Defining relations for quantum symmetric pair coideals of Kac-Moody type

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as quantum symmetric pairs, replace the fixed point Lie subalgebras by one-sided coideal subalgebras of the quantized enveloping algebra $U_q(\mathfrak{g})$. We provide a complete presentation by generators and relations for these quantum symmetric pair coideal subalgebras. These relations are of inhomogeneous $q$-Serre type and are valid without restrictions on the generalized Cartan matrix. We draw special attention to the split case, where the quantum symmetric pair coideal subalgebras are generalized $q$-Onsager algebras.


Introduction
A classical symmetric pair consists of a Lie algebra g together with its subalgebra k of fixed points under a Lie algebra involution θ. Quantum analogs of this construction, known as quantum symmetric pairs, have emerged in the beginning of the 1990's. They replace g by its quantized universal enveloping algebra U q (g) and k by a one-sided coideal subalgebra B c,s of U q (g), which is called a quantum symmetric pair (QSP) coideal subalgebra. The algebras B c,s were first constructed by Noumi, Sugitani and Dijkhuizen [Nou96,NS95,NDS97] under the name quantum Grassmannians. A different approach, based on the Drinfeld-Jimbo presentation of U q (g), was pursued by Letzter. She developed a comprehensive theory of quantum symmetric pairs for semisimple Lie algebras g in an elaborate series of papers [Let99,Let02,Let03]. This has allowed to identify the zonal spherical functions on quantum symmetric spaces as Macdonald-Koornwinder polynomials [Let04]. This whole theory was later extended to symmetrizable Kac-Moody algebras g by Kolb in [Kol14], which treats the structure theory of the Kac-Moody QSP coideal subalgebras B c,s in great detail.
Over the years, it has become increasingly apparent that quantum symmetric pairs play a crucial role in quantum integrability, notably of the reflection equation [Che84,Skl88]. The latter replaces the quantum Yang-Baxter equation when reflecting boundary conditions are imposed by K-matrices. Such boundaries break the quantum symmetry down to a coideal subalgebra of the quantum affine algebra which encodes the symmetries in the bulk of a quantum spin chain [DM03]. It has been suggested that representations of the QSP coideal subalgebras B c,s give rise to universal solutions of the reflection equation, just like solutions of the quantum Yang-Baxter equation arise naturally from representations of quantum affine algebras [DM06]. This has been worked out for several specific QSPs [KS09,Kol15,RSV16]. Invaluable tools in this respect are the definition of a bar involution for quantum symmetric pairs by Balagović and Kolb in [BK15], and the construction of quasi K-matrices as QSP analogs of Lusztig's quasi R-matrices, both in recursive [BK19] and in factorized form [DK19].
In view of the same idea, many more fundamental concepts in representation theory have been extended from quantum affine algebras to quantum symmetric pairs. A special class of QSP coideal subalgebras arises in the categorification of U q (g)-representations based on skew Howe duality, as was established by Ehrig and Stroppel in [ES18]. A thorough theory of canonical bases and Schur-Jimbo duality for quantum symmetric pairs was set up by Bao and Wang in [BW18a,BW18b], leading them to prove longstanding conjectures on irreducible characters of orthosymplectic Lie superalgebras.
A first step towards a classification of finite-dimensional B c,s -modules was recently set in [Let19], which establishes a quantum Cartan decomposition of QSP coideal subalgebras for semisimple Lie algebras g. Moreover, it was shown in [Kol20] that the finite-dimensional representations of B c,s form a braided module category over the braided monoidal category of finite-dimensional U q (g)-representations. It is well-known that this braided monoidal category encodes the structure of the universal R-matrix of U q (g). A similar categorical framework encoding the universal K-matrix of a quantum symmetric pair was given in [Wee19] under the name Z 2 -braided pair. In [JM11] the quantum analog of the classical symmetric pair (gl N , gl p × gl N −p ) is used to relate representations of (double) affine braid groups and (double) affine Hecke algebras of type C ∨ C n . Finally, generalizations of quantum symmetric pairs have been considered in [RV18], which replace the g-involution θ by a more general semisimple automorphism of g.
In this paper we will adopt the notational conventions of [Kol14]. We will write g = g(A) for the Kac-Moody algebra associated to a symmetrizable generalized Cartan matrix A of dimension n. We take I to be the set {0, . . . , n − 1}, such that we can write A = (a ij ) i,j∈I . We will use Kolb's definition of admissible pairs, as will be repeated later in Definition 2.1, to parametrize the involutive automorphisms of g of the second kind. To each such admissible pair one can associate a quantum symmetric pair and hence a coideal subalgebra B c,s of U q (g), which also depends on a multiparameter (c, s).
The QSP coideal subalgebras B c,s can be presented in terms of generators and relations. The set of generators depends on the choice of admissible pair, but always contains certain elements B i , with i ∈ I. A set of defining relations which describe these algebras abstractly in terms of their generators was given by Kolb in [Kol14,Theorem 7.1] and will be repeated in the upcoming Theorem C. One of these relations states that (1) can be written as a lower-degree polynomial in B i and B j which depends on the entry a ij of A. However, Kolb's theorem does not provide a precise form for this polynomial, which he denotes by C ij (c). Up to present, expressions for C ij (c) were only known for a few possible values of a ij , namely a ij ∈ {0, −1, −2, −3}.
These have been obtained in [Kol14] and [BK15] by explicit calculations, which follow similar results in [Let03] for finite-dimensional g. It was suggested by Kolb that the same rationale could lead to expressions for C ij (c) valid for all a ij , but this has not been explicitized before. This paper provides for the first time closed expressions for the polynomials C ij (c), valid without restrictions on the Cartan matrix or the admissible pair. It thereby completes the presentation of the quantum symmetric pair coideal subalgebras by generators and relations. Such a presentation is highly desirable in view of the representation theory of the algebras B c,s . This was already indicated in [BK15], where the definition of a new bar involution for quantum symmetric pairs was validated by showing that it respects the defining relations of B c,s . By the absence of such relations beyond the case |a ij | ≤ 3, this could only be done for a limited class of Cartan matrices. Our results allow to extend this to any quantum symmetric pair.
Our approach will be as follows. We will rewrite (1) as a complicated expression in U q (g) ⊗2 , where one of the tensor components is acted upon with a projection operator. This leads to the upcoming expressions (39) and (42), which were essentially already contained in [Kol14]. The main novelty of our approach lies in how we expand these expressions further. We will distinguish two cases, based on the behavior of i and j with respect to the admissible pair, each leading to a different expression for C ij (c). In Propositions 3.3 and 3.7 we will perform a binary distributive expansion to rewrite (1) as a polynomial which, in the first of these two cases, is of the form whereas in the second case one finds where the elements Z i , W ij and K j are well-defined in terms of the generators and where all sums are finite. The major difficulty lies in the determination of the coefficients ρ m,m ′ , ρ m,m ′ ,t and σ m,t , which we will refer to as the structure constants of the algebra B c,s . Initially, we will describe these in terms of monomials in U q (g) acted upon with a projection operator. Closed expressions for the actions of these projection operators and hence for the structure constants are consequently derived in Theorems 3.13 and 3.19. It may not surprise that the formulae we obtain there turn out to be rather computationally extensive. Indeed, even the expressions obtained in [Kol14] and [BK15] for small values of a ij , as displayed in the upcoming Tables 1, 2 and 3, were already quite intricate. Nevertheless, our formulae contain nothing but finite sums and products, which can easily be carried out either by hand or by a computer.
Many interesting examples of quantum symmetric pairs and their coideal subalgebras have appeared in the literature. The paper [AKR17] studies a QSP coideal subalgebra associated to the quantum group U q (su(3)). Its finite-dimensional irreducible representations are classified and its highest weight vectors are expressed in terms of dual q-Krawtchouk polynomials. For finite-dimensional g one can easily extend the g-involution θ to the loop algebra L(g) or the untwisted affine Lie algebra g. Quantum analogs of the fixed point Lie subalgebra of L(g) under this extended automorphism are known as twisted quantum loop algebras (of the second kind). These comprise the twisted q-Yangians of [MRS03] and the QSPs of type AIII of [CGM14]. A unified approach to the structure theory of these twisted quantum loop algebras is contained in [Kol14,Section 11]. Other remarkable examples of the subalgebras B c,s for non-affine g are the quantized GIM Lie algebras of [LT13], treated by Kolb in [Kol14,Section 12].
In this paper, we will draw special attention to the QSP coideal subalgebras in the split case, corresponding to the trivial admissible pair (∅, id). These are known as generalized q-Onsager algebras. Their name has been derived from the algebra defined by Onsager in [Ons44] as a tool towards his analytic solution of the planar Ising model in zero magnetic field. This algebra was presented in [DG82] and [Per89]  It has received much attention in special function theory and integrable lattice models [Dav90,KP03,HT07,Bax09,VOF19]. It can be embedded in the affine Lie algebra sl 2 as its subalgebra of fixed points under the Chevalley involution [Roa91], and hence together with sl 2 it forms a (split) classical symmetric pair. The theory of quantum symmetric pairs thus offers a solid framework to deform the Onsager algebra to a quantum algebra. The resulting q-Onsager algebra [Bas05,BK05] is abstractly defined by the q-Dolan-Grady relations where [A, B] q = qAB − q −1 BA is the q-commutator and ρ is a scalar depending on q. The q-Onsager algebra has become an important object of study in quantum integrability [Bas05,BF12,KP19] and in connection with q-orthogonal polynomials [BVZ17] and Leonard pairs [IT09]. Upon adding a defining relation in its equitable presentation, the q-Onsager algebra is refined to the Askey-Wilson algebra [Zhe91], as was shown in [Ter00]. A central extension of the latter, known as the universal Askey-Wilson algebra [Ter11], was also identified as a quotient of the q-Onsager algebra [Ter18]. This Askey-Wilson algebra provides an algebraic framework for the q-Askey scheme of orthogonal polynomials [BMVZ19,KM18], see also [DDV19, DD19, Gro19] for some recent multivariate generalizations. The left-hand side of (4) can be rewritten as 1}. This coincides with the expression (1) for n = 2 and a 01 = a 10 = −2, i.e. for g = sl 2 . It is hence apparent that the q-Onsager algebra coincides with the quantum symmetric pair coideal subalgebra B c,s of U q ( sl 2 ) for the trivial admissible pair and a special choice of the parameters c, s. Kac-Moody generalizations of the q-Onsager algebra were constructed by Baseilhac and Belliard in [BB10]. A presentation with generators and relations was given for affine Lie algebras g, again for a limited set of Cartan matrices. The relations we will derive in Theorem 3.13 extend this to symmetrizable Kac-Moody algebras without restrictions on the Cartan matrix. Moreover, we will use a recent result by Chen, Lu and Wang [CLW19] to obtain alternative, transparent expressions of quantum Serre type for the defining relations of these generalized q-Onsager algebras. These relations, which will be given in Theorem 4.7, turn out to hold even for any quantum symmetric pair provided the indices i and j satisfy the conditions of the aforementioned Case 1. In addition, these allow to prove symmetry properties of the structure constants ρ m,m ′ from (2).
For q = 1, such inhomogeneous Serre relations for generalized Onsager algebras had already been obtained by Stokman in [Sto19]. His classical generalized Onsager algebras extend those of [Roa91, UI96, DU04, NSS12] to arbitrary root systems. The defining relations he provides, involve a set of coefficients which are defined in a recursive fashion. Our approach now allows to derive closed expressions for these coefficients and thus solve the recursion relations, by taking the limit q → 1 of the analogous expressions in the quantum case. This will be performed in Theorem 4.14.
The paper is organized as follows. In Section 2 we recall the necessary prerequisites on quantum symmetric Kac-Moody pairs in the notation of [Kol14]. We treat the classical symmetric pairs (g, k) in Subsection 2.1 and their quantum analogs (U q (g), B c,s ) in Subsection 2.2. In Subsection 2.3 we state some of the results obtained by Kolb in [Kol14], which we will need in what follows. The main body of work is contained in Section 3, where the missing defining relations for B c,s will be derived. In Subsection 3.1 we will perform a binary distributive expansion to reduce the computation of the polynomials C ij (c) to an easier problem, namely determining the coefficients in (2) and (3) through the action of the counit and a certain projection operator on monomials in U q (g). This problem will be solved in Subsections 3.2 and 3.3 treating Cases 1 and 2 respectively. The principal results are presented in Theorems 3.13 and 3.19. To conclude, we will derive alternative and more accessible expressions for the polynomials C ij (c) in Case 1 based on the work [CLW19] in Theorem 4.7. Finally, we turn our attention to the generalized q-Onsager algebras and their classical counterparts. We repeat the obtained relations applied to the split case and reconsider them in the limit q → 1 to solve the recursion relations of [Sto19] in Theorem 4.14.

Construction of the generators
Let us start by recalling some crucial concepts and notations introduced in [Kol14].
Let K be an algebraically closed field of characteristic 0. Let A be an indecomposable generalized Cartan matrix of dimension n and let us denote by I the set {0, 1, . . . , n − 1}. This means that A = (a ij ) i,j∈I satisfies the properties i. a ii = 2, for all i ∈ I, ii. a ij ∈ Z − , if i = j ∈ I, iii. a ij = 0 ⇔ a ji = 0, for any i, j ∈ I iv. For every non-empty proper subset I ′ ⊂ I there exist i ∈ I ′ , j ∈ I \ I ′ such that a ij = 0. Moreover, we assume A to be symmetrizable, i.e. there exists a diagonal matrix D = diag(ǫ i : i ∈ I), with mutually coprime and nonzero entries ǫ i ∈ N, such that DA is symmetric.
In Subsection 2.1, we will construct the classical symmetric pair (g, b s ), where g = g(A) is the Kac-Moody algebra associated to A. This construction will motivate the definition of the quantum symmetric pair (U q (g), B c,s ) inside the corresponding quantum group U q (g), which will be given in Subsection 2.2.
be a minimal realization of A. This means that h is a K-vector space of dimension 2n − rank(A) and that Π ∨ and Π are linearly independent subsets of h and its dual h * respectively, subject to α j (h i ) = a ij for any i, j ∈ I. Let Q = ZΠ be the corresponding root lattice.
The Kac-Moody algebra g = g(A) associated to A is the Lie algebra over K generated by h and 2n Chevalley generators e i , f i with i ∈ I, with defining relations for all i, j ∈ I and h, h ′ ∈ h. Here we denoted by ad the adjoint mapping (7) ad : g → gl g : x → ad x, ad x : g → g : y → [x, y].
The derived Lie subalgebra g ′ = [g, g] of g is generated by h ′ = i∈I Kh i and the elements e i , f i with i ∈ I. As usual, we will write g β = {x ∈ g : [h, x] = β(h)x, ∀h ∈ h} for any β ∈ h * and Φ = {β ∈ h * : g β = {0}} for the corresponding root system.
For any i ∈ I we denote by r i ∈ GL(h) the fundamental reflection which acts on h ∈ h by The subgroup W of GL(h) generated by all such r i stands as the Weyl group of g. Via duality, W can also act on h * and hence in particular on Q, via for any α ∈ h * . Consider a subset X ⊆ I. Let g X be the corresponding Lie subalgebra of g, generated by the elements e i , f i and h i with i ∈ X. Write Φ X ⊆ Φ for its root system and ρ ∨ X for half the sum of the positive coroots of Φ X . We will write W X for the parabolic subgroup of the Weyl group W associated to X, and w X for its longest element. Finally, let us denote by Aut(A, X) the set of permutations σ of I subject to σ(X) = X and a σ(i),σ(j) = a ij , ∀i, j ∈ I.
Any σ ∈ Aut(A, X) extends to an automorphism of g by taking and defining the action of σ on h ∈ h \ h ′ as described in [KW92,Section 4.19]. Similarly, σ ∈ Aut(A, X) extends to an automorphism of Q upon setting This terminology allows to repeat the definition of an admissible pair, as given in [Kol14, Definition 2.3].
(2) The action of τ on X coincides with the corresponding action of −w X , i.e. for any j ∈ X one has h τ (j) = −w X (h j ) and α τ (j) = −w X (α j ), where we have used the interpretations of τ and w X according to (8)-(9)-(10). (3) For any i ∈ I \ X satisfying τ (i) = i, one has α i (ρ ∨ X ) ∈ Z. An important motivation for introducing admissible pairs is that they arise naturally as Kac-Moody generalizations of Satake diagrams [Ara62]. Moreover they parametrize the so-called involutive automorphisms of g of the second kind [Lev88,KW92] up to conjugation by elements of Aut(g), as was shown in [Kol14, Theorem 2.7]. The automorphism θ(X, τ ) corresponding to an admissible pair (X, τ ) can be constructed using the following four key concepts.
The first is the element τ ∈ Aut(A, X), interpreted as an automorphism of g according to (9). Furthermore, we will need the Chevalley involution ω ∈ Aut(g) given by for any i ∈ I and h ∈ h. Moreover, the longest element w X of W X can be lifted to an element m X of the Kac-Moody group of g ′ , with corresponding automorphism Ad(m X ) ∈ Aut(g). For details we refer to [KW92, Section 1.3] and [Kol14, Section 2].
Finally, one can define a group morphism s(X, τ ) : Q → K × from the root lattice Q to the multiplicative group K × , by if j ∈ I \ X and τ (j) > j, if j ∈ I \ X and τ (j) < j, where i ∈ K is a square root of −1. The corresponding automorphism Ad(s(X, τ )) ∈ Aut(g) is defined by for all h ∈ h and v ∈ g α , α ∈ Φ. These four ingredients can now be combined to yield the following involutive automorphism θ(X, τ ).
Definition 2.2. To each admissible pair (X, τ ) we associate the automorphism θ(X, τ ) of g given by It is an involutive g-automorphism of the second kind by [Kol14, Theorem 2.5].
Let us from now on fix an admissible pair (X, τ ) and write θ for the above defined automorphism θ(X, τ ). Then θ gives rise to an algebra which will be of special interest in this paper.
Definition 2.3. For any vector s = (s i ) i∈I\X ∈ K I\X we define b s = b s (X, τ ) to be the subalgebra of U (g ′ ) generated by the elements The couple (g, b s ) stands as the (classical) symmetric pair associated to the admissible pair (X, τ ).
In [Kol14], the algebra b s was denoted by U (k ′ ). We have chosen to adopt this alternative notation, to emphasize that b s is a classical counterpart of the quantum algebra B c,s , which we will define in Subsection 2.2. The defining relations of b s will then follow as a limit q → 1 of the quantum Serre relations for B c,s we will derive in Section 3.
2.2. The quantum case. Let q be an indeterminate, assumed not to be a root of unity in the field K. We denote by K(q) the field of rational functions in q.
Recall the matrix D = diag(ǫ i : i ∈ I) we have introduced above. For each i ∈ I we set q i = q ǫi . For any m ∈ N, we define the q i -number [m] qi and the q i -factorial [m] qi ! as with the convention that [0] qi ! = 1. For N, m ∈ N with N ≥ m, we define the q i -binomial coefficient as Like in the classical case, one has We will often use the following polynomial in two non-commutative variables x and y, which we will refer to as the quantum Serre polynomial: for i, j ∈ I we write with m ∈ Q. Notice that ad q m i reduces to ad defined in (7) in the limit q → 1, for any m ∈ Q. It can easily be shown by induction that one has for any A, B ∈ U q (g) and any r ∈ N, which, upon substituting r = 1 − a ij , becomes  Hence in the limit q → 1, the expression F ij (A, B) reduces to where a and b are the specializations of A and B respectively, and so (20) indeed translates to (6). A detailed account on this notion of specialization, which is a formal way to implement this limiting process q → 1, can be found in [DK90], [HK02, Sections 3.3 and 3.4] and [Kol14, Section 10].
The quantum group U q (g ′ ) has the structure of a Hopf algebra, with the following expressions for the coproduct ∆, the counit ǫ and the antipode S: Now let us once more fix an admissible pair (X, τ ). A quantum analog of the automorphism θ(X, τ ) defined in (14) can be built from five fundamental constituents, one of which is the mapping τ ∈ Aut(A, X) viewed as an automorphism of U q (g ′ ) by . Secondly, one can extend Ad(s(X, τ )) ∈ Aut(g) to an automorphism of U q (g) by Ad(s(X, τ ))(v) = s(X, τ )(α)v, for all v ∈ U q (g) α = {u ∈ U q (g) : K i u = q (αi,α) uK i , ∀i ∈ I}, α ∈ Q. Here by (·, ·) we denote the bilinear form on h * satisfying (α i , α j ) = ǫ i a ij .
Next, we will need a q-deformation of the Chevalley involution (11), which we will again denote by ω. It is given by and classifies as a coalgebra antiautomorphism of U q (g ′ ).
To obtain a quantum analog of the element Ad(m X ) in (14) one needs the Lusztig automorphisms T i , i ∈ I, which appeared in [Lus94, Section 37.1] under the name T ′′ i,1 . Let w X = r i1 r i2 . . . r i k be a reduced expression for the longest element w X of the parabolic subgroup W X of W , then denote by T wX the corresponding Finally, define another automorphism ψ : These are all the tools needed to q-deform θ(X, τ ).
Definition 2.4. To each admissible pair (X, τ ) we associate the automorphism θ q (X, τ ) of U q (g ′ ) given by Note that θ q = θ q (X, τ ) is no longer involutive. Finally, let us denote by Q Θ the set {α ∈ Q : −w X τ (α) = α}. Here, we interpret both τ ∈ Aut(A, X) and w X ∈ W X as automorphisms of Q according to (8) and (10). Moreover, if β = i∈I m i α i ∈ Q, we will write K β for i∈I K mi i . This brings us to the definition of the quantum analog B c,s of the algebra b s defined in (15).
Definition 2.5. For any vector c = (c i ) i∈I\X ∈ (K(q) × ) I\X and s = (s i ) i∈I\X ∈ K(q) I\X , we define B c,s = B c,s (X, τ ) to be the subalgebra of U q (g ′ ) generated by the elements When applying the coproduct ∆ described in (22) on the generators (24), one can make the following observation.
Upon comparing (24) with (15) in the light of Remark 1, it is immediately clear that B c,s is a q-deformation of the algebra b s under certain conditions on the parameters c i and s i , and that it reduces to the latter under the specialization q → 1. The precise conditions are described in the following theorem. Moreover, let us assume that the vector (c, s) is specializable, i.e. lim q→1 (c i ) = 1 for any i ∈ I and all c i , s i lie in the localization K[q] (q−1) of the polynomial ring K[q] with respect to the ideal generated by q − 1. Then B c,s reduces to the algebra b s under the formal specialization q → 1 and is maximal with this property.
Although the assumption of specializability is required to obtain b s as an exact limit of B c,s for q → 1, it is still commonly accepted to view B c,s as a quantum analog of b s even if the latter condition is not fulfilled. Hence Proposition A suggests the following terminology.
Definition 2.6. For (c, s) ∈ C × S, the algebra B c,s is called a quantum symmetric pair coideal subalgebra.
Throughout the rest of this paper, we will fix a vector of parameters (c, s) ∈ C × S and work with the corresponding quantum symmetric pair coideal subalgebra B c,s .

Kolb's projection technique.
In this section, we repeat some of the results obtained by Kolb in [Kol14], which we will use in Section 3 to derive the defining relations of the quantum symmetric pair coideal subalgebras B c,s . For ease of notation, we will write M + X and U 0 Θ ′ for the subalgebras of U q (g ′ ) generated by the sets {E i : i ∈ X} and {K β : β ∈ Q Θ } respectively, and set B j := F j for j ∈ X. Let U + , U − and U 0 ′ be the subalgebras of U q (g ′ ) generated by Let us also denote by J i,j the set of multi-indices given by With this notation, one can write down the following theorem.
Theorem C ([Kol14, Theorem 7.1]). For any distinct i, j ∈ I there exist elements can be expressed as a polynomial in B i and B j of smaller total degree with coefficients in M + X U 0 Θ ′ , possibly depending on c but not on s. Moreover, the algebra B c,s is abstractly defined by the relations together with the relations (18) and (20).
Our main goal in this paper will be to find explicit expressions for these lower degree polynomials C ij (c), which, up to present, had not been written down in general. A few special cases had however already been treated by Kolb.
Another case was treated by Balagović and Kolb in [BK15]. It requires us to introduce some more notation. We will denote by ad the left adjoint action of U q (g) on itself: for every x, u ∈ U q (g) one has where we have used the Sweedler notation, i.e. ∆(x) = x (1) ⊗ x (2) . It is not to be confused with the adjoint map of the Kac-Moody algebra g, which we have introduced in (7) under the same notation. Recall also the notation T wX for the product of Lusztig automorphisms corresponding to a reduced expression of w X .

Lemma E ([Kol14, equation (4.4), Theorem 4.4]).
For any i ∈ I \ X there exists a monomial For any i ∈ I \ X we may now define and v i are as defined in Lemma E. It follows immediately from (30) and the expression (22) for Furthermore, we will use the notation This enables us to state the following theorem by Balagović and Kolb.
By Theorems D and F, it only remains to compute C ij (c) in 2 cases, namely Case 1: i ∈ I \ X, j ∈ I \ X and τ (i) = i, Case 2: i ∈ I \ X, j ∈ X and τ (i) = i. These cases turn out to be remarkably complicated. In [Kol14] and [BK15], explicit calculations have led to expressions for C ij (c) for a ij ∈ {0, −1, −2, −3} in Case 1 and for a ij ∈ {0, −1, −2} in Case 2, but no attempt has been made to write down relations valid without restrictions on a ij . In Section 3, we will derive such relations for both cases. As could be expected from the above mentioned calculations, these expressions will be rather intricate, but nevertheless easily computable, as they involve only finite sums and products of elements of K(q).
The key tool to obtain such relations is the projection P −λij introduced by Kolb. The classical triangular decomposition for quantum groups can be deformed to where the isomorphism is given by multiplication, and consequently denote the corresponding projection with respect to the decomposition (33), where Then one can prove the following statements.
Lemma G ([Kol14, equation (5.14)]). P −λij is a homomorphism of left U q (g ′ )-comodules: Proposition H ([Kol14, Proposition 5.16]). For any distinct i, j ∈ I one has Combining Lemma G, Proposition H and the fact that ∆ is an algebra morphism, we find that Since K −λij is invariant under P −λij and sent to 1 by the counit ǫ, the expression (36) asserts Our main purpose in Section 3 will be to expand the right-hand side of (37) as a polynomial in To do so, we will need an expression for the ∆(B i ) and ∆(B j ) in (37). These follow from the following lemma.
Note that the formulation of this lemma is somewhat stronger than the original one in [Kol14], but one readily verifies the correctness of this extra restriction on Υ i upon computing ∆(ad(Z + i )(E i )). Finally, let us note that the following relations follow immediately from (28)-(29).
Lemma J. Let i ∈ I \ X be such that τ (i) = i, then for any j ∈ I \ X one has

Quantum Serre relations for the algebras B c,s
We are now ready to derive closed expressions for the quantum Serre relations (27) by expanding the right-hand side of (37). Crucial in this respect is the presence of the morphism id ⊗ ǫ, which by (22) tells us that no term containing a nontrivial element of U + U − in the second tensor component will survive in (37). This allows us to eliminate some of the terms in (38).
We will first focus on Case 1.
Proof. First, let us note that the polynomial F ij is of degree 1 and hence linear in its second argument. Since j / ∈ X, the expression (38) for ∆(B i ) contains no factors E j or F j in its second tensor component. Since ǫ(E j ) = ǫ(F j ) = 0, the expression for ∆(B j ) obtained from Lemma I, together with the relation (37), asserts . When expanding ∆(B i ) according to (38), there will be no contribution from the two latter terms with k ∈ X, since each term in (41) contains at least one factor E k ′ , k ′ ∈ X, in its second tensor component, and again ǫ( The same simplification can be performed for Case 2. Lemma 3.2. Let i ∈ I \ X be such that τ (i) = i and let j ∈ X. Then one has Proof. Since j ∈ X, we have B j = F j . Hence it follows from (37), (22) and the linearity of F ij in its second argument that and Υ i will not contribute, since each of their terms contains at least one factor E k with k = i in the second tensor component, and ǫ(E k ) = 0. For the second occurrence of ∆(B i ), the situation is different. The term will effectively contribute, since when expanding j and the last term in this expansion will turn out to be significant, as will be explained in what follows. The term Υ i in ∆(B i ) however, will still not contribute. Indeed, each term in Υ i contains either a factor E 2 j or a factor E k with k ∈ X \ {j}, and both F j E 2 j and F j E k cannot be expanded to yield a non-vanishing term under ǫ. This proves the claim.
One observes immediately that the right-hand side of (42) equals the right-hand side of (39), added with a second term. In what follows, we will treat both terms separately and thereby obtain explicit expressions for each of the two cases.
3.1. Binary expansions. In this section, we will expand the right-hand sides of (39) and (42). We will first treat the right-hand side of (39), which occurs in (42) as well and which can, to a large extent, be rewritten irrespective of whether or not j lies in X. The second term, which appears only in (42), i.e. for j ∈ X, will be addressed afterwards.
Our main strategy will be to perform a "binary" distributive expansion, which requires summation over binary tuples ℓ ∈ {0, 1} N , N ∈ N. For any such tuple ℓ, we will use the notation Throughout this paper, we will often meet finite sums and products over natural numbers. We will use the convention that a sum vanishes if its lower bound exceeds its upper bound or equivalently if it ranges over the empty set, whereas a product reduces to one in this situation. Otherwise stated, for any function a of r and any M > N we take N r=M a(r) = r∈∅ a(r) = 0 and N r=M a(r) = r∈∅ a(r) = 1. Note also that in our convention 0 is a natural number, i.e. 0 ∈ N, and 0 0 = 1. Proposition 3.3. Let i ∈ I \ X be such that τ (i) = i and let j ∈ I be distinct from i. Then one has where The rationale of this expansion is that for ℓ r = 1, we get the contribution of B i ⊗ K −1 i , for ℓ r = 0 and s r−|ℓ|1;r = 1 we find 1 ⊗ F i , whereas for ℓ r = 0 and s r−|ℓ|1; i . Note that the indexation of the s-variables was chosen in such a way that there is only a summation over these variables in case the corresponding ℓ r = 0. Indeed, if we were to write s r instead of s r−|ℓ|1;r and sum over all s 1 , . . . , s 1−aij −k ∈ {0, 1}, then the terms corresponding to ℓ r = 1 would contribute twice.
Since B i commutes with Z i by Lemma J, we obtain with T i ℓ,s,r as in (46). Performing a similar expansion for the term as in (45). It remains only to observe that the term corresponding to |ℓ| = 1 − a ij , i.e. ℓ = (1, 1, . . . , 1), yields Many of the s in the sum in (44) will have a vanishing contribution. One can make the following observation.
Lemma 3.4. Let i ∈ I \ X be such that τ (i) = i and let j ∈ I be distinct from i. Let ℓ ∈ {0, 1} 1−aij with |ℓ| = 1 − a ij , s ∈ {0, 1} 1−aij−|ℓ| and k ∈ {0, . . . , 1 − a ij }. Then one has if one of the following conditions is fulfilled: (a) a ij + |ℓ| is even, Proof. To acquire the action of ǫ • P −λij on p in a standard ordering, namely as a K(q)-linear combination of elements of the form We may do so by applying the U q (g ′ )-relations (18)-(19). Each such element will be projected to either itself or 0 by P −λij . But when applying ǫ, such a term can only survive if N 1 = N 2 = 0, by (22). Suppose now s is such that p (i,j,aij ) ℓ,s,k contains an unequal number of factors F i and E i . Then each term in its standard ordering will still contain an unequal number of factors F i and E i , as follows from (19). Hence the standard ordering will consist of terms If a ij + |ℓ| is even, then the total number of factors in p (i,j,aij ) ℓ,s,k with ℓ r = 0 will be odd and hence the number of factors F i and E i in p (i,j,aij ) ℓ,s,k will always be unequal, for any s. Finally, suppose p ∈ {1, . . . , 1 − a ij − |ℓ|} is such that |s| 1;p < p 2 . This means that up to position p, the number of factors E i will exceed the number of factors F i . As the difference between these numbers is not altered by the relation (19), this means that the standard ordering of the corresponding term will consist only of terms This result will help us to simplify the notation used in Proposition 3.3. Indeed, by Condition (b) in Lemma 3.4, we know that for any (ℓ, s, k) contributing non-trivially to (44), we have and hence also Moreover, we will need the notion of the even and an odd part of an integer number d ∈ Z, denoted by d e and d p respectively, and defined as Note that for any d ∈ Z one has d = 2d e + d p .
This will now help us to rewrite C ij (c) for Case 1.
Corollary 3.5 (Case 1). Let i ∈ I \ X be such that τ (i) = i and let j ∈ I \ X be distinct from i. Then one has as in (45) and Proof. Upon combining Lemma 3.1, Proposition 3.3, the equation (47) and the fact that We can restrict the sum over ℓ to one over L m,m ′ ,k , by setting This requires an additional summation over m and m ′ . A priori, we have m ) will vanish for any s, by Condition (a) of Lemma 3.4. This explains the presence of (a ij + m + m ′ ) p in (51) and the fact that in the sum in (50) we restrict to m + m ′ ≤ −1 − a ij . Note also that the requirements (53) imply that Similarly, the sum over s may be restricted to S m,m ′ by Conditions (b) and (c) of Lemma 3.4.
For Case 2, the first line of the right-hand side of (42) is identical to the right-hand side of (39), and hence the first part of C ij (c) can be expanded as above. Nevertheless, we have to take into account that in this case Z i and B j no longer commute, which effects our notation.
Corollary 3.6 (Case 2). Let i ∈ I \ X be such that τ (i) = i and let j ∈ X. Then one has where as in (45), L m,m ′ ,k as in (52) and Proof. Upon combining Lemma 3.2, Proposition 3.3 and the equations (47) and (48), we obtain The sum over ℓ can be restricted to L m,m ′ ,k , with an additional summation over m, m ′ , just like in the proof of Corollary 3.5. Setting t ℓ,s,k equal to a parameter t, over which we sum as well, determines the condition as follows from (57). This restriction, together with Conditions (b) and (c) of Lemma 3.4, determines the definition of S m,m ′ ,k,t .
We will now perform a similar binary expansion for the last line of (54).
Proposition 3.7. Let i ∈ I \ X be such that τ (i) = i and let j ∈ X. Then one has where with T i ℓ,s,r as in (46). Proof. By the definition (17) of F ij , the left-hand side of (58) can be written as In the term ( does not need to be taken into account. Indeed, the standard ordering of the expansion with respect to this term would consist of terms with M ≥ 1. But of course each such term vanishes under ǫ. In the term ( it does need to be taken into account. More precisely, in the whole sum we obtain when expanding the k-th power, each term must contain exactly one to obtain a non-zero contribution. Indeed, if we were to take more than one such factor, then we would end up with a normal ordering consisting of terms of the form (60) with M ≥ 1 and with M ≥ 1, which again disappear under ǫ, whereas if we were to take 0 such factors, then we would find in the normal ordering, which also yields 0 under ǫ by the presence of F j . This also explains why we can The claim now follows upon expanding binarily the powers of i , as in the proof of Proposition 3.3. Note that this time, we will need a total of 1 − a ij − k + d + (k − d − 1) = −a ij variables ℓ r . Observe also that we have used Lemma J to obtain the factor q aij |ℓ| 1;1−a ij −k+d i . Once more, many of the s in the sum in (58) will not contribute. In analogy to Lemma 3.4, one can formulate the following result.
As before, this means that we can determine Just like in the previous situation, this now leads to a complete description of C ij (c) in Case 2.
Corollary 3.9 (Case 2). Let i ∈ I \ X be such that τ (i) = i and let j ∈ X. Then one has with ρ where Proof. This follows from Corollary 3.6, Proposition 3.7 and the equations (61) in exactly the same fashion as we have derived Corollaries 3.5 and 3.6, i.e. upon setting Again, |ℓ| = m cannot equal −a ij , since then a ij + m would be even, which is excluded by Condition (a) in Lemma 3.4. So m runs from 0 to −1 − a ij . It follows immediately that m ′ runs from 0 to m. The conditions in Lemma 3.8 determine the definition of S ′ m,m ′ ,k,t,d .
The relations we have obtained in Corollaries 3.5 and 3.9 comply with the explicit calculations performed in [Kol14] and [BK15] by Balagović and Kolb. They also obtained explicit values for the structure constants for a limited set of possible a ij : they computed ρ   The main purpose of this paper will be to find expressions in K(q) for the structure constants ρ , valid without any restrictions on a ij . By Corollaries 3.5, 3.6 and 3.9, this amounts to deriving how ǫ • P −λij acts on p (i,j,aij) ℓ,s,k and r (i,j,aij ) ℓ,s,k,d . This computation will be performed in the next two subsections.
3.2. Case 1: τ (i) = i ∈ I \ X and j ∈ I \ X. Let us now fix i ∈ I \ X such that τ (i) = i and j ∈ I distinct from i. A priori, we don't specify whether or not j ∈ X. Let us also fix m, m ′ ∈ N such that a ij + m + m ′ is odd and m + m }, ℓ ∈ L m,m ′ ,k and s ∈ S m,m ′ ,k,t . Note that this automatically implies that s ∈ S m,m ′ , by (52) and (56). Hence by (51) and (55) it suffices to compute the action of ǫ • P −λij on p (i,j,aij ) ℓ,s,k , defined in (45), in order to obtain the full polynomial C ij (c) for Case 1, as well as the first of the two parts of this polynomial for Case 2. This computation will now be performed. Let us introduce the notation P i N , with N ∈ Z, for the projection operator with respect to the decomposition (33). Let us also renormalize the element E i as Then we can state the following result.
Proof. As argued in the proof of Lemma 3.4, the total number of factors F i and E i in p (i,j,aij ) ℓ,s,k must be equal and must yield When shifting the factor K −1 j through the second term between brackets in (45) using (18), we will induce a factor q −aij x i , with By (70) this is reduced to We will perform the same shifting process for the factors K −ℓr i with ℓ r = 1. For each such r this will induce a factor q −2x ′ i with Applying the same reasoning as in (71)-(72), we obtain x ′ = ζ (r−1) ℓ,s , such that one can write Finally, we will do the same for the K −1 i occurring in a factor (E i K −1 i ) (1−ℓr)(1−s r−|ℓ| 1;r ) with ℓ r = 0 and s r−|ℓ|1;r = 0. This will give rise to a factor q −2x ′′ i with x ′′ = #(factors E i succeeding (E i K −1 i ) (1−ℓr )(1−s r−|ℓ| 1;r ) )−#(factors F i succeeding (E i K −1 i ) (1−ℓr )(1−s r−|ℓ| 1;r ) ). The same reasoning now shows that x ′′ yields where the extra −1 comes from the E i inside (E i K −1 i ) (1−ℓr )(1−s r−|ℓ| 1;r ) . The total power of q i we can put in front hence becomes where we have applied (47), and with β ℓ,s,k as in (68).
Finally, we will perform the renormalization E i = (q i − q −1 i )E i , which, again taking into account the formula (70), leads to (73)
We have now reduced the computation of (ǫ • P −λij )(p (i,j,aij ) ℓ,s,k ) to a simpler problem, namely computing acts on a product of an equal number of factors F i and E i , which is balanced in the sense that up to each position in the product, the number of factors F i exceeds or equals the number of factors E i , as imposed by Condition (c) of Lemma 3.4. This action can be deduced from the following lemma. We will need the notation (N ) q 2 i for the modified q 2 i -number Note that it relates to the ordinary q i -number as Lemma 3.11. Let M ∈ N be such that M ≥ 1. Let Y ∈ U q (g ′ ) be a product of M factors F i and M factors E i , appearing in any order, but with F i as the first factor. Let N ∈ N be maximal such that the first N factors of Y are F i , such that we can write Y = F N i E i X, for some X ∈ U q (g ′ ). Then we have Proof. We will prove this by induction on N . Our strategy will be to rewrite Y in its standard ordering, i.e. as a K(q)-linear combination of terms of the form E i m1 F m2 i K m3 i , and then observe that for any M ′ ∈ Z one has The first term will have a standard ordering consisting of terms E i m1 F m2 i K m3 i with m 1 ≥ 1, which will all be killed by ǫ. For the second term, observe that X contains M − 1 factors F i and the same number of factors E i , since N = 1. Each factor F i , when taken together with a factor E i , can contribute at most one factor K −1 i by (19). Hence the lowest possible power of K i occurring in the normal ordering of K i X will be 1 − (M − 1) = −M + 2 > −M . Hence the second term will not contribute either. For the third term, we have K −1 i X = XK −1 i , since X contains an equal number of factors F i and E i . So we have , in agreement with (75). Now suppose the claim has been proven for N − 1 ≥ 1, then we have As before, the second term will not contribute: the coefficient of K −M i in its standard ordering will vanish, as the lowest power of K i that can occur will again be −M + 2. Consider now the third term in this sum. When shifting K −1 i through X, we will induce a factor q −2x Note that F N −1 i E i X ′ still contains M factors F i and M factors E i , and has F i as its first factor, since N − 1 ≥ 1. Hence we may apply the induction hypothesis to write The statement now follows from F N −2 i X ′ = F N −1 i X and upon observing that Let once more Y, X ∈ U q (g ′ ) and M, N ∈ N be as in the statement of Lemma 3.11. As already observed, the element F N −1 i X is again of the type described in Lemma 3.11: it is a product of M − 1 factors F i and the same number of factors E i , and has F i as its first factor, provided X has F i as its first factor or N − 1 ≥ 1. If N ′ ≥ N − 1 is the maximal number such that the first N ′ factors of F N −1 i X are F i , then we may write Consequently, Lemma 3.11 asserts and thus This process will only terminate if at some position p in the product, the number of factors E i preceding p exceeds the number of factors F i preceding p. In that case, we would at some point be left with N ′ = 1 and a corresponding X ′ starting with E i instead of F i .
Let us now assume that this is not the case, i.e. up to each position p in Y , the number of factors F i preceding p exceeds or equals the number of factors E i preceding p. Then this process of applying Lemma 3.11 consecutively will continue until we have applied it M times and we have reached N ′ = 1 and X ′ = 1, and of course (ǫ • P i 0 )(1) = 1. Each factor E i can now be assigned a level, which is the exponent N of F i that will occur in front of E i at the moment this factor is cancelled when applying the formula (75) in this consecutive process. Then our reasoning in fact asserts where the product runs over all factors E i in Y . Now note that each application of the formula (75) cancels one factor F i and one factor E i , hence each F i is in fact coupled to exactly one factor E i . Thus instead of running over all E i in Y , we might as well run over all factors F i in Y and assign to each F i a level, which equals the level of the E i to which it is coupled. We find Now say the element Y contains a factor E i at position p in the product, which, in the process above, is coupled to a factor F i at position r, with of course r < p. From the definition, it follows that the level of the E i at position p is the total number of factors F i preceding it, minus the number of factors E i preceding it, again since each application of (75) kills one F i and one E i . So where the +1 comes from the F i at position r itself. Moreover, we have that (78) #(factors F i between r + 1 and p − 1) = #(factors E i between r + 1 and p − 1).
Indeed, suppose not, then after coupling all possible E i between positions r + 1 and p − 1 with an F i , there would either still be F i 's left, hence position p would be coupled to some position r ′ > r, or else there would still be E i left, so position r would be coupled to p ′ < p. Inserting (78) into (77), we obtain (79) level(F i at position r) = #(factors F i preceding r) + 1 − #(factors E i preceding r).
Let us now return to the statement of Proposition 3.10. The element is a product of an equal number of factors F i and E i , namely where we have applied (47). Moreover, at each position p in Y ℓ,s , the number of factors F i preceding p, i.e. |s| 1;p , exceeds or equals the number of factors E i preceding p, i.e. p − |s| 1;p , by Condition (c) of Lemma 3.4. Hence the formula (76) is applicable.
Combining this with (80), we immediately obtain the following result.
Corollary 3.12. For i, j, m, m ′ , k, ℓ and s as fixed before, one has ℓ,s as in (69).
Alternatively, one can also iterate over the factors E i in Y ℓ,s rather than the factors F i , as initially established. Since (77) implies level( E i at position p) = ζ (p−1) ℓ,s , we also have . This formula will be of use in Subsection 3.3.
Corollaries 3.5 and 3.12 and Proposition 3.10 now lead to an explicit expression for the structure constants ρ (i,j,aij ) m,m ′ for Case 1.
Theorem 3.13 (Case 1). For any i ∈ I \ X such that τ (i) = i and any j ∈ I \ X distinct from i, one has where the structure constants are given by where with L m,m ′ ,k and S m,m ′ as in (52) and ζ (r) ℓ,s as in (69). Proof. This follows upon combining Corollary 3.5, Proposition 3.10 and Corollary 3.12. Note that for each k, ℓ and s, the exponent of q i becomes Similarly, this leads us to an explicit expression for the structure constants ρ (i,j,aij ) m,m ′ ,t for the first part of C ij (c) for Case 2.
Corollary 3.14. Let i ∈ I \ X be such that τ (i) = i, j ∈ X and m, m ′ and t as fixed before. Then the structure constants ρ Proof. This follows upon comparing (51) with (55).
It can readily be checked that these expressions comply with the values computed in [Kol14] and [BK15], as displayed in Tables 1 and 2. 3.3. Case 2: τ (i) = i ∈ I \ X and j ∈ X. Consequently, we will obtain the second part of the polynomial C ij (c) for Case 2, as described by the last line of (62). To this end, let us fix i ∈ I \ X such that τ (i) = i, j ∈ X, m ∈ {0, . . . , −1−a ij }, t ∈ {0, . . . , ℓ,s,k,d , defined in (59). This will be the subject of the present subsection.
As a first step, we will again shift all factors K −1 i in r (i,j,aij ) ℓ,s,k,d to the back, as we have done for p (i,j,aij ) ℓ,s,k in Proposition 3.10. Recall the notation E i = (q i − q −1 i )E i and let us write, as an extension of (65), (84) , for the projection operator with respect to the decomposition (33), where M, N ∈ Z.
Proposition 3.15. For i, j, m, t, k, d, m ′ , ℓ and s as fixed before, we have with ζ (r) ℓ,s as in (69).
Proof. Let us start by shifting the factor K −1 Reasoning as in the proof of Proposition 3.10, this induces a factor q −2x i . Now let us perform the same shifting for the factors K −ℓr i with ℓ r = 1, which leads to a factor q x ′ r i , where this time x ′ r depends on r. In general, we have again since r (i,j,aij ) ℓ,s,k,d contains an equal number of factors F i and E i , and precisely 1 factor F j and 1 factor E j . For r ∈ {1, . . . , 1 − a ij − k} we have For r ∈ {2 − a ij − k, . . . , 1 − a ij − k + d} on the other hand, by the same reasoning this becomes where the −1 arises from the factor E i in E j E i − q aij i E i E j . Finally, this shifting process for the factor K −1 i in (E i K −1 i ) (1−ℓr )(1−s r−|ℓ| 1;r ) with ℓ r = 0 and s r−|ℓ|1;r = 0 induces a factor q x ′′ r i , with, reasoning as above, for r ∈ {2 − a ij − k + d, . . . , −a ij }.
In total, this shifting gives rise to a factor q η i , with in agreement with (88), where we have used (61) and the definition (64) of S ′ m,m ′ ,k,t,d . Finally, the renormalization (66) gives rise to a factor (q We have hence reduced our problem to computing how ǫ • P i,j ℓ,s,k,d . Each of the latter terms is a product of an equal number of factors F i and E i and precisely one factor F j and E j , which is balanced in the sense that up to each position in the product, the number of factors F i exceeds or equals the number of factors E i , and that the factor F j preceeds the factor E j . The presence of F j and E j now complicates matters substantially in comparison to the situation in Case 1, because F i does not commute with F j and similarly for E i and E j . We will need to derive an analog of Lemma 3.11 which takes into account the presence of these factors. Recall the notation (N ) q 2 i for the modified q 2 i -number (74) and let us also define for M, N ∈ N. Write also α N = 0 for N < 0. Then one can prove the following result.
Lemma 3.16. Let M ∈ N be such that M ≥ 1. Let Y ∈ U q (g ′ ) be a product of M factors F i , M factors E i , 1 factor F j and 1 factor E j , appearing in any order but with F i as its first N 0 factors, for some N 0 ∈ N, followed by a factor F j . Let N 1 ∈ N be maximal such that the first N 1 factors of Y succeeding F j are F i , so that we can write Y = F N0 i F j F N1 i E i X, for some X ∈ U q (g ′ ). Then we have Proof. We prove this by induction on N 1 . As before, our strategy will be to write Y in its standard ordering, with m 1 , . . . , m 4 ∈ N, m 5 ∈ Z, δ ∈ {0, 1} and δ ′ ∈ {−1, 0, 1}, and then observe that for any M ′ ∈ Z one has by (84) and (22).
in its standard ordering. For A straightforward generalization of Lemma 3.11 then asserts which yields the claim since X ′ = F j X and by the definition (89) of α N0 and the fact that γ N0,N0 = 0. Suppose now the claim has been proven for N 1 − 1 ≥ 0. Note that by (19) and (66) we have The second term will not contribute, since its standard ordering cannot contain a multiple of K −M i K −1 j . Indeed, this term contains M − 1 factors F i and the same number of factors E i , and each factor F i can only contribute one factor K −1 i to the normal ordering upon combining it with a factor E i , by (19). Hence the lowest possible power of K i occurring in the standard ordering of this term will be −(M − 1) + 1 > −M .
The third term contains again as many F i as E i and can whence be rewritten as q As F N0 i F j F N1−1 i E i X ′ still contains M factors F i and the same number of factors E i , and meets all other requirements of the statement as well, we may apply the induction hypothesis to write The statement now follows from X ′ = F i X and the fact that The formula obtained in Lemma 3.11 can easily be iterated, since its right-hand side consists of only one term, leading to a product iteration of the form (76). The formula obtained in Lemma 3.16 however, is much more complicated, since its right-hand side consists of two different terms, each containing a projection operator and the counit ǫ. One iteration of Lemma 3.16 hence leads to a right-hand side containing three terms.
A second iteration will then lead to four terms in the right-hand side and so on. Meanwhile, the occurring coefficients become increasingly intricate at each further iteration. To describe the full outcome after T iterations, for any T ∈ N, let us introduce the notation where a < b ∈ N, N = (N 0 , N 1 , . . . , N b+1 ) ∈ N b+2 and |N | 0;p = N 0 + N 1 + · · · + N p . We also set c (a) a,N = 1. Proposition 3.17. Let M ∈ N be such that M ≥ 1. Let Y ∈ U q (g ′ ) be a product of M factors F i , M factors E i , 1 factor F j and 1 factor E j , of the form for some X ∈ U q (g ′ ), where N = (N 0 , N 1 , . . . , N T ) ∈ N T +1 and T ≥ 1. Then we have Proof. We will prove this by induction on T . For T = 1 we have Y = F N0 i F j F N1 i E i E j X and so its follows from Lemma 3.16 that where we have used the fact that F i and E j commute. When rewriting F j E j in its standard ordering via only the last term will contribute by (91), so we may replace F j E j by K −1 j in the equation above. Since

Hence it follows from (65) and (84) that
which agrees with the claim since c (0) 0,(N0,N1) = 1 and T − |N | 1;T − 1 ≤ 0 for T = 1. Suppose now the claim has been proven for T ≥ 1 and set Then Lemma 3.16 asserts Both Y ′ and Y ′′ satisfy the requirements of the statement: they each contain M − 1 factors F i , the same number of factors E i , 1 factor F j and 1 factor E j , and they are of the form (93) with respectively. Both N 0 − 1 and N 1 + N 2 − 1 might become negative, but in this case the corresponding coefficients α N0 and γ N0,N0+N1 will vanish. We may hence assume that N ′ , N ′′ ∈ N T +1 and apply the induction hypothesis to obtain where Θ N ′ ,N ′′ is given by where we have used the fact that |N ′ | 0;T − T = |N ′′ | 0;T − T = |N | 0;T +1 − (T + 1). It now suffices to show that Upon replacing the summation index u in the first line in (94) by u ′ = u + 1, which we thereafter rename to u again, this term becomes and it is immediate that α N0 u−2 r=0 α N0−1−r = u−1 r=0 α N0−r . Replacing the first line of (94) by (96) and separating the term corresponding to u = 0 in the second line and the one with u = T in the first line, we with T,N = 1, such that the last line in (97) agrees with the term in the right-hand side of (95) corresponding to u = T . Hence it suffices to prove the following two claims: where we have used the fact that N ′ 0 = N 0 − 1. Now since for every occurring r one has r ≤ T − u and p r ≤ u − 1, we have that T − p r − r ≥ 1 and hence |N ′ | 0;T −pr −r = |N | 0;T −pr −r+1 − 1, such that It is evident that one has Hence (100) implies where in the last line we have separated the factor in the product corresponding to r = T − u, since here we have set p T −u = u. One immediately recognizes the first line as c (T ) u,N , and moreover one has Again every T − p r − r ≥ 1 and hence |N ′′ | 0;T −pr−r = |N | 0;T +1−pr−r − 1, such that (102) coincides with the sum in the last line in (101). We conclude that and so we have shown (99). This concludes the proof.
The question now remains how one can apply Proposition 3.17 to compute the action of ǫ ℓ,s,k,d , as defined in (85)-(86). This will be addressed in the following proposition.
This agrees with (105)-(106). The total number T of factors E i in ℓ,s,r can be found as . . , 1 − a ij − k + d} such that ℓ r = 0 and s r−|ℓ|1;r = 1) in agreement with (103). With these notations one may now write for some N T ℓ,s,k,t +1 ∈ N, where we have used the fact that [F i , E j ] = 0. The analogous term with λ = 1 becomes for the same unknown N T ℓ,s,k,t +1 ∈ N. By Proposition 3.17 we thus have, for λ = 0 Here we have observed that Proposition 3.17 is only applicable for T ≥ 1, which explains the power δ T ℓ,s,k,t ,0 . The analogous term with Note also that we have , which determines the unknown N T ℓ,s,k,t +1 , in agreement with (107). By Lemma 3.11 this also implies It now remains only to apply the formula (81) and find With this result, we now have all necessary tools in hand to write down the polynomial C ij (c) for Case 2.
Theorem 3.19 (Case 2). For any i ∈ I \ X such that τ (i) = i and any j ∈ X, one has with ρ (i,j,aij ) m,m ′ ,t as obtained in Corollary 3.14 and Here we have used the notations (64) Proof. This follows upon combining Corollary 3.9 and Propositions 3.15 and 3.18, after expanding η ℓ,s,k,d,t,m ′ using (64) and observing that These expressions for the structure constants σ (i,j,aij ) m,t comply with the values computed in [Kol14], as displayed in Table 3. Moreover, Theorems 3.13 and 3.19 and Corollary 3.14 make it possible to compute the structure constants for higher values of |a ij |. For example, it follows from Theorem 3.13 that for a ij = −4 one has if i, j ∈ I \ X are distinct such that τ (i) = i, where the structure constants ρ (i,j,−4) m,m ′ are given in Table 4. Similarly, for a ij = −3, i ∈ I \ X with τ (i) = i and j ∈ X one has m,m ′ ,t and σ (i,j,−3) m,t as in Tables 5 and 6. (1, 1)  for a ij = −3 Theorems 3.13 and 3.19, together with the previously obtained Theorems C, D and F now yield a complete set of defining relations for the quantum symmetric pair coideal subalgebras B c,s .

Alternative expressions for Case 1
In this section, we will derive alternative expressions for the polynomial C ij (c) in Case 1. We will start from a result by Chen, Lu and Wang. In [CLW19] these authors provide defining relations of q-Serre type for what they call ι-quantum groups, which are in fact quasi-split quantum symmetric pair coideal subalgebras, coinciding with the algebras B c,s in the special case X = ∅. These correspond to Satake diagrams without black nodes. Since Z i = −1 in this situation, the polynomial C ij (c) will be given by if τ (i) = i, as follows from Corollary 3.5, and where the structure constants ρ (i,j,aij ) m,m ′ were obtained in Theorem 3.13. In this section we will derive equivalent expressions for ρ (i,j,aij ) m,m ′ , based on the results of [CLW19]. These expressions will also be valid beyond the quasi-split case. Indeed, our result (51) shows that ρ (i,j,aij ) m,m ′ is independent of X and can be obtained solely from the U q (g ′ )-relations (18)-(19)-(20). Hence the expressions for ρ (i,j,aij ) m,m ′ we will derive in this section will be valid not only for X = ∅, but for any admissible pair (X, τ ) provided we restrict to Case 1, i.e. τ (i) = i ∈ I \ X and j ∈ I \ X distinct from i.
Before we can state the result from [CLW19], we need to introduce the following notation.
where we have again used the notation (49).
Using Lusztig's theory of modified quantum groups [Lus94, Section 23.1] and a class of intricate q-binomial identities, Chen, Lu and Wang were able to prove a result, which, translated to our notations, can be formulated as follows.
Theorem K ([CLW19, Theorem 3.1]). Consider the quantum symmetric pair coideal algebra B c,s corresponding to an admissible pair (X = ∅, τ ). For any i ∈ I satisfying τ (i) = i and any j ∈ I distinct from i, the ι-divided powers satisfy the relations 4.1. Quantum Serre relations from ι-divided powers. Let us now fix τ ∈ Aut(A, ∅) such that (X = ∅, τ ) is an admissible pair, and fix also a corresponding quantum symmetric pair coideal subalgebra B c,s . In the present section, we will rewite the relations (113) and (114) as inhomogeneous quantum Serre relations, so as to derive two new expressions for the structure constants ρ (i,j,aij ) m,m ′ for Case 1. Let us start by introducing the following notation.
These α Proof. Expanding (112) distributively, it is clear that in agreement with the proposed formula. Similarly, for r odd one has whereas for r even, we have But of course, since [0] qi = 0, we have that which again agrees with the statement of the lemma.
In the upcoming proofs, we will often be required to switch the order of summation in a particular kind of nested sums. Below, we propose a general strategy for this resummation. Proof. We will derive this identity in several steps, which are explained below.   We will now rewrite the results of Theorem K using Lemma 4.3.
Proposition 4.5. The relations (113)-(114) can equivalently be expressed as k,N as in Definition 4.2. Proof. Let us consider the case where a ij is even. We will start by splitting the sum over m in (113) into a sum over m even and one over m odd: Substituting the expressions for B Referring to the notation (116), we may write the terms between brackets above as Θ In a similar fashion, one can combine the relation (113) for a ij odd with the relation (114) for a ij even. This gives rise to the following expressions. Comparing the relations (115) and (118) with (110), we obtain alternative expressions for the structure constants ρ (i,j,aij ) m,m ′ . As explained in the introduction of Section 4, these will not only be valid for the quasisplit case, but for any admissible pair, provided we restrict to Case 1. Hence from now on we may again assume (X, τ ) to be an arbitrary admissible pair and consider the corresponding quantum symmetric pair coideal subalgebra B c,s .
Theorem 4.7. For any distinct i, j ∈ I \ X such that τ (i) = i, one has where the structure constants are given by and D i,j lies in the set E i,j of K(q)-linear combination of monomials in U q (g ′ ) containing at most −1 − a ij factors F i , and either one factor K −1 j , or one factor F j and at least one factor K −1 i . Since the U q (g ′ )relations (18)-(19)-(20) imply E i,j ∩ U − = {0}, both D i,j and F i,j must vanish. The assertion F i,j = 0 is a polynomial equation of degree 1 in F j and at most of degree −1 − a ij in F i . But such a polynomial must have trivial coefficients, since the lowest degree K(q)-linear combination of F j and powers of F i with non-trivial coefficients that vanishes, is precisely the quantum Serre polynomial F ij (F i , F j ), which is of degree 1 − a ij in F i . Hence we find m,m ′ , for any m, m ′ with m+m ′ ≤ −1−a ij . This holds for the special case X = ∅, and since ρ (i,j,aij ) m,m ′ is independent of X as explained above, this establishes the same relation for admissible pairs with X = ∅.
Remark 2. It follows from Propositions 4.5 and 4.6 that Θ (1,i,aij ) m,m ′ for any m, m ′ ∈ N with m + m ′ ≤ −1 − a ij and any distinct i, j ∈ I \ X with τ (i) = i. Hence the expressions (116) and (119) must be equal, which determines a non-trivial identity of q-binomial type.
To conclude, we will show that the structure constants ρ (i,j,aij ) m,m ′ exhibit certain symmetry properties, as suggested by the values in Table 1. In practical calculations, this significantly reduces the number of couples (m, m ′ ) for which the structure constants must be computed.
with i ∈ I, and where (c, s) takes values in the set C × S defined in (25)-(26). By Theorem C, Corollary 3.5 and the fact that in this case Z i = −1, it is abstractly defined by the relations for i = j ∈ I, with ρ (i,j,aij ) m,m ′ as obtained in (83) or equivalently in (120).
In the special case g = sl 2 , this algebra coincides with the q-Onsager algebra [Bas05], which is typically described as generated by two elements B 0 and B 1 subject to the q-Dolan-Grady relations which, by the presence of nested q-commutators, can be considered relations of q-Dolan-Grady type. To conclude, we will consider the limit of the generalized q-Onsager algebra O q (g) under the specialization q → 1 described in Remark 1, which is precisely the algebra b s = b s (X, τ ) from Definition 2.3 in the special case X = ∅ and τ = id. It follows immediately that in this case Ad(s(X, τ )) = Ad(m X ) = id, and hence the automorphism θ(X, τ ) coincides with the classical Chevalley involution ω defined in (11). Moreover, Definition 2.3 asserts that we may state the following. For ease of notation, we have chosen to retain only the case s = 0 from Definition 2.3, which, in the terminology of [Let02, Section 7], is known as the standard case.
The algebras O(g) were studied by Stokman in [Sto19], where a complete set of defining relations of inhomogeneous Serre type or Dolan-Grady type was given. To describe these relations, we will need the following recursively defined coefficients.