A Q-operator for open spin chains I: Baxter's TQ relation

We construct a Q-operator for the open XXZ Heisenberg quantum spin chain with diagonal boundary conditions and give a rigorous derivation of Baxter's TQ relation. Key roles in the theory are played by a particular infinite-dimensional solution of the reflection equation and by short exact sequences of intertwiners of the standard Borel subalgebras of $U_q(\widehat{\mathfrak{sl}_2})$. The resulting Bethe equations are the same as those arising from Sklyanin's algebraic Bethe ansatz.

Sklyanin developed the algebraic Bethe ansatz for the example of the open XXZ model with diagonal boundary conditions in the founding paper [14] (the formulation is presented in the notation of the current paper in Appendix C). Again, there has been a lot of effort since then to produce Bethe equations for large classes of other open integrable quantum spin chains. There are two main approaches: the first produces eigenvalue TQ relations by fusing transfer matrices associated with finite-dimensional auxiliary spaces and then conjecturing that a suitable asymptotic limit in the auxiliary space dimension exists [20]; the second, and more widely used, is the analytic Bethe ansatz method [21].
There is a some existing work on the explicit construction of open Q-operators. The paper most related to the current work is that of Frassek and Szécsényi [22] who consider the XXX case with open boundary conditions. Their work is similar in spirit but relies more on linear algebra relations rather than module homomorphisms (intertwiners). Another related paper is that of Baseilhac and Tsuboi [23] who consider the same XXZ model with the same boundary conditions, but take the asymptotic approach: namely they construct monodromy matrices and K-matrices associated with an infinite-dimensional auxiliary space by taking the asymptotic limit of the Verma module. This work stops before constructing an explicit open Q-operator.
In this paper we give an algebraic construction of the Q-operator for the open XXZ model with diagonal boundary conditions. The idea is straightforward: we construct the Q-operator as a trace of the double-row monodromy matrix over infinite-dimensional representations of the Borel subalgebra as for the closed spin chain. The execution is complicated by two issues: the first is that we must construct infinite-dimensional K-matrix solutions of the reflection equation. In fact this is relatively simple in the diagonal boundary condition case and the solution is given by Proposition 2.5. The second issue is that there are now five algebras in play: the quantum group U q ( sl 2 ); the two Borel subalgebras U q (b ± ) (each associated with one of the rows of the double-row transfer matrix); and a coideal subalgebra for each boundary. The convergent open Q-operator (4.14) constructed in this paper does not need the regularizing twist that is required in the closed case. The TQ relation is proven in Theorem (5.3) using the short exact sequences (3.3) and (3.5) for the two different Borel subalgebras along with the compatibility of the associated homomorphisms with the boundary fusion relations given by Lemma 3.2. The finite-dimensional counterpart of these relations is given in [24,Eq. (4.7)] which in itself goes back to the original notion of K-matrix fusion [16,25]. The boundary fusion relations are the only key relations in this paper for which we do not have a full representation-theoretic understanding, but nevertheless they hold as linear algebra relations. In addition we prove the important commutativity statement [T (y), Q(z)] = 0 in Theorem 5.1.
In Section 6 we show that the Q-operator satisfies a simple formula given in Theorem 6.6, known as crossing symmetry, and use it, combined with the TQ relation, to recover the same Bethe equations as in the algebraic Bethe ansatz approach of Sklyanin [14] as reproduced in our Appendix C. The crossing symmetry property and hence the derivation of the Bethe equations rely on Conjecture 6.3, which states that family {Q(y)} is commutative and the matrix entries of the Q-operator are polynomial in z; we cannot yet prove this conjecture in generality, but we do prove polynomiality for all diagonal entries in Appendix A and verify the full conjecture for two lattice sites in Appendix B.
2. Level-0 representation theory of U q ( sl 2 ) 2.1. Quantum affine sl 2 . We denote by g the (derived) affine Lie algebra sl 2 and by σ the associated nontrivial diagram automorphism, i.e. the permutation of the set {0, 1}. For p ∈ C × := C\{0} and elements x, y of any algebra 1 we denote [x, y] p = xy − pyx. In this section, and in particular in the following definition, we allow q ∈ C\{−1, 0, 1} (later on we will restrict q). The (derived) affine quantum group U q (g) is the algebra [26,27,28] with generators e i , f i , k ±1 i (i ∈ {0, 1}) and relations for all i ∈ {0, 1}. The following assignments define a coproduct on U q (g): i which restricts to a coproduct on the following important subalgebras: Equivalently, we can define U q (b + ) and U q (b − ) as the algebras generated by e 0 , e 1 , k ±1 0 , k ±1 1 and f 0 , f 1 , k ±1 0 , k ±1 1 respectively, subject to only those relations among (2.1-2.3) which only contain those symbols; the respective identifications of generators extend to algebra embeddings U q (b ± ) → U q (g). This is important since we will be looking at representations of U q (b + ) and U q (b − ).

2.2.
Level-0 representations of quantum affine sl 2 and its Borel subalgebras. Note that k 0 k 1 is central in U q (g), and for any representation ρ of U q (g), U q (b + ) or U q (b − ), the level λ is the unique complex number such that ρ(k 0 k 1 ) acts on the image of ρ as the scalar q λ . Let I be the two-sided ideal of U q (g) generated by k 0 k 1 −1. If a representation has level 0 then it factors through a representation of the quotient U q (g)/I, which is isomorphic to the quantum loop algebra of sl 2 . In particular, all finite-dimensional representations of U q (g) have level 0. We will restrict our attention to level-0 representations from now on; on the other hand we widen the scope somewhat by also including representations of the subalgebras U q (b ± ) where k 0 k 1 acts as the identity. For a more comprehensive discussion about level-0 representations of U q (g) as evaluation modules constructed from Verma modules for U q (sl 2 ) see for example [11,Section 3] and [24,Section 2].
All vector spaces under consideration and their tensor products are defined over C. Let U, U ′ be any two vector spaces. We shall write Hom(U, U ′ ) to mean the set of C-linear maps from U to U ′ and End(U ) = Hom(U, U ). We denote by P U,U ′ ∈ Hom(U ⊗ U ′ , U ′ ⊗ U ) the map that naïvely swaps tensor factors: P U,U ′ (u ⊗ u ′ ) = u ′ ⊗ u for all u ∈ U and u ′ ∈ U ′ . If there is no cause for confusion, we will simply write P . In this paper we will focus on two vector spaces in particular and tensor products of them.
Let z ∈ C × . Also, let V = C 2 and choose any ordered basis (v 0 , v 1 ) of V . Expressing linear operators on V as 2 × 2 matrices with respect to this basis, the following assignments extend to an algebra homomorphism π z : U q (g) → End(V ) (a level-0 representation of U q (g) on V ).
Consider the infinite-dimensional vector space Cw j and let F denote the commutative algebra of functions: Z ≥0 → C. We define certain linear maps on W as follows: for all j ∈ Z ≥0 and f ∈ F. The linear maps a, a † and f (D) for all f ∈ F satisfy the defining relations of (an extension of) the q-oscillator algebra, to wit We denote the subalgebra of End(W ) generated by a, a † and the commutative algebra F(D) := {f (D) | f ∈ F} by osc q . As a consequence of (2.7) we have the linear decomposition For more details on the q-oscillator algebra, the reader may consult for instance [29] and [30,Sec. 5].
(i) These two families of representations depend on an additional parameter r which does not appear in representations of U q ( sl 2 ). We need this parameter in order to have a short exact sequence of U q (b + )-intertwiners, see Lemma 3.1. The Q-operator does not depend on r. (ii) In order to define the representations ρ ± z,r one does not need to work with the whole of F(D), but only by the subalgebra generated by q 2D and q −2D . However for the various intertwiners we will need a larger algebra; namely for infinite-dimensional solutions of the Yang-Baxter equation (L-operators) we need the subalgebra of F(D) generated by q D and q −D , see (2.16).
For infinite-dimensional solutions of the reflection equation it is convenient to allow all of F(D), see (2.30).
Proof of Proposition 2.1. We give the proof for ρ + z,r ; the proof for ρ − z,r is analogous. We only need to verify the relations Only the last is nontrivial; we have , [e 1 , e 0 ] q 2 ] 1 ) = 0, from which the result follows.
It can be checked that the algebra maps ρ ± z,r with domain U q (b ± ) cannot be extended to algebra maps of U q (g).

R-matrices and their infinite-dimensional counterparts.
Let R be the parameterindependent universal R-matrix of U q (g), see e.g. [31,32]. It satisfies and as a consequence (2.14) From the quantum double construction it follows that R lies in a completion of U q (b + ) ⊗ U q (b − ); in particular the following linear maps are well-defined: These operators depend rationally on the quotient z 1 /z 2 , see e.g. [33,Lecture 9]. We denote suitable scalar multiples of these operators by R(z 1 /z 2 ), L(z 1 /z 2 , r) and L − (z 1 /z 2 , r), respectively. In order to write down explicit expressions for these linear operators it is customary to apply the appropriate representation to (2.11), restricting to a subalgebra where appropriate, and solve the resulting linear equations, which are This leads us to the following solutions (unique up to overall scalar multiples): Note that L(z, r) is invertible if z 2 = 1, with the inverse given by As a consequence of (2.14) we obtain the following Yang-Baxter equations: in terms of partial transpositions · t 1,2 with respect to the first and second factors in the tensor products V ⊗ V and W ⊗ V respectively. Since R(z) t 1 , R(z) t 2 , L(z, r) t 2 depend polynomially on z and are invertible for z = 0 it follows that R(z) and L(z, r) are well-defined for all but finitely many values of z. As a consequence of [34,Thm. 4.1], R(z) −1 can be expressed in terms of R(q 2 z).
In particular, in our case due to our choice of normalization of R(z) we have The operator L(z, r) enjoys a similar identity; in fact, the proof of [34,Thm. 4.1], which is stated only for tensor products of finite-dimensional representations of U q (g), applies in this setting as well. Indeed, straightforward computations give Combining this with (2.18) we readily obtain (2.24) L(z, r) −1 = 1 − q 2 z 2 1 − q 4 z 2 L(q 2 z, r).

K-matrices and their infinite-dimensional counterparts.
Let ξ ∈ C × . The matrix is a solution of the (finite-finite) right reflection equation see [14]. If we include the "limit cases" 1 0 then this family provides all invertible diagonal solutions K V (z) ∈ End(V ) of (2.26) up to an overall scalar factor, see e.g. [35,Lemma 2.2].
Remark 2.3. Consider the unique involutive Lie algebra automorphism θ of g such that e i ↔ −f σ(i) for i ∈ {0, 1} and let k = g θ be the corresponding fixed-point subalgebra. The following subalgebra of U q (g) quantizes U (k) ⊂ U (g): tends to 1 as q → 1; see [36] for a more general theory of quantized fixed-point subalgebras of Kac-Moody Lie algebras. Note that, unlike U (k), the coproduct on U q (g) does not restrict to one on U q (k). Instead, U q (k) is a left coideal: Note that U q (k) possesses an independent description as a U q (g)-comodule algebra (in terms of generators and relations) and in such a context is also known as the augmented q-Onsager algebra, see [37]. The matrix K V (z) ∈ GL(V ) is a U q (k)-module homomorphism. More precisely, it intertwines the representations (V, π z | Uq(k) ) and (V, π 1/z | Uq(k) ), i.e.: for all x ∈ U q (k).
The simple linear relation (2.28) defines K V (z) uniquely, up to a scalar. In general it is known that, under certain technical assumptions, such intertwiners satisfy the reflection equation (2.26), see [17,38] and [18, Sec. 6.2].
We will also consider an element K W (z, r) ∈ End(W ) which satisfies the (infinite-finite) right reflection equation . Remark 2.4. As opposed to K V (z), the map K W (z, r) does not satisfy a simple relation of the form (2.28) for the representations ρ ± z,r , since U q (k) is not contained in either We denote the q 2 -deformed Pochhammer symbol by where x ∈ C and j ∈ Z ∪ {∞} -note that this depends holomorphically on q in the unit disk and, for j ≥ 0, holomorphically on x.
Proposition 2.5. The unique solution of (2.29) such that K W (z, r)(w 0 ) = w 0 is given by That is, Proof. It is convenient to re-write (2.29) in the form by virtue of (2. 16-2.18). Considering that the diagonal entries are identical and how the off-diagonal entries are related we obtain the result.
Remark 2.6. Interestingly, the infinite-dimensional K-matrices in [23, (4.9)-(4.12)], constructed as solutions of certain intertwining relations, appear to be different from those in equation (2.30) which are constructed directly as solutions of reflection equations. There is no contradiction, since the reflection equation [23, (4.16)] involves two different L-operators whereas in the corresponding equation in our case (2.29) the L-operators are the same.
We will also be interested in solutions K V (z) ∈ End(V ) and K W (z, r) ∈ End(W ) of the left reflection equations Consider (2.32). Employing (2.24), inverting and reparametrizing (y, z) → (q −1 y, q −1 z), we see that it is equivalent to . Letξ ∈ C × be arbitrary. Comparing (2.33) with (2.29) we obtain that satisfies (2.32) with K V and K W given by (2.25) and (2.30). Here f V , f W are any scalars depending meromorphically on z andξ ∈ C × is arbitrary.
Following the same argument, one obtains that (2.31) is satisfied.
Note that K W (z, r) ∈ End(W ) has a well-defined limit (qr) −D z 2D ∈ End(W ) as ξ → 0; similarly K W (z, r) ∈ End(W ) tends to q D 2 r D 0 D = δ D0 ∈ End(W ) asξ → 0. Since the matrices K V (z) and K V (z) are well-defined and the reflection equations (2.26-2.29) and (2.31-2.32) are preserved in these limits, from now on we will allow ξ,ξ ∈ C.

Short Exact Sequences and Fusion
In this section we construct U q (b + )-intertwiners which take part in short exact sequences relating the module (W ⊗V, ρ + z,r ⊗π z ) to the module (W, ρ + z ′ ,r ′ ) for certain shifted parameters z ′ , r ′ . Similarly, we construct U q (b − )-intertwiners involved in short exact sequences relating the U q (b − )-modules (V ⊗ W, π z ⊗ ρ − z,r ) and (W, ρ − z ′′ ,r ′′ ) for certain shifted parameters z ′′ , r ′′ . For r ∈ C × , consider the following linear maps called fusion intertwiners: Here we interpret elements of W ⊗ V and elements of V ⊗ W as vectors with two entries in W , using the ordered basis (v 0 , v 1 ) of V . In other words, we have Lemma 3.1. The maps ι(r) and τ (r) are U q (b + )-intertwiners as follows: and we have the following short exact sequence of U q (b + )-intertwiners: Similarly, the maps P ι(r −1 ) and τ (r −1 )P are U q (b − )-intertwiners: and we have the following short exact sequence of U q (b − )-intertwiners: Proof. The requirements (3.2) and (3.4) that these maps are intertwiners between the indicated modules is equivalent to Solving (3.6-3.7) for u = k 0 and u = k 1 we obtain, using (2.8), that respectively, where f 0 , f 1 , g 0 , g 1 ∈ F are arbitrary. Now solving (3.6-3.7) for u ∈ {e 0 , e 1 } and using (2.9), as well as for u ∈ {f 0 , f 1 } and using (2.10), we deduce (3.1).
To prove the short exact sequence statements, we straightforwardly verify that ι(r) is injective, τ (r) is surjective and the image of ι(r) equals the kernel of τ (r), respectively.
We also have boundary fusion relations similar to the finite-dimensional relation given in [24, Eqn. (4.7)]. Boundary fusion was first discussed in the papers [16,25] without explicitly using short exact sequences.
We have the following identities in Hom(W, W ⊗V ) and Hom(W ⊗V, W ), respectively: and the following identities in Hom(W, W ⊗ V ) and Hom(W ⊗ V, W ), respectively: as required. We also have as required. This gives (3.10-3.11). Now (3.12-3.13) straightforwardly follow by applying (2.24) and (2.34).

Transfer matrices and the Q-operator
In this section we define, and prove properties of, the main objects of this paper: the transfer matrix and the Q-operator for the open XXZ chain with diagonal boundaries. Initially we will define the Q-operator as a formal power series and in Section 6 we will show it converges for suitable parameters and hence determines a well-defined linear map.
We will construct operators acting on V ⊗N by first considering operators called quantum monodromy matrices acting on a larger tensor product V ⊗ V ⊗N or W ⊗ V ⊗N . The additional factor in the tensor product is called the auxiliary space -in the construction of the transfer matrix this space will be traced out. The quantum monodromy matrices are compositions of operators each acting on the auxiliary space and at most one other space, as follows: Here we have labelled the auxiliary space by a (in general we will use lowercase roman letters to label auxiliary spaces). Note that due to the explicit expressions (2.15-2.16) we have . These quantum monodromy matrices are particular to open chains; the operator M V (z) was first considered by Sklyanin [14]. In particular he showed it satisfies a version of the reflection equation and M W (z, r), see (4.5) below. In such identities there are two auxiliary spaces which are labelled a and b, respectively. The following lemma is vital to the proof of the main theorem of this paper, Theorem 5.3.

Lemma 4.2. We have the following identities in
where we have introduced the coefficient polynomials Proof. Take (4.3) in the special case y = z. Right-multiplying it by ι(r) ⊗ Id V ⊗N , using (3.8) N times, (3.10) once and (3.8) another N times we arrive at On the other hand, left-multiplying (4.3) with y = z by τ (r) ⊗ Id V ⊗N , using (3.9) N times, (3.11) once and (3.9) another N times, we arrive at Combining (4.9-4.10) with (3.12-3.13) we obtain (4.6-4.7).
We record some more properties of M W (z, r) which we will need later on. As an immediate consequence of (2.15-2.16), (2.25) and (2.30) we obtain which implies that for all r ∈ C × and y, z ∈ C we have (4.11) From (2.16) we obtain L(z, r) = L(z, 1) r 0 0 1 = r 0 0 1 r −D L(z, 1)r D and from (2.30) and (2.36) we have and hence we have the following factorization for all r ∈ C × , z ∈ C: The transfer matrices (with respect to auxiliary spaces V and W , respectively) are the following linear maps on V ⊗N : where the subscript of the trace labels the space over which we trace. Moreover the Q-operator is the linear map For now we treat these operators as formal power series in z.
Remark 4.4. Note that we have set r = 1 in the definition of T W (z) and hence Q(z). The relations (4.6-4.7), where r varies, will be used to derive Baxter's functional relation, see (5.8), and to compensate for this it is necessary, as will become apparent, to introduce the extra diagonal matrix in the definition of Q(z).
As a consequence of the properties and cyclicity of the trace we obtain that the matrix-valued formal power series Q(z), T W (z), T V (z) are invariant under z → −z and hence we deduce that the matrix-valued formal power series Q(z), Similarly, as a direct consequence of (4.11), the properties and cyclicity of the trace we obtain understood as an equation of matrix-valued formal power series in y and z. As a consequence, Note that T V (z) is a well-defined linear operator on V ⊗N whose entries depend polynomially on z (they are finite sums of certain entries of products of matrices whose entries depend polynomially on z). We summarize the above discussion in the following Lemma. The polynomiality of T W (z) and Q(z) is not obvious. For now we show that for suitable values of q, ξ andξ and all but finitely many z the matrix entries of T W (z) and Q(z), are well-defined (i.e. the series associated with the trace converges). In order to do this, for q, x inside the unit disk, a, b ∈ C and c ∈ C\q 2Z ≤0 we consider the basic hypergeometric function (see e.g.
which converges absolutely, as a consequence of the ratio test.
Proof. By (4.14) it suffices to prove the result for T W (z). Claim: Any matrix entry of K W a (z, 1)M W a (z, 1) is of the form for some c L,M (z 2 ; t) ∈ C which depends holomorphically on each of q, ξ, ξ, Laurent polynomially on each t n and meromorphically on z 2 , with all poles, if any, simple and at z 2 = q −2ξ−1 and Note that owing to the ratio of q-Pochhammer symbols there may be simple poles at z 2 = q −2kξ−1 for k ∈ Z ≥2 . The claim implies that an arbitrary matrix entry of T W (z) is of the form so that the condition on the norm of ξξ now guarantees convergence and the statement of the theorem follows.
Re-arranging the powers of q we obtain that, modulo terms in Ker(Tr W ), any matrix entry of M W (z, 1) is a linear combination, with coefficients depending polynomially on z 2 , Laurent polynomially on q and Laurent polynomially on each t n , of expressions of the form Also note the basic property (x) j+k = (x) j (q 2j x) k for all j, k ∈ Z, see [39, (1.2.33)]. Hence we deduce that an arbitrary matrix entry of K W (z, 1)M W (z, 1) is a linear combination of expressions of the form modulo Ker(Tr W ). Note that (q 2 z 2 /ξ) −j(0,1,0)−j(0,1,1)+j(1,0,0)+j(1,0,1) has simple poles at we obtain bounds for the summation variables and arrive at the claim.
We obtain the desired conclusion.

Baxter's TQ relation
In this section we derive the major result of this paper, namely Baxter's relation for the matrices T V (z) and Q(z). We start with proving some commutativity properties of the families {T V (z)} z∈C and {T W (z)} z∈C . 5.1. Commutativity. Sklyanin's argument of commuting two-row transfer matrices involves extending the tensor product V ⊗N , the domain of the operators T V,W (z) by a tensor product of two auxiliary spaces labelled a and b. This auxiliary tensor product is W ⊗ V in the case of [T W (y), T V (z)] and V ⊗ V in the case of [T V (y), T V (z)]. The proof we give here for (5.2) is based on the well-documented one discussed in [14,42,43,44,45] but arranged, following Frassek and Szécsényi's approach in [22], in such a way that partial transpositions are not taken with respect to the infinite-dimensional vector space W .
Theorem 5.1. Let y ∈ U and z ∈ C\{±q −1 y −1 }. Then where in the tensor product W ⊗ V ⊗ V ⊗N we have labelled the factor W by a and the first factor V by b, and written Tr a,b for the partial trace with respect to W ⊗ V . Furthermore, for all y ∈ U , z ∈ C we have and for all y, z ∈ C we have Proof. We will establish (5.2) by proving [T W (y), T V (z)] = 0; in the process we will also prove (5.1). The proof of (5.3) is analogous to the proof of (5.2) and was already given in [14]. Let y ∈ U and z ∈ C\{±q −1 y −1 }. Then L(yz, 1) t b ∈ End(W ⊗ V ) is invertible, see (2.23). We have where we have inserted L(yz, 1) t b L(yz, 1) t b = Id W ⊗V and used cyclicity of the trace twice. Standard properties of the partial transpose yield and we obtain (5.1). Now additionally assume that z = ±y so that L(y/z, 1) ∈ End(W ⊗ V ) is invertible. We insert L(y/z, 1) −1 L(y/z, 1) = Id W ⊗V and obtain here we have used (2.32) and (4.5). Using cyclicity of the trace and restoring the partial transpositions we obtain and we use cyclicity of the trace again. We arrive at as required. For y ∈ U and z ∈ C the matrix D(y, z) := [T W (y), T V (z)] ∈ End(V ⊗N ) is welldefined and depends polynomially on z. For all y ∈ U we have shown that D(y, z) = 0 for all but finitely many z; hence D(y, z) = 0 for all z ∈ C. This completes the proof. R +− (y/z) = (ρ + y,1 ⊗ ρ − z,1 )(R) ∈ End(W ⊗ W ). However R +− (z) t b is not a well-defined linear operator; some columns of the matrix of R +− (z) t b with respect to the ordered basis (w 0 ⊗w 0 , w 0 ⊗w 1 , . . . , w 1 ⊗w 0 , w 1 ⊗w 1 , . . . , . . .) of W ⊗W have infinitely many nonzero entries. Therefore we cannot insert the identity

5.2.
Baxter's TQ relation through trace decomposition in a short exact sequence. Having established that T V (z) and Q(z) are well-defined elements of End(V ⊗N ) for generic values of z which mutually commute, we consider the crucial functional relation satisfied by them. With our preparations, this is now a simple consequence of the fact that short exact sequences of vector spaces are split.  where B is assumed to have a basis. Then there exist ι ′ ∈ End(B, A) and τ ′ ∈ End(C, B) such that ι ′ • ι = id A , τ • τ ′ = id C and ι ′ • τ ′ = 0. Given θ ∈ End(B) we have Proof. Since this is a short exact sequence of vector spaces and B has a basis, there exists a right inverse for τ i.e. τ ′ ∈ Hom(C, B) such that τ • τ ′ = id C . By the Splitting Lemma, see for instance [47,Prop. 3.2], we have Now define ι ′ ∈ Hom(B, A) by ι ′ (ι(a)) = a for all a ∈ A and ι ′ | Im(τ ′ ) = 0. Automatically we have The second term maps Im(τ ′ ) to Im(ι) and Im(ι) to 0 and the third term maps Im(ι) to Im(τ ′ ) and Im(τ ′ ) to 0. Also, the first and fourth terms are supported on Im(ι) and Im(τ ′ ), respectively. Hence where we have used that ι is an isomorphism from A to ι(A) ⊆ B with inverse ι ′ and τ ′ is an isomorphism from C to τ ′ (C) ⊆ B with inverse τ . This proves (5.5), from which (5.6) immediately follows.
We now arrive at the main theorem of the paper. Proof. Recall the definition (4.14). From (5.1) with y = z we obtain Because of the short exact sequence (3.3) and Lemma 4.2, both in the case r = 1, we may apply the formula (5.6). It yields By virtue of (4.12) and basic properties of scalar multiples of convergent series, these traces are well-defined. Hence, by (4.15) we have

Crossing symmetry and Bethe equations
In this section we will establish a functional equation for the matrix Q(z) (the so-called crossing symmetry) and derive the Bethe equations.
6.1. Diagonalizability and crossing symmetry of T V (z). Lemma 6.1. For all z ∈ C and for generic values of q,ξ, ξ, t 1 , . . . , t N , the matrices T V (z) are diagonalizable.
Proof. This can be done analogously to the method outlined in [40,Sec. 4.1] for the closed chain. Up to an overall scalar, T V (z) depend holomorphically on each of q, ξ,ξ ∈ C and each t n ∈ C × . This is a direct consequence of the definition (4.13) and properties of its constituents. By the results in [41, Sec. II], the number of eigenvalues and their algebraic and geometric multiplicities is constant outside a discrete set. Hence T V (z) is generically diagonalizable or generically non-diagonalizable.
We will now establish that we are in the former case, as required, by showing this matrix is normal with respect to a suitable inner product, and hence diagonalizable, for uncountably many values of each of the indicated parameters, namely for Consider the usual inner product on V defined by (v i , v j ) = δ ij for i, j ∈ {0, 1}. We extend this inner product multiplicatively over tensor factors to define inner products on V ⊗N and V ⊗ V ⊗N . With respect to these inner products, the adjoint X * of a linear map X is given by the conjugate transpose and taking adjoints commutes with taking traces.
The assumptions (6.1) on the parameters imply and hence By (5.3) we conclude that for uncountably many values of the parameters q, ξ,ξ, t 1 , . . . , t N the matrix T V (z) is normal, as required.
Compared to transfer matrices of closed spin chains, those of open spin chains typically have an additional symmetry, namely one of the form T (pz −1 ) = (scalar)T (z) for some p ∈ C × . We will combine this later with Theorem 5.3 to deduce an analogous property of Q(z). Lemma 6.2. We have Proof. We make three observations to facilitate the proof. Abbreviate σ = . Straightforward computations yield the following identities for generic z: Furthermore, consider the Yang-Baxter equation (2.19); by partially transposing with respect to the first tensor factor, left-and right-multiplying by (R 13 (z 1 /z 3 ) t 1 ) −1 and partially transposing the result with respect to the third tensor factor we obtain the following identity in End(V ⊗ V ⊗ V ): Finally, linear algebra in the tensor product V ⊗ V ⊗ V and the definition of R(z) imply (6.5) Tr Having made these preparatory steps, we note that by cyclicity of the trace, where we have given the auxiliary space the label c. Now applying (6.3) (using the labels b and a for the two additional auxiliary spaces, respectively) we obtain where we have combined the traces, pulled the factor P ac all the way to the left and the factor K V a (z) ta all the way to the right. Repeatedly using (6.4) we arrive at where we have moved the first N -fold product of partially transposed R-matrices all the way to the left. Owing to (6.5) the partial trace over the space labelled c amounts to the identity map and as a consequence so does the trace over the space labelled b. We are left with Now (6.2) follows from standard properties of the partial transpose.
6.2. The polynomiality and commutativity conjecture for the Q-operator. The results of the remainder of this section are conditional on the following conjecture. In Appendix A we prove that diagonal entries of T W (z) depend polynomially on z 2 . Moreover, in Appendix B we show that the conjecture is true for N = 2. Conjecture 6.3 implies that the set U defined by equation (4.20) can be replaced by C.

Diagonalizability and crossing symmetry of Q(z).
Lemma 6.4. For all z ∈ C and generic values of q,ξ, ξ, t 1 , . . . , t N , the matrices T W (z) and Q(z) are diagonalizable.
Proof. Note that the claim for Q(z) follows from the claim for T W (z). We can follow the same steps as in the proof of Lemma 6.1. Owing to the definitions (4.13) and properties of its constituents, as well as Theorem 4.6, T W (z) depends holomorphically on each of q, ξ,ξ ∈ C and each t n ∈ C × , up to an overall factor, and as before T W (z) is either generically diagonalizable or generically non-diagonalizable. Again, we will show that T W (z) is normal and hence diagonalizable if the parameters satisfy (6.1); since these conditions still allows for uncountably many values for each parameter, it follows that T W (z) is generically diagonalizable.
Note that the commutative ring C is a * -ring with the involution given by complex conjugation z → z. The algebra osc q becomes a * -algebra over this * -ring if we define The space W is a Hilbert space with respect to the inner product defined by Moreover, it is easy to check that X * is the Hermitian adjoint of X ∈ osc q with respect to this inner product. We extend the inner products on V and W multiplicatively over tensor factors to define inner products on V ⊗V ⊗N and W ⊗V ⊗N , so that taking adjoints commutes with taking traces (provided the traces converge).
Using the commutativity of {T V (z)} z∈C ∪ {T W (z)} z∈C , we now deduce that this family of operators is simultaneously diagonalizable. Hence we may restrict Baxter's relation (5.8) and the polynomiality of T V (z) and Q(z) to joint eigenspaces and these relations descend to statements about eigenvalues. We summarize this in the following lemma. Lemma 6.5. All eigenvalues of T V (z) and Q(z) are polynomial function of z 2 ∈ C. For z ∈ C, let T V (z) and Q(z) be simultaneous eigenvalues of T V (z) and Q(z), respectively. Then for all z ∈ C we have Hence, if y ∈ C is such that Q(y) = 0 or y 4 = q −2 we have (6.8) p + (y)Q(qy) + p − (y)Q(q −1 y) = 0.
We will relate equation (6.8) to the Bethe equations known from Sklyanin's algebraic Bethe ansatz [14]. First, we state and prove the promised analogon of Lemma 6.2 for Q(z). Theorem 6.6 (Crossing symmetry of Baxter's Q-operator). For all z ∈ C × we have Proof. Define the auxiliary z-dependent matrix .
In the first part of the proof we will show that A(z) = 0. By Lemma 6.2, comparing (5.8) as-is to (5.8) with z → q −1 z −1 and using (5.2) yields .

From (4.8) we derive the functional relations
and hence (6.9) is equivalent to (6.10) Any matrix entry of A(z) satisfies the same q-difference equation (6.10). Also, as a consequence of Conjecture 6.3, the matrix entries of A(z) are rational functions in z 2 . Finally, note that the coefficient on the right-hand side of (6.10) depends non-trivially on z becauseξξ = q 2 and (ξ 2 ,ξ 2 ) = (q 2 , q 2 ); these inequalities follows straightforwardly from the assumption |ξξ| < |q| 2N .
Writing an arbitrary rational scalar-valued solution of (6.10) as a quotient of polynomials without common factors, we deduce that it has infinitely many zeroes and hence is the zero function. It follows that the matrix A(z) is the zero matrix.
In analogy with [22,Eq. (5.14)], it is tempting now to define a new auxiliary matrix A(z) = Q(z) −1 Q(q −1 z) and from A(z) = 0 derive the functional equation A(q −1 z −1 ) = q 4N A(z). However because of the factor q 4N = 1 this has no solutions in rational functions, so it is necessary to provide an alternative approach. Namely we consider the auxiliary matrix which is well-defined for generic z by virtue of Lemma 4.7. Combining A(z) = 0 with (6.6) we obtain the functional relation for generic z. Again, by descending to the matrix entries and using that the entries of B(z) depend rationally on z 2 , we obtain that B(z) is independent of z. Hence there exists a constant matrix B such that, for all nonzero z, Replacing z by q −1 z −1 for nonzero z we obtain that B 2 = Id, so that B is diagonalizable with its spectrum contained in {−1, 1}. Because Q(z) and Q(q −1 z −1 ) are simultaneously diagonalizable it follows that the eigenspaces of B coincide with the eigenspaces of Q(z) which are independent of z.
It remains to prove that B = Id. If Q(q −1/2 ) is invertible then (6.11) at z = q −1/2 immediately implies the desired result. Now assume on the contrary that Q(q −1/2 ) is not invertible, i.e. that it has a zero eigenvalue. Then B = Id follows from the following claim.
then we have Q(q k−1/2 ) = 0 for all k ∈ Z ≥0 . Indeed, since Q(z) depends polynomially on z, the claim implies that Q(z) = 0 for all z ∈ C which contradicts the generic invertibility of Q(z). It follows that B has eigenvalue 1 on all eigenspaces of Q(z) whose eigenvalue vanishes at z = q −1/2 , for generic values of ξ, ξ and each t n . Hence B = Id for generic values of ξ, ξ and each t n . Because Q(z) depends analytically on ξ and ξ and Laurent polynomially on each t n , B depends meromorphically on these parameters. Hence B = Id for all parameter values which proves the theorem.
6.4. Bethe equations. The following technical result together with the eigenvalue version of Baxter's relation (6.8) will allow us to derive the Bethe equations. Proposition 6.7 (Zeroes of the eigenvalues of Q(z)). Let Q(z) be an eigenvalue of Q(z). There exist M ∈ {0, 1, . . . , N } and f, y 1 , . . . , y M ∈ C × such that (6.14) Proof. It suffices to prove this for generic values of z. Since the eigenvalues of Q(z) are not generically zero and depend polynomially on z 2 we have for some M ′ , N ′ ∈ Z ≥0 , f, y 1 , . . . , y M ′ ∈ C × . Combining this with Theorem 6.6 we obtain This is equivalent to the following identity of polynomials: The powers of z yield that M ′ = 2M is even and N ′ = N − M . Therefore M ≤ N and The first equation of (6.15) requires that the multiset {y 2 j } 2M j=1 is invariant under the involution ψ of C × defined by ψ(Y ) = q −2 Y −1 . Combined with the second equation and the fact that |q| < 1 it follows that the multiplicities of the two ψ-fixed points ±q −1 are even. It follows that {y 2 1 , . . . , y 2 2M } splits up into nontrivial ψ-orbits and an even number of ψ-fixed points. Up to relabelling we may assume y 2 M +j = q −2 y −2 j for j ∈ {1, . . . , M } and we obtain (6.14). Given the factorization (6.14), setting y = y i in (6.8) immediately yields the following result. where p ± (z) and Q(z) are given by Equations (4.8) and (6.14).
In Appendix C we develop Skylanin's formulation of the Algebraic Bethe Ansatz for the open XXZ model in the conventions of this paper and show it coincides precisely with (6.16).

Discussion
For the closed XXZ chain, the Q-operator can be viewed as a 'fundamental' transfer matrix in the following sense [5]: the infinite-dimensional Verma module transfer matrix T + µ (z) can we written as a product of the form where µ ∈ R is the highest weight of the Verma module, and Q(z) and Q(z) are Q-operators given as traces of the monodromy matrix over two different q-oscillator representations associated with the vector spaces W = Cw j respectively. If we choose µ ∈ Z ≥ 0, then the transfer matrix T µ (z) of the µ + 1 dimensional representation (which is a quotient of the Verma module of weight µ by that of weight −µ − 2) is given by In particular, choosing µ = 0 and using (7.1) and (7.2) gives the quantum-determinant expression involving the operator Q. The key equation (7.1) can be proven either by writing the tensor product of the two q-oscillator representation in terms of a filtration (see Appendix C of [46]) or by using the R-matrix factorization of Derkachov (see [48,49]). The connection between these two approaches is explained in [50]. The closed Baxter T Q relations can in turn be proved by combining the quantum determinant expression with the µ = 1 case of equation (7.2).
In this paper we have taken the alternative route to the T Q relations that involves the tensor product structure of Lemma 3.1. In the second paper in this series we will derive analogues of (7.1) and (7.2) for open chains, thus yielding expressions for the quantum determinant and higher-spin transfer matrices in terms of the Q operator of this paper.
One question is the following: do the results of this paper, in particular the boundary fusion relation of Lemma 3.2, extend to non-diagonal K-matrices K V (z)? (The general solution to the reflection equation associated to the vector representation of U q ( sl 2 ) was found in [51].) We do not yet have an answer to this question; the main complication is the difficulty of finding a solution K W (z) of the reflection equation (2.29) in the non-diagonal case.
A. Polynomiality of the diagonal entries of the Q-operator In Appendices A and B we use some auxiliary notation. For α = (α 1 , . . . , α N ), β = (β 1 , . . . , β N ) ∈ {0, 1} N and t ∈ (C × ) N we consider the matrix entries M(z) α β ∈ osc q defined by for all w ∈ W , and the matrix entries T (z) α β ∈ C defined by Moreover, in this section we explicitly write the dependence of T W , K W and M W on the parametersξ and t = (t 1 , . . . , t N ) where applicable. In particular, and we indicate these parameters also for the matrix entries defined by (A.1-A.2). Consider the Z-linear map s : As a consequence of the decomposition (2.8) and the commutativity (4.15), for all t = (t 1 , . . . , t N ) ∈ (C × ) N and α, β ∈ {0, 1} N there exists f (z; t; ·) α β ∈ F such that Note that if s(α) = s(β) then the two expressions coincide, as required.
Together with the boundary values the system (A.5) defines f (z; t; ·) β α ∈ F uniquely. We will need (A.5) only for α = β and γ = δ, in which case it simplifies to As a consequence of (A.4) we have The key step in showing polynomiality is the following recursion for the entries of T W (z).
Proposition A.2. Let ξ ∈ C,ξ ∈ C, u ∈ C × , t ∈ (C × ) N +1 and α, β ∈ {0, 1} N . We have the following identities: Proof. Without loss of generality we may assume that s(α) = s(β). We will start our derivation of the recursion relations with the recursion (A.9) and the expression (A.10). In particular, we have Shifting the summation variable in the second term leads to as required. To derive the recursion for T (z;ξ; u, t) 1,β 1,α , we have where we have shifted the summation variable in the second term. Using the identity we arrive at We are now in a position to prove the desired properties of the diagonal entries of T W (z).
Proof. For N = 0 a direct computation gives We recall the q-Gauss summation formula for a, b ∈ C × and c ∈ C such that | c ab | < 1, see e.g. [39, (1.5.1)], we obtain that T W (z) = (1− ξξ) −1 , clearly polynomial in z 2 . Now induction with respect to N , using the fact that the coefficients in the recurrences given in Proposition A.2 depend polynomially on z 2 , proves the theorem for T W (z). By (4.14) the corresponding statement for Q(z) follows immediately. B. The case N = 2 In this section we demonstrate for N = 2 that the matrix entries T W (z) depend polynomially on z 2 and that [T W (y), T W (z)] = 0. It then follows from (4.14) that the Q-operator has the same properties. Recall that we assume |ξξ| < |q| 2N = |q| 4 and 0 < |q| < 1.
B.1. Polynomiality. Our strategy is straightforward: we use the definitions (4.2) and (4.13) directly. As a consequence of Theorem A.3, it suffices to do this for the two non-zero off-diagonal entries T W (z) 0,1 1,0 and T W (z) 1,0 0,1 . We have Using (A.11) we obtain After collecting terms that are manifestly proportional to ξ − z 2 we obtain The first two terms can be combined, yielding Note the appearance of the overall factor ξ − z 2 . The two terms inside parentheses can be combined in the same way, yielding , so that also an overall factor 1 − q 2ξ z 2 has emerged. Comparing with (B.1), we see that the two z-dependent factors in the denominator are cancelled, yielding the following polynomial in z 2 : ξξ) .

B.2.
Commutativity. Considering the locations of the zero entries of T W (z), we see that the property [T W (y), T W (z)] = 0 is equivalent to the following ratios being independent of z: As a consequence of the explicit formula (B.3) and the fact that T W (z) 1,0 0,1 can be obtained from T W (z) 0,1 1,0 by inverting t 1 and t 2 , it is sufficient to prove that the first ratio of (B.4) is independent of z. To this end, we compute the difference T W (z) 0,1 0,1 − T W (z) 1,0 1,0 ∈ C following the procedure of the diagonal entries. We simply present the result of the first part of the computation: Using the relations (A.11) and (B.2) this yields and the required property of the ratios (B.4) readily follows.