor product, we obtain (1) where the coefficients are defined by (5.10) and We remark that rational analogues of (B18) and (B19) were previously considered in [30]. Moreover, the diagonal parts of (B18) and (B19), which are essential in the proof of Baxter TQ-relations, appeared in [32]. Define permutation operators by Applying p 13 • p 12 to (2.9), we obtain Then we evaluate (B22) under ρ (1) x ⊗ π y ⊗ 1 (x, y ∈ C × ), to get L 23 (y)L 13 (xq Now we can show the relation (5.9) for a = 1 step by step as follows 18 . (1)

12Ǩ
(1) 12 (x 2 q 1 s )K 2 (x)G 12 apply (B18) 18 Here tr 1 = tr W1 ⊗ 1 ⊗ 1, tr 2 = 1 ⊗ tr ⊗ 1, and the third component of the tensor product is in U q ( sl 2 ). The parts which contribute to the trace tr 1 are linear combinations of the terms of the from e n 1 f n 1 q ξh1 , n ∈ Z ≥0 , ξ ∈ C. They are invariant under the anti-involution (2.32): (e n 1 f n 1 q ξh1 ) t = e n 1 f n 1 q ξh1 . We will also use the invariance g t = g,Ǩ 13 (x −1 q where (. . . ) are the parts which do not contribute to the trace. Applying the map ζ • σ to (5.9) for a = 1, one can show (5.9) for a = 2.

Appendix D: Generic unitarity relations of Roperators
It is known that the R-matrices ((3.4) and (3.5)) of the 6-vertex model satisfy the unitarity relation where I is the 2 × 2 unit matrix. Here we reconsider this type of relations in general situation. We define the generic R-operators by R(xy −1 ) = R(x, y) = (ev x ⊗ ev y )R, R(xy −1 ) = R(x, y) = (ev x ⊗ ev y )R, where x, y ∈ C × . For any finite dimensional irreducible representations χ 1 , χ 2 of U q (sl 2 ), we set Then the following relation holds [67] R χ 1 ,χ 2 (x)R χ 1 ,χ 2 (x) = R χ 1 ,χ 2 (x)R χ 1 ,χ 2 (x) = S(x)(I 1 ⊗ I 2 ), where S(x) is a scalar function 20 on x, and I 1 and I 2 are unit matrices. This implies the following generic unitarity relation: where C(x) is central on U q (sl 2 ) ⊗ U q (sl 2 ). Derivation of (D5) from (D4) is given as follows [68]. Let g be a finite dimensional Lie algebra. The following proposition is well known.
Then one can show the following.
Corollary If u ∈ U q (g) is scalar on all finite dimensional irreducible U q (g)-modules, then u belongs to the center of U q (g).
Proof. Take any element a ∈ U q (g) and apply Proposition to [u, a]. Then one finds [u, a] = 0. This means that u is central since a is arbitrarily. Let us regard (D4) as a matrix with respect to the second component of the tensor product. The matrix elements of this matrix are scalar (or 0) for any finite dimensional irreducible representation χ 1 of U q (sl 2 ). Then Corollary suggests (D5).