Factorisation of quasi K-matrices for quantum symmetric pairs

The theory of quantum symmetric pairs provides a universal K-matrix which is an analogue of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is sl(n) or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.

1. Introduction 1.1. Background. Let g be a complex semisimple Lie algebra and U q (g) the corresponding Drinfeld-Jimbo quantised enveloping algebra with positive and negative parts U + and U − , respectively. The quasi R-matrix for U q (g) is a canonical element in a completion of U − ⊗ U + which plays a pivotal role in many applications of quantum groups. In the theory of canonical or crystal bases for U q (g) developed by G. Lusztig and M. Kashiwara, the quasi R-matrix appears as an intertwiner of two bar involutions on ∆(U q (g)), where ∆ denotes the coproduct of U q (g). The quasi R-matrix is used to define canonical bases of tensor products of U q (g)-modules and of the modified quantised enveloping algebraU, see [Lus94,Part IV].
Moreover, the quasi R-matrix for U q (g) is used in [Lus94, Chapter 32] to construct a family of commutativity isomorphisms. These maps turn suitable categories of U q (g)-modules into braided monoidal categories and hence have applications in low-dimensional topology, in particular the construction of invariants of knots and links, see [RT90]. Up to completion, the commutativity isomorphisms come from a universal R-matrix for U q (g).
As in [BK16] we denote the quasi R-matrix of U q (g) by R. One of the key properties of R is that it admits a factorisation as a product of quasi R-matrices for U q (sl 2 ). Let {α i | i ∈ I} be the set of simple roots for g. The quasi R-matrix corresponding to i ∈ I is given by where E i , F i ∈ U q (g) are generators of the copy of U q (sl 2 ) labelled by i, and q i = q (αi,αi)/2 . Let s i for i ∈ I be the generators of the Weyl group W of g, and let T i : U q (g) → U q (g) for i ∈ I be the corresponding Lusztig automorphisms. For any reduced expression w 0 = s i1 · · · s it of the longest word w 0 ∈ W define for j = 1, . . . , t.
In the present paper we aim to find a similar factorisation in the theory of quantum symmetric pairs. Let θ : g → g be an involutive Lie algebra automorphism and let k = {x ∈ g | θ(x) = x} be the corresponding fixed Lie subalgebra. We refer to (g, k) as a symmetric pair. A comprehensive theory of quantum symmetric pairs was developed by G. Letzter in [Let99], [Let02]. This theory provides families of subalgebras B c,s ⊂ U q (g) with parameters c and s, which are quantum group analogues of U (k). Crucially, B c,s is a right coideal subalgebra of U q (g), that is ∆(B c,s ) ⊆ B c,s ⊗ U q (g).
Initially, the theory of quantum symmetric pairs was used to perform harmonic analysis on quantum group analogues of symmetric spaces, see [Nou96], [Let04]. Recent pioneering work by H. Bao and W. Wang [BW13] and by M. Ehrig and C. Stroppel [ES13] has placed quantum symmetric pairs in a much broader representation theoretic context. Both papers consider a bar involution for specific quantum symmetric pair coideal subalgebras of type AIII/AIV . Moreover, Bao and Wang construct an intertwiner X (denoted by Υ in [BW13]) between the bar involutions on B c,s and on U q (g). The intertwiner X is an analogue for quantum symmetric pairs of the quasi R-matrix for U q (g). Using the intertwiner, Bao and Wang show that large parts of Lusztig's theory of canonical bases [Lus94,Part IV] extend to the setting of quantum symmetric pairs. While [BW13] restricts to the specific quantum symmetric pairs of type AIII/AIV , the more recent work [BW16] develops a theory of canonical bases for all quantum symmetric pairs of finite type.
Following the program outlined in [BW13], the existence of the bar involution and the intertwiner X was proved for general quantum symmetric pairs in [BK15] and [BK16], respectively. The intertwiner X was then used in [BK16] to construct a universal K-matrix for B c,s which is an analogue of the universal R-matrix for U q (g). For this reason we call the intertwiner X the quasi K-matrix for B c,s . The universal K-matrix gives suitable categories of B c,s -modules the structure of a braided module category and allows similar applications in low-dimensional topology as braided monoidal categories, see [Kol17].
In [BW13] and [BK16] the quasi K-matrix is defined recursively by the intertwiner property for the two bar involutions. It is noted at the end of [BW16,Section 4.4] that this recursion can be solved in principle. However, to this date no closed formula for X is known.
1.2. Results. In the present paper we provide a general closed formula for the quasi K-matrix X in many cases, and we conjecture that our formula holds for all quantum symmetric pairs of finite type. In doing so, we take guidance from the known factorisation (1.2) of the quasi R-matrix of U q (g).
Recall that the involutive automorphism θ is determined up to conjugation by a Satake diagram (I, X, τ ). Here, X is a subset of I and τ : I → I is a diagram automorphism. The τ -orbits in I \ X correspond to rank one subdiagrams of the Satake diagram (I, X, τ ). Associated to the symmetric Lie algebra (g, θ) is a restricted root system Σ with Weyl group W = W (Σ) which can be considered as a subgroup of W . The Coxeter generators s i of W are parametrised by the τ -orbits in I \ X. We introduce the notion of a partial quasi K-matrix X w for any w ∈ W with a reduced expression w = s i1 . . . s it . More precisely, for j = 1, . . . , t let X i denote the quasi K-matrix corresponding to the rank one Satake subdiagram ({i, τ (i)} ∪ X, X, τ | {i,τ (i)}∪X ) of (I, X, τ ). The Lusztig automorphisms T w : U q (g) → U q (g) for w ∈ W allow us to define automorphisms T i := T si for all i ∈ I \ X. For j = 1, . . . , t we set where Ψ denotes an algebra automorphism of an extension of U + = µ∈Q + (2Σ) U + µ defined in Equation (3.45). In analogy to (1.2) we now define the partial quasi Kmatrix corresponding to w ∈ W by w . The following theorem is the main result of the present paper. It gives an explicit formula for X in the case that s = 0 = (0, 0, . . . , 0) for certain Satake diagrams. Let w 0 ∈ W denote the longest element.
Theorem A. (Corollary 3.22) Let g be of type A n or X = ∅. Then the quasi K-matrix X for B c,0 is given by X = X w0 for any reduced expression of the longest element w 0 ∈ W .
We conjecture that Theorem A holds for all Satake diagrams of finite type, see Conjecture 3.23. The proof of Theorem A proceeds in three steps. First we construct the quasi K-matrices corresponding to all rank one Satake diagrams of type A n in the case where s = 0. The difficulty here is that there are many rank one cases, see Table 1. Secondly, we prove Theorem A in rank two by direct calculation. In rank two the longest element w 0 ∈ W has two reduced expressions. We show for each reduced expression that the element X w0 defined by (1.3) satisfies the defining recursive relations for the quasi K-matrix. This involves tedious calculations which we have banned to Appendix A. The calculations require explicit knowledge of the rank one quasi K-matrices. The restriction to type A n or X = ∅ in Theorem A hence stems from the fact that the rank one quasi K-matrices are only known in type A n .
In the cases considered in Appendix A, the calculations imply in particular that X w0 is independent of the chosen reduced expression for w 0 . We conjecture that this is true in general.
Conjecture B. (Conjecture 3.15) Assume that (I, X, τ ) is a Satake diagram of rank two. Then the element X w defined by (1.3) depends only on w ∈ W and not on the chosen reduced expression.
Conjecture B is all that is needed to prove Theorem A for all Satake diagrams of finite type. Indeed, assume that Conjecture B holds for all rank two Satake subdiagrams of the given Satake diagram. Then we can use braid relations for the operators T i to show that X w is independent of the chosen reduced expression for w ∈ W . In the case of the longest element w 0 ∈ W we choose different reduced expressions for w 0 to show that X w0 satisfies the defining recursive relations for the quasi K-matrix for B c,0 . In summary, we obtain the following result in the case s = 0.
Theorem C. (Theorems 3.18, 3.21) Let (I, X, τ ) be a Satake diagram such that all rank two Satake subdiagrams satisfy Conjecture B. Then the following hold: (1) The partial quasi K-matrix X w depends only on w ∈ W and not on the chosen reduced expression.
(2) The quasi K-matrix X for B c,0 is given by X = X w0 where w 0 ∈ W denotes the longest element.
In the case s = 0 it is harder to give an explicit formula for the quasi Kmatrix X c,s of B c,s . However, we can make use of the fact that B c,s is obtained from B c,0 via a twist by a character χ s of B c,0 . We consider the element R θ c,s = ∆(X c,s )R(X −1 c,s ⊗1) which was introduced in [BW13] under the name quasi R-matrix for B c,s , and which lives in a completion of B c,s ⊗ U + , see also [Kol17, Section 3.3]. We show that the quasi K-matrix X c,s for B c,s satisfies the relation . Hence the explicit formulas (1.2) and (1.3) for R and X c,0 , respectively, provide a formula for the quasi K-matrix of B c,s also in the case s = 0. However, in this case we do not obtain a factorisation as in Equation (1.3).
1.3. Organisation. In Section 2 we recall background material on the restricted root system Σ and its Weyl group W (Σ). We show in particular how W (Σ) can be identified with a subgroup W of the Weyl group W of g, see Proposition 2.7. Section 3 forms the heart of the paper. In Sections 3.1 and 3.2 we fix notation for quantised enveloping algebras and quantum symmetric pairs, respectively. In Section 3.3 we determine explicit closed formulas for the quasi K-matrices X for B c,0 in all cases where the Satake diagram is of rank one and of type A n . We plan to return to the remaining rank one cases in the future. The theory of partial quasi K-matrices X w for w ∈ W is developed in Section 3.4. Here the rank two case is crucial. The explicit calculations which prove Theorem A in rank two are left for Appendix A. Based on the rank two results we prove in Theorem 3.18 that X W is independent of the chosen reduced expression for w. This gives the first part of Theorem C, and the second part as well as Theorem A follow, see Theorem 3.21. The case s = 0 is treated in Section 3.5 where Formula (1.4) is proved in Proposition 3.26.

The restricted Weyl group
In this section we recall the construction of involutive automorphisms θ : g → g of a semisimple Lie algebra g following [Kol14]. This allows us to construct a subgroup W Θ of the Weyl group W consisting of elements fixed under the corresponding group automorphism of W . Of particular importance is a subgroup W of W Θ which has an interpretation as the Weyl group of the corresponding restricted root system. The results in this section do not claim originality, but are all in some form contained in [Lus76], [Lus03] and also [GI14].
2.1. Involutive automorphisms of semisimple Lie algebras. Let g be a finite dimensional complex semisimple Lie algebra. Let h ⊂ g be a Cartan subalgebra and Φ ⊂ h * the corresponding root system. Choose a set of simple roots Π = {α i | i ∈ I} where I is an index set labelling the nodes of the Dynkin diagram of g. Let Q = ZΠ be the root lattice and set Q + = N 0 Π. Let Φ + be the set of positive roots corresponding to Π and set V = RΦ. For i ∈ I, let s i : V → V denote the reflection at the hyperplane orthogonal to α i . We write W to denote the Weyl group generated by the simple reflections s i . We fix the W -invariant non-degenerate bilinear form (−, −) on V such that (α, α) = 2 for all short roots α ∈ Φ in each component. Let {e i , f i , h i | i ∈ I} be the Chevalley generators for g.
Involutive automorphisms of g are classified up to conjugation by pairs (X, τ ) where X ⊂ I and τ : I → I is a diagram automorphism. More precisely, for any subset X ⊂ I let g X denote the Lie subalgebra of g generated by Let Q X denote the sublattice of Q generated by {α i | i ∈ X}. This is the root lattice for g X . Let ρ X ∈ V and ρ ∨ X ∈ V * denote the half sum of positive roots and positive coroots for g X . The Weyl group W X of g X is the parabolic subgroup of W generated by {s i | i ∈ X}. Let w X ∈ W X denote the longest element of W X .
Definition 2.1. ([Sat60,p. 109], see also [Kol14, Definition 2.3]) Let X ⊂ I and τ : I → I a diagram automorphism such that τ (X) = X. The pair (X, τ ) is called a Satake diagram if it satisfies the following properties: 2) The action of τ on X coincides with the action of −w X , that is 3) If j ∈ I \ X and τ (j) = j, then α j (ρ ∨ X ) ∈ Z. Remark 2.2. When we need to identify the underlying Lie algebra, we write (I, X, τ ) to indicate the Satake diagram. With this convention, if (I, X, τ ) is a Satake diagram and i ∈ I \ X then ({i, τ (i)} ∪ X, X, τ | {i,τ (i)}∪X ) is also a Satake diagram.
Graphically, the ingredients of a Satake diagram are recorded in the Dynkin diagram of g. The nodes labelled by X are coloured black and a double sided arrow is used to indicate the diagram automorphism. See [Ara62, p.32/33] for a complete list of Satake diagrams for simple g.
We can now construct an involution θ corresponding to the Satake diagram (X, τ ) as in [Kol14, Section 2.4]. Let ω : g → g be the Chevalley involution given by The diagram automorphism τ can be lifted to a Lie algebra automorphism τ : g → g denoted by the same symbol. Recall from [Ste67, Lemma 56] that the action of the Weyl group W on h can be lifted to an action of the corresponding braid group Br(W ) on g. We denote the action of w ∈ Br(W ) on g by Ad(w). Let s : I → C × be a function such that This map extends to a group homomorphism s Q : Q → C × such that s Q (α i ) = s(i) for each simple root α i . Let Ad(s) : g → g denote the Lie algebra automorphism such that the restriction to any root space g α is given by multiplication by s Q (α).
Given a Satake diagram (X, τ ) we define a Lie algebra automorphism θ = θ(X, τ ) of g by Here, the longest element w X ∈ W X is considered as an element in the braid group Br(W ). The Lie algebra automorphism θ = θ(X, τ ) is involutive, that is θ 2 = id. Any involutive Lie algebra automorphism of g is Aut(g)-conjugate to an automorphism of the form θ(X, τ ), see for example [Kol14, Theorems 2.5, 2.7].
2.2. The subgroup W Θ . For any Satake diagram (X, τ ), the automorphism θ = θ(X, τ ) satisfies θ(h) = h. More explicitly, the definition (2.4) implies that θ| h = −w X • τ . The dual map Θ : h * → h * is given by the same expression where now w X and τ act on h * . We obtain a group automorphism In this section we recall the structure of the subgroup W Θ following [Lus76] and [GI14]. For any i ∈ I one has This formula and property 2) in Definition 2.1 imply that W X is a subgroup of W Θ . Recall that for any subset J ⊂ I we write w J to denote the longest element in the parabolic subgroup W J . For all i ∈ I \ X define Recall that there exists a diagram automorphism τ 0 : I → I such that the longest element w 0 ∈ W satisfies (2.9) w 0 (α i ) = −α τ0(i) for all i ∈ I. By inspection of the list of Satake diagrams in [Ara62, p. 32/33] one sees that the set X is τ 0 -invariant. By Remark 2.2 the triple ({i, τ (i)} ∪ X, X, τ | {i,τ (i)}∪X ) is a Satake diagram for any i ∈ I \ X. In analogy to (2.9) there exists hence a diagram automorphism for all j ∈ {i, τ (i)} ∪ X. With this notation we get and hence This proves that w X and w {i,τ (i)}∪X commute in W for any i ∈ I \ X. Hence Equation (2.7) implies that (2.12) s i ∈ W Θ for all i ∈ I \ X.
Let W ⊂ W Θ denote the subgroup of W generated by all s i for i ∈ I \ X. Let l : W → N 0 denote the length function with respect to W . Let be the set of minimal length left coset representatives of W X . By [Lus03,25.1] the subset (2.14) is a subgroup of W . By (2.8) and (2.11) we have s i ∈ W for all i ∈ I \ X. Let Lemma 2.3. Any w ∈ W τ may be written as w = s i1 · · · s i k such that s i1 , . . . , s i k ∈ W and l(w) = l( s i1 ) + · · · + l( s i k ). Remark 2.6. By (2.11) the group W acts on W X by conjugation. Moreover W X ∩ W = {id}. One can show that W Θ = W X ⋊ W .
2.3. The restricted root system. The group W has an interpretation as the Weyl group of the restricted root system of the symmetric Lie algebra (g, θ). This fact is implicit in [Lus76] but we feel that it is beneficial to explain this connection in some detail. As θ(h) = h, we obtain a direct sum decomposition The restricted root system Σ ⊂ a * is obtained by restriction of all roots in Φ to a, that is As Θ(Φ) = Φ we have Θ(V ) = V . Moreover, the inner product (−, −) is Θinvariant. Hence we obtain an orthogonal direct sum decomposition Indeed, any α ∈ V can be written as where (α + λΘ(α))/2 ∈ V λ for λ = ±1. For any β ∈ V −1 and h ∈ h 1 , we have β(h) = 0 as β(h) = β(Θ(h)) = Θ(β)(h) = −β(h). Hence we may consider V −1 as a subspace of a * and V −1 = RΣ. For any β ∈ V we define (2.20) Hence the group W Θ acts on Σ.
We have an inner product on V −1 by restriction of the inner product on V . As the inner product on V is W -invariant and V −1 is a W Θ -invariant subspace, the restriction of the inner product on V −1 is W Θ -invariant. For any i ∈ X, we have α i ∈ V +1 . Hence the direct sum decomposition (2.17) implies that s i ( β) = β for all β ∈ Σ, i ∈ X. The group W on the other hand can be interpreted in terms of the restricted root system Σ. We include a pedestrian proof, avoiding the more sophisticated setting in [Lus76].
(1) The reflections at the hyperplanes perpendicular to elements of Σ generate a finite reflection group W (Σ).
(2) There is an isomorphism of groups ρ : W → W (Σ) which sends s i to the reflection at the hyperplane orthogonal to α i for any i ∈ I \ X.
Proof. For any i ∈ I \ X we have Now suppose β ∈ V −1 such that ( β, α i ) = 0. Using the W Θ -invariance of the bilinear form on V −1 , we obtain On the other hand, by the definition of s i we have s i (β) = β + n i α i + n τ (i) α τ (i) + j∈X n j α j for some n j ∈ Q. From this we obtain The inner product is positive definite so it follows that m i = 0. Hence s i ( β) = β. This together with (2.21) implies that s i is the reflection at the hyperplane orthogonal to α i . As seen above, the action of W on V −1 gives a group homomorphism Adapting the proof of [Hum90, Theorem 1.5] one shows that ρ is surjective which implies part 1. To prove part 2 it remains to show that ρ is injective. This is a consequence of Lemma 2.8 below.
Lemma 2.8. The action of W on Σ is faithful.
Proof. Assume that there exists w ∈ W such that w = 1 W and w( α i ) = α i for all i ∈ I.
We can rewrite this formula as For all i ∈ X we have w(α i ) > 0 as l(ws i ) = l(w) + 1. Hence there exists i ∈ I \ X such that w(α i ) < 0. In this case also w(α τ (i) ) < 0 since elements of W are fixed under τ . Consider Equation (2.22) for this i: The right hand side lies in Q + and is of the form where n j ∈ N 0 for each j ∈ X. We can write the left hand side as where m j ∈ N 0 for each j ∈ X. Hence inserting (2.23) and (2.24) into (2.22), we get Now we apply the tilde map to the above equation. The terms involving α j for j ∈ X vanish, because the tilde map is zero on Q X and w commutes with Θ. We get The right hand side lies in Q + (Σ). The left hand side lies in −Q + (Σ) because w(α i ) and w(α τ (i) ) lie in −Q + . Hence both sides of the equation must vanish. However, this is not possible, in particular for the right hand side which is 2 α i . We have a contradiction.
Proposition 2.7 has the following consequence which we note for later reference.
Corollary 2.9. For any i ∈ I \ X and µ ∈ Q(Σ) the relation

Factorisation of quasi K-matrices
As explained in the introduction, the quasi R-matrix for U q (g) [Lus94, Chapter 4] has a deep connection to the Weyl group W . This was first observed by Levendorskiȋ and Soibelman [LS90], and independently by Kirillov and Reshetikhin [KR90]. In this section, we establish a similar connection between the quasi K-matrix X for a quantum symmetric pair and the restricted Weyl group W . In particular, we will see in many cases that the quasi K-matrix X factorises into a product of quasi K-matrices for Satake diagrams of rank one.
We first fix notation for quantised enveloping algebras and quantum symmetric pairs in Sections 3.1 and 3.2. Recall that the construction of quantum symmetric pairs depends on an additional choice of parameters c ∈ C and s ∈ S, see Definition 3.1. In Sections 3.3 and 3.4 we find explicit formulas for X in the case s = (0, . . . , 0). Section 3.5 then deals with the case of general parameters s.
3.1. Quantised enveloping algebras. In this section we fix notation for quantum groups mostly following the conventions in [Lus94] and [Jan96]. Let q be an indeterminate and K a field of characteristic zero. Denote by K(q) the field of rational functions in q with coefficients in K. The quantised enveloping algebra U q (g) is the associative K(q)-algebra generated by elements E i , F i , K ±1 i for all i ∈ I satisfying the relations given in [Lus94, 3.1.1] or [Jan96, 4.3]. The algebra U q (g) has the structure of a Hopf algebra with coproduct ∆, counit ε and antipode S given by for all i ∈ I. Let U + , U − and U 0 denote the subalgebras of U q (g) generated by The elements K λ for λ ∈ Q form a vector space basis of U 0 . For any U 0 -module M and any µ ∈ Q let denote the corresponding weight space. In particular, both U + and U − are U 0modules via the left adjoint action so we may apply the above notation. This gives algebra gradings By [Lus94,39.4.3] the braid group Br(W ) acts on U q (g) by algebra automorphisms analogously to the action of Br(W ) on g. For any i ∈ I let T i be the algebra where m ij denotes the order of s i s j ∈ W . Hence for any w ∈ W , there is a well-defined algebra automorphism T w : In Sections 3.2 and 3.4 it is necessary to consider a completion U of U q (g). We recall the construction of U following [BK16, 3.1]. Let O int denote the category of finite dimensional U q (g)-modules of type 1, and let Vect denote the category of vector spaces over K(q). Both categories are tensor categories and the forgetful functor F or : O int → Vect is a tensor functor. We let U = End(F or) be the K(q)algebra of all natural transformations from the functor F or to itself. Observe that U q (g) and U + = µ∈Q + U + µ may be considered as subalgebras of U , see [BK16, Section 3.1]. We usually write elements in U + additively as infinite sums µ∈Q + u µ with u µ ∈ U + µ . By [Lus94, 1.2.13] for any i ∈ I there exist uniquely determined linear maps i r, for all j ∈ I, x ∈ U + µ and y ∈ U + ν where µ, ν ∈ Q + . We may extend the skew derivation i r : U + → U + to a linear map Similarly we may extend the skew derivation r i : U + → U + to a linear map r i : Finally, recall that the bar involution for U q (g) is the K-algebra automorphism 3.2. Quantum symmetric pairs. We recall the definition of quantum symmetric pair coideal subalgebras B c,s as introduced by G. Letzter in [Let99]. Here we follow the conventions in [Kol14]. Let (X, τ ) be a Satake diagram and let s : I → K × be a function satisfying equations (2.2) and (2. Quantum symmetric pair coideal subalgebras depend on a choice of parameters c = (c i ) i∈I\X ∈ (K(q) × ) I\X and s = (s i ) i∈I\X ∈ (K(q) × ) I\X with added constraints. Define where a ij = 2 (αi,αj ) (αj ,αj ) for i, j ∈ I are the entries of the Cartan matrix of g. Define the parameter sets All through this paper we assume that the parameters c = (c i ) i∈I\X ∈ C and s = (s i ) i∈I\X ∈ S satisfy the additional relations The map B is called the bar involution for B c,s and plays a similar role as the bar involution (3.12) for U q (g). In particular there exists a quasi K-matrix X ∈ U which resembles the quasi R-matrix for U q (g). More explicitly, following a program outlined by H. Bao and W. Wang in [BW13], it was proved in [BK16, Theorem 6.10] that there exists a uniquely determined element X = µ∈Q + X µ ∈ µ∈Q + U + µ with X 0 = 1 and X µ ∈ U + µ such that the equation For symmetric pairs of type AIII/IV with X = ∅ the existence of the quasi K-matrix satisfying (3.22) was first observed in [BW13]. The quasi K-matrix is an essential building block for the construction of the universal K-matrix in [BK16]. To unify notation we define c i = s i = 0 and B i = F i for i ∈ X. Moreover, we write By [BK16, Proposition 6.1] the quasi K-matrix X = µ∈Q + X µ is the unique solution to the recursive system of equations with the normalisation X 0 = 1. Using the extension of the skew derivation i r to U + given by (3.11) we can rewrite (3.25) in the compact form In Section 3.3 and in Appendix A we will use the above formula to perform uniform calculations with X. Equation (3.25) implies in particular that as c j = s j = 0 for all j ∈ X. This property has the following consequence which was already observed in [BW16,Proposition 4.15]. Recall that w 0 ∈ W denotes the longest element. Moreover, for any w ∈ W recall the definition of the subalgebra U + [w] of U + given in [Jan96,8.21,8.24].
Proof. In view of (3.27), Equation (4) of [Jan96,8.26 which completes the proof of the Lemma.
We write X c,s for X if we need to specify the dependence on the parameters. Any diagram automorphism η : I → I with η(X) = X induces a map η : This notation allows us to record the effect of diagram automorphisms on the quasi K-matrix X.
Lemma 3.3. Let η : I → I be a diagram automorphism with η • τ = τ • η and η(X) = X. Then for any c ∈ C, s ∈ S we have η(c) ∈ C, η(s) ∈ S and Proof. The relations η(c) ∈ C, η(s) ∈ S follow from the assumptions on η and the definitions (3.14) and (3.15) of C and S. By [BK16, Proposition 6.1], property (3.22) is equivalent to the relation where we write B c,s i instead of B i to denote the dependence on c and s.
and η : U q (g) → U q (g) commutes with the bar involution U on U q (g). Hence applying η to relation (3.30) we obtain 3.3. Quasi K-matrices for Satake diagrams of rank one. For the remainder of this paper, following Remark 2.2, we denote Satake diagrams as triples (I, X, τ ) to also indicate the underlying Lie algebra.
Definition 3.4. A subdiagram of a Satake diagram (I, X, τ ) is a triple (J, X ∩ J, τ | J ) with J ⊆ I, such that τ (J) = J and J ∩ X consists of all connected components of X which are connected to a white node in J.
Remark 3.5. Any subdiagram (J, X ∩ J, τ | J ) of a Satake diagram satisfies the properties of Definition 2.1 and hence is itself a Satake diagram. It is possible to give a slightly more general definition of a Satake subdiagram which allows J ∩ X to contain connected components of X which are not connected to any white node in J. Here we exclude such components for convenience.
Let I be the set of τ -orbits of I \ X. There is a projection map (3.31) π : I \ X −→ I that takes any white node to the τ -orbit it belongs to.
In other words, a Satake diagram has rank n if there are n distinct orbits of white nodes. By Proposition 2.7 the rank of a Satake diagram coincides with the rank of the corresponding restricted root system Σ.
For any i ∈ I \ X let X i denote the union of all connected components of X which are connected to i. It follows by inspection of the list of Satake diagrams in [Ara62, p. 32/33] that X i = X τ (i) . Given a Satake diagram (I, X, τ ), any i ∈ I \ X determines a subdiagram ({i, τ (i)} ∪ X i , X i , τ | {i,τ (i)}∪X i ) of rank one. Let X i be the quasi K-matrix corresponding to this rank one subdiagram. For any w ∈ W we define U + [w] = µ∈Q + U + [w] µ . As U [w] + is a subalgebra of U + we obtain that U + [w] is a subalgebra of U + and hence of U . Formulating Lemma 3.2 in the present setting we obtain (3.32) In the following lemma we consider the case τ (i) = i and make the dependence of X i on the parameter c i more explicit.
The quasi K-matrices of rank one are the building blocks for quasi K-matrices of higher rank. In the following we give explicit formulas for rank one quasi Kmatrices of type A shown on the left hand side of Table 1 in the case s = (0, . . . , 0). The additional rank one Satake diagrams on the right hand side of Table 1 are not tackled in this paper.
Recall the definition of the q-number For n = 0 we set {0} i ! = 1 and {0} i !! = 1. Again, we omit the index i if all roots are of the same length. Further we use the following conventions. For any x, y ∈ U q (g), a ∈ K(q) we denote by [x, y] a the element xy − ayx. For any i, j ∈ I we write T ij = T i • T j : U q (g) → U q (g) and we extend this definition recursively.
3.3.1. Type AI 1 . Consider the Satake diagram of type AI 1 .

1
Lemma 3.8. The quasi K-matrix X of type AI 1 is given by Proof. By Equation 3.26, we need to show that 1 r(X) = (q − q −1 )(q 2 c 1 )E 1 X. Using the recursive formula (3.9) for 1 r, we see that 3.3.2. Type AII 3 . Consider the Satake diagram of type AII 3 .
Lemma 3.10. The quasi K-matrix X of type AIII 11 is given by Proof. By symmetry, we only need to show that 3.3.4. Type AIV for n ≥ 2. Consider the Satake diagram of type AIV for n ≥ 2. Lemma 3.11. The quasi K-matrix X of type AIV is given by Proof. We have i r(X) = 0 for i ∈ X. Hence by symmetry we only need to show that Remark 3.12. Let A = Z[q, q −1 ] and let A U + be the A -subalgebra of U + generated by E This integrality property is crucial for the theory of canonical bases of quantum symmetric pairs developed in [BW16].
We observe that the integrality of the quasi K-matrix in rank one can in some cases be read off the explicit formulas given in this section. Indeed, Lemma 3.8, 3.10 and 3.11 imply that X ∈ A U + in the rank one cases of type AI, AIII and AIV . The rank one case AII 3 is more complicated, and Lemma 3.9 does not give an obvious way to see that X ∈ A U + . Nevertheless, X is also integral in this case, as shown in [BW16,A.5]. Based on the present remark, the integrality of X in higher rank is discussed in Remark 3.24.
3.4. Partial quasi K-matrices. All through this section we make the assumption that s = 0 = (0, 0, . . . , 0) ∈ S. In Section 3.5 we discuss the case of general parameters s ∈ S. Recall that the Lusztig automorphisms T i of U q (g) for all i ∈ I give rise to a representation of Br(W ) on U q (g). Since W is a subgroup of W , we obtain algebra automorphisms of U q (g) defined by By Theorem 2.4 and Proposition 2.5 the algebra automorphisms T i give rise to a representation of Br( W ) on U q (g).
Hence we may consider the quasi K-matrix X as an element in µ∈Q + (2Σ) U + µ ⊂ U . For any w ∈ W define and set U + = µ∈Q + (2Σ) U + µ . Then U + and U + [w] are K(q)-subalgebras of U + and U + [w], respectively. In particular by Equation (3.32) we have for any i ∈ I \ X.
Let K ′ be a field extension of K(q) which contains q 1/2 and elements c i such that . Define an algebra automorphism Ψ : U + 1/2 → U + 1/2 by (3.45) Ψ(E 2 αi ) = q ( αi, αi) c i E 2 αi for all E 2 αi ∈ U + 2 αi . For each i ∈ I \ X define an algebra homomorphism We consider the restriction of the algebra homomorphism Ω i to the subalgebra U + [ s i w 0 ], and we denote this restriction also by Ω i . Crucially, by the following proposition, the image of the restriction Ω i belongs to U + and does not involve any of the adjoined square roots.
Proposition 3.13. For every i ∈ I \ X the map Proof. It remains to show that the image of Ω i is contained in U + . Observe that for all µ ∈ Q + (2Σ). By Corollary 2.9 we have Hence Equation (3.45) implies that Since µ ∈ Q + (2Σ) it follows that the exponent −(µ, α i ) is an integer. Moreover, Corollary 2.9 implies that the exponent −(µ, α i )/( α i , α i ) is an integer.
If i = τ (i) then Equation (3.44) and condition (2.2) imply that c i = ±c i . This implies that the image of Ω i is contained in U + in this case.
Finally, we consider the case that i = τ (i) and (α i , Θ(α i )) = 0. We are then in Case 3 in [Let08, p. 17] and hence the restricted root system Σ is of type (BC) n for n ≥ 1 and ( α i , Hence the image of Ω i is contained in U + in all cases as required.
Consider w ∈ W and let w = s i1 s i2 . . . s it be a reduced expression. For k = 1, . . . , t let for l = 2, . . . , k − 1 and hence the elements X

[k]
w are well-defined. Moreover, by Proposition 3.13 we have X When clear, we omit the subscript w and write Definition 3.14. Let w ∈ W and let w = s i1 s i2 . . . s it be a reduced expression. The partial quasi K-matrix X w associated to w and the given reduced expression is defined by We expect that the partial quasi K-matrix X w only depends on w ∈ W and not on the chosen reduced expression. As we will see in Theorem 3.18 it suffices to check the independence of the reduced expression in rank two. If the Satake diagram is of rank two then the restricted Weyl group W is of one of the types A 1 × A 1 , A 2 , B 2 or G 2 . In each case, only the longest word for W has distinct reduced expressions.
Conjecture 3.15. Assume that (I, X, τ ) is a Satake diagram of rank two. Then the element X w ∈ U defined by (3.47) depends only on w ∈ W and not on the chosen reduced expression.
In Appendix A, we prove the following Theorem which confirms Conjecture 3.15 in many cases. The proof is performed by showing that for both reduced expressions of the longest element w 0 ∈ W the resulting elements X w0 satisfy the relations (3.26).
Remark 3.17. The Hopf algebra automorphism Ψ in the definition of Ω i turns out to be necessary for the rank two calculations in Appendix A which prove Theorem 3.16. The conjugation by Ψ affects the coefficients in the partial quasi K-matrix associated to a reduced expression of an element w ∈ W . In rank two the two partial quasi K-matrices associated to the longest word w 0 ∈ W coincide only after this change of coefficients. The effect of the conjugation by Ψ can be seen in particular in Sections A.3 and A.4 of the appendix which treat type AIII n for n ≥ 3.
Once the rank two case is established, we can generalise to higher rank cases.
Theorem 3.18. Suppose that (I, X, τ ) is a Satake diagram such that all subdiagrams (J, X ∩J, τ | J ) of rank two satisfy Conjecture 3.15. Then the element X w ∈ U depends on w ∈ W and not on the chosen reduced expression.
Proof. Let w and w ′ be reduced expressions which represent the same element in W . Assume that w and w ′ differ by a single braid relation. The following are the possible braid relations: The argument for each relation is the same, so we only consider the second case. Assume that w and w ′ differ by relation (3.48), that is w = s i1 · · · s i k−1 s p s r s p s i k+3 · · · s it , w ′ = s i1 · · · s i k−1 s r s p s r s i k+3 · · · s it for some k = 1, . . . , t − 2. For l = 1, . . . , k − 1, we have Since the algebra automorphisms T i satisfy braid relations, we have for l = k + 3, . . . , t. Finally, consider the rank two subdiagram (J, X ∩ J, τ | J ) obtained by taking J = J 1 ∪ J 2 , where J 1 = {r, p, τ (r), τ (p)} and J 2 ⊂ X is the union of connected components of X which are connected to a node of J 1 . By assumption, It follows from this that Hence we have X w = X w ′ as required.
If w and w ′ differ by more than a single relation, then we can find a sequence of reduced expressions w = w 1 , w 2 , . . . , w n = w ′ such that for each i = 1, . . . , n − 1, the expressions w i and w i+1 differ by a single relation. We repeat the above argument at each step and obtain X w = X w ′ .
In this case the rank one Satake subdiagram is either of type AIII 11 or of type AIV for n ≥ 2 as in Table 1.
Lemma 3.20. Let w 0 = s i1 · · · s it be a reduced expression for the longest word in W . Then X w0 ∈ U + [ s k w 0 ] for i = 1, . . . , t − 1 and k = τ 0 (i t ). Proof. We have By definition of U + [w] for each w ∈ W and Proposition 2.5 we have for j = 1, . . . , t − 1. Now the claim of the lemma follows from Equation (3.32), Proposition 3.13 and the fact that With the above preparations we are ready to prove the main result of the paper.
Theorem 3.21. Suppose that (I, X, τ ) is a Satake diagram such that all subdiagrams (J, X ∩ J, τ | J ) of rank two satisfy Conjecture 3.15. Then X w0 coincides with the quasi K-matrix X.
Proof. It suffices to show that X w0 for all i ∈ I \ X. By Theorem 3.18, we can choose any reduced expression w 0 = s i1 · · · s it of the longest element of W . Proposition 3.19 implies that Suppose τ 0 (i t ) ∈ I is a representative of the τ -orbit {k, τ (k)} for some k ∈ I \ X.
Combining Theorems 3.16 and 3.21 we obtain the following result.
Corollary 3.22. Let g be of type A or X = ∅. Then the quasi K-matrix X is given by X = X w0 for any reduced expression of the longest word w 0 ∈ W .
Conjecture 3.23. The statement of Corollary 3.22 holds for any Satake diagram of finite type.
Remark 3.24. We continue the discussion of the integrality of the quasi K-matrix X from Remark 3.12 under the assumption that c i s(τ (i)) ∈ ±q Z for all i ∈ I \ X. In this case X w ∈ A U + for k = 1, . . . , t if w ∈ W has a reduced expression w = s i1 s i2 . . . s it . Indeed, the discussion in the proof of Proposition 3.13 shows that X [k] w differs from T i1 . . . T i1 (X i k ) by a factor in ±q Z . Hence we obtain X w ∈ A U + for all w ∈ W . By Corollary 3.22, choosing w = w 0 , we obtain X ∈ A U + whenever g is of type A or X = ∅. In these cases we have hence reproduced [BW16,Theorem 5.3] for s = 0 without the use of canonical bases. The case of general Satake diagrams hinges on Conjecture 3.23 and the integrality in rank one from [BW16, Appendix A].
3.5. Quasi K-matrices for general parameters. We now give a description of the quasi K-matrix X for general parameters s ∈ S. By [Let03, Theorem 7.1], [Kol14,Theorem 7.1] there exists an algebra isomorphism ϕ s : B c,0 → B c,s given by This algebra isomorphism allows us to define a one dimensional representation χ s : B c,0 → K(q) by χ s = ε • ϕ s . By definition we have For later use we observe the following compatibility with the bar involution.
Lemma 3.25. For all b ∈ B c,0 we have Proof. As (χ s ⊗ id) • ( Bc,0 ⊗ U ) • ∆ and U • ϕ s are K-algebra homomorphisms, it suffices to check Equation U which proves (3.61) in this case. Finally, if i ∈ {j ∈ I ns | a jk ∈ −2N 0 for all k ∈ I ns \ {j}}, then the definition (3.13) of I ns implies that Hence, using s i = s i U from (3.19), we get which completes the proof of the lemma.
As in [BK16, 3.2] we consider the algebra and observe that µ∈Q see also [Kol17, Section 3.3]. In [BW13] the element R θ is called the quasi R-matrix for B c,s . By [BW13, Proposition 3.2] it satisfies the following intertwiner property Similarly to the notation X c,s introduced at the end of Section 3.2, we write R θ c,s if we need to specify the dependence on the parameters. Observe that once we have an explicit formula for X c,0 , Equation (3.62) provides us with an explicit formula for R θ c,0 . This in turn provides a formula for the quasi K-matrix X c,s for general parameters s ∈ S. Indeed, by Equation (3.64) we can apply the character χ s to the first tensor factor of R θ c,0 to obtain an element X ′ = (χ s ⊗ id)(R θ c,0 ) which can be written as Moreover, Equation (3.64) implies that X ′ 0 = 1. By the following proposition the element X ′ ∈ U is the quasi K-matrix for B c,s . Proposition 3.26. For any c ∈ C, s ∈ S we have X c,s = (χ s ⊗ id)(R θ c,0 ).
Proof. We keep the notation X ′ = (χ s ⊗ id)(R θ c,0 ) from above. By Equation (3.63) we have Applying χ s ⊗ id to both sides of this relation, we obtain in view of Equation (3.60) the relation By Lemma 3.25 the above relation implies that This gives in particular B c,s i X ′ = X ′ B c,s i U for all i ∈ I and bX ′ = X ′ b for all b ∈ M X U Θ 0 . This means that X ′ satisfies the defining relation (3.22) of X c,s and hence, in view of the normalisation X ′ 0 = 1⊗1 observed above, we get X ′ = X c,s .
Remark 3.27. The existence of the quasi K-matrix X c,s was established in [BK16, Theorem 6.10] by fairly involved calculations. It was noted in [BK16, Remark 6.9] that these calculations simplify significantly if one restricts to the case s = 0.
Proposition 3.26 now shows that in the presence of (3.64) the existence of X c,0 implies the existence of X c,s for any s ∈ S satisfying (3.19). Relation (3.64) was established in [Kol17] for g of finite type.
Appendix A. Rank two calculations In rank two, there are two distinct reduced expressions for the longest word w 0 ∈ W . All irreducible rank two Satake diagrams for simple g are shown in Table  2. Using the explicit formulas from Section 3.3, we confirm that the partial quasi K-matrices for the two reduced expressions of w 0 coincide with the quasi K-matrix if the rank two Satake diagram is of type A n or B 2 . We have also performed the calculation in type G 2 . As this case involves significantly longer calculations, we do not include it here, but the details will appear in [Dob18]. The calculations in this appendix and [Dob18] prove Theorem 3.16. Since Θ = −id the restricted Weyl group W coincides with the Weyl group W . The longest word of the Weyl group has two reduced expressions given by w 0 = s 1 s 2 s 1 w ′ 0 = s 2 s 1 s 2 . Proposition A.1. In this case, the partial quasi K-matrices X w0 and X w ′ 0 coincide with the quasi K-matrix X. Hence X w0 = X w ′ 0 . Before we prove this, we need to know how the Lusztig skew derivations 1 r and 2 r act on certain elements and their powers, and also some commutation relations. These are given in the following two lemmas, whose proofs are obtained by straightforward computation.
Lemma A.2. For any n ∈ N, the relations Lemma A.3. For any n ∈ N, the relations Proof of Proposition A.1. Consider first the element X w0 . Using (3.47) and Lemma 3.8, we write is the occurrence of a q-power in each summand of the infinite series. By Equation (3.26), to show that X w0 coincides with the quasi K-matrix X we show that 2 r(X w0 ) = (q − q −1 )(q 2 c 2 )E 2 X w0 . By Lemma A.3 and 3.8, we see that By the property (3.9) of the skew derivative 1 r, we have We use Lemma A.2 to bring the E 1 in the above summand to the front. We have Hence, It follows from (A.2) that 1 r(X w0 ) = (q − q −1 )(q 2 c 1 )E 1 X w0 , so X w0 coincides with the quasi K-matrix X. Instead of repeating the same calculation for X w ′ 0 , we use the underlying symmetry in type AI 2 , which implies that X w ′ 0 also coincides with the quasi K-matrix X.
A.2. Type AII 5 . Consider the Satake diagram of type AII 5 . In this case the involutive automorphism Θ : h * → h * is given by There are two τ -orbits of white nodes given by the sets {2} and {4}. The restricted root system is of type AI 2 since the restricted roots have the same length. The subgroup W ⊂ W Θ is generated by the elements s 2 = s 2 s 1 s 3 s 2 , s 4 = s 4 s 3 s 5 s 4 .
The longest word of the restricted Weyl group has two reduced expressions given by w 0 = s 2 s 4 s 2 , w 0 ′ = s 4 s 2 s 4 .
By Lemma 3.9 we have Proposition A.4. The partial quasi K-matrices X w0 and X w0 ′ coincide with the quasi K-matrix X.
We have the following relations needed for the proof of Proposition A.4. These are proved by induction.
Lemma A.5. For any n ∈ N the relations hold in U q (sl 6 ).
Lemma A.6. For any n ∈ N the relation holds in U q (sl 6 ).

Proof of Proposition A.4.
We only confirm that X w0 coincides with the quasi Kmatrix X. By the underlying symmetry in type AII 5 , the calculation for X w0 ′ is the same up to a change of indices. By definition, we have Using Lemma 3.9 and [Jan96, 8.26, (4)] we see that We want to show that 2 . By property (3.9) of the skew derivative 2 r and [Jan96, 8.26, (4)] we have The second summand of Equation (A.6) is of the form (q−q −1 )c 2 X [3] X [2] T 13 (E 2 )X [1] . Using Lemma A.5, we bring the T 13 (E 2 ) term in this expression to the front. We have It follows from (A.6) that 2 r(X w0 ) = (q − q −1 c 2 T 13 (E 2 )X w0 as required.
A.3. Type AIII 3 . We consider the diagram of type A3 with non-trivial diagram automorphism τ and no black dots.

2 3
Here, we see that there are 2 nodes in the restricted Dynkin diagram, corresponding to the restricted roots A quick check confirms that 1/2(α 1 + α 3 ) is the short root, and hence the restricted root system is of type B 2 . The subgroup W is generated by the elements The longest word of the restricted Weyl group has two reduced expressions given by w 0 = s 1 s 2 s 1 s 2 , w ′ 0 = s 2 s 1 s 2 s 1 . The definition (3.14) and condition (3.18) imply that c 1 = c 3 = c 1 . By Lemmas 3.8 and 3.10 we have Proposition A.7. The partial quasi K-matrix X w0 coincides with the quasi Kmatrix X.
The following relations are needed for the proof of Proposition A.7. They are checked by induction.
Lemma A.8. For any n ∈ N, the relations Lemma A.9. For any n ∈ N, the relations Proof of Proposition A.7. Take w 0 = s 1 s 2 s 1 s 2 . Then we have where X [4] (3.50) = X τ0(2) = X 2 , By Lemma 3.8, property (3.9) of the skew derivative 2 r and [Jan96, 8.26, (4)], we see that 2 r(X w0 ) = (q − q −1 )q 2 c 2 E 2 X w0 . Due to the underlying symmetry in this case, we only need to show that Then by the property (3.9) of the skew derivation 1 r, we have Using Lemmas 3.10 and A.9, it follows that It follows that by equations (A.16), (A.17) and (A.18). Hence, it follows that 1 r(X w0 ) = (q − q −1 )c 1 E 3 X w0 , as required.
Proposition A.10. The partial quasi K-matrix X w0 ′ coincides with the quasi Kmatrix X.
The following relations are needed for the proof of Proposition A.10. They are checked by induction.
Lemma A.11. For any n ∈ N, the relations hold in U q (sl 4 ).
Lemma A.12. For any n ∈ N, the relations Proof of Proposition A.10. For w 0 ′ = s 2 s 1 s 2 s 1 we have By Lemma 3.10, property (3.9) of the skew derivative 1 r and [Jan96, 8.26, (4)] we have 1 r(X w0 ′ ) = (q − q −1 )c 1 E 3 X w0 ′ . We want to show that 2 r(X w0 ′ ) = (q − q −1 )(q 2 c 2 )E 2 X w0 ′ . For i = 1, 2, 3, 4, we let . By the property (3.9) of the skew derivative 2 r, we have We want to write the last summand in terms of X [2] . To do this, we use the fact that {n + 1} = 1 + q 2 {n} for n ≥ 1. This is a useful fact that will be used again in future calculations. Using this, we have Inserting this equation into the expression for 2 r(X [2] ), we obtain We use these to obtain the following.
. We gather terms now and get where we use the fact that E 1 T 2 (E 3 ) and E 3 T 2 (E 1 ) both commute with T 213 (E 2 ), and hence with X [3] . It follows that A.4. Type AIII n for n ≥ 4. Consider the Satake diagram of type AIII n for n ≥ 4. In this case the restricted root system is of type B 2 with α 1 = α 1 + α n 2 , α 2 = α 2 + α 3 + · · · + α n−1 2 .
The longest word of the restricted Weyl group has two reduced expressions given by w 0 = s 1 s 2 s 1 s 2 , w ′ 0 = s 2 s 1 s 2 s 1 . The definition (3.14) and condition (3.18) imply that c 1 = c n = c 1 . By Lemmas 3.10 and 3.11 we have Proposition A.13. The partial quasi K-matrix X w0 coincides with the quasi Kmatrix X.
We have the following relations needed in the proof of Proposition A.13, proved by induction.
Lemma A.14. For any k ∈ N the relations Lemma A.15. For any k ∈ N the relations One can show that the above Lemmas still hold if we consider the case where we have no black dots. In this situation, the calculations differ slightly but the results still hold.
Proof of Proposition A.13. We write where, recalling the notation c 2 2 = c 2 c n−1 s(n − 1)s(2), we have 2 r(X w0 ) = (q − q −1 )c 2 s(n − 1)T wX (E n−1 )X w0 . The underlying symmetry implies that we only need to show that 1 r(X w0 ) = (q − q −1 )c 1 E n X w0 . By Property (3.9) of the skew derivative 1 r and [Jan96,8.26 To write an expression for 1 r(X [2] ), we use the splitting of X We would like the last summand of this expression to be in terms of X [2] . To this end, we use Equation (A.34) in Lemma A.14 to obtain + q 2 (q − q −1 )c 1 c 2 s(n − 1)T n(n−1) T wX (E 2 )T 12n T wX (E n−1 )X [2;1] .
We now consider the reduced expression w ′ 0 = s 2 s 1 s 2 s 1 . Proposition A.22. The partial quasi K-matrix X w ′ 0 coincides with the quasi Kmatrix X.
The following relations are needed for the proof and are obtained by induction.