Quasitriangular coideal subalgebras of $U_q(\mathfrak{g})$ in terms of generalized Satake diagrams

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagovi\'c the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data first considered by A. Heck. A generalized Satake diagram naturally defines a semisimple automorphism $\theta$ of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. It also defines a subalgebra $\mathfrak{k}\subset \mathfrak{g}$ satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$, but not necessarily a fixed-point subalgebra. The subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a universal K-matrix in the sense of Kolb and Balagovi\'c. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way.


Introduction
Given a finite-dimensional semisimple complex Lie algebra g and an involutive Lie algebra automorphism θ ∈ Aut(g), a symmetric pair is a pair (g, k) where k = g θ is the corresponding fixed-point subalgebra of g, see [Ar62,Sa71]. Quantum symmetric pairs are their quantum analogons. That is, the enveloping algebra U (g) can be quantized to a quasitriangular Hopf algebra, the Drinfeld-Jimbo quantum group U q (g) endowed with the universal R-matrix R, see [Ji85,Dr87]. The quantum analogon of g θ is a coideal subalgebra B ⊆ U q (g) [Le99,Le02,Ko14] having a compatible quasitriangular structure, the universal K-matrix K [ BK19,Ko20] (see also [BW18a,Sec. 2.5] for the case of quantum symmetric pairs of type AIII/AIV). Quantizations of symmetric pairs appeared earlier in a different approach in [NDS95,NS95] (also see [KS09]). A prior notion of a universal K-matrix, not directly linked to a quantum symmetric pair, appeared in [DKM03].
The map θ, the fixed-point subalgebra k, the coideal subalgebra B and the universal object K are all defined in terms of combinatorial information, the so-called Satake diagram (X, τ ). Here X is a subdiagram of the Dynkin diagram of g and τ is an involutive diagram automorphism stabilizing X and satisfying certain compatibility conditions, see [Le02,Ko14].
It is the aim of this paper to extend some of this work to a more general setting. A direct motivation for this is the fact that the correct quantum group analogue of the fixed-point subalgebra in the Letzter-Kolb theory is not a fixed-point subalgebra itself, but merely tends to one as q → 1, see [Le99,Sec. 4] and [Ko14,Ch. 10]. This suggests that there may be a generalization of this theory that does not require a fixed-point subalgebra as input.
A careful analysis of [Ko14,BK15,BK19] indeed indicates that the compatibility conditions for X and τ can be weakened. Indeed, in [BK15, Rmks. 2.6 and 3.14] it is explicitly suggested that some key passages of the theory are amenable for generalizations. This leads to the notion of a generalized Satake diagram, see Definition 1, and the whole theory survives in this setting with minor adjustments. The resulting Lie subalgebra k = k(X, τ ) is given in Definition 2 and the corresponding coideal subalgebra B = B(X, τ ) in Definition 4. For g of type A, all generalized Satake diagrams are Satake diagrams. For other g, the generalized Satake diagrams that are not Satake diagrams are listed in Table 1.
Our proposed generalization of Satake diagrams can be traced back to the work of A. Heck [He84]. These diagrams classify involutions of the root system of g such that the corresponding restricted Weyl group is the Weyl group of the restricted root system. The characterization in terms of the restricted Weyl group is relevant in the context of the universal R-and K-matrices for quantum symmetric pairs. The universal Rmatrix R has a distinguished factor called quasi R-matrix playing an important role in the theory of canonical bases for U q (g), see [Ka90] and [Lu94,Part IV]. The quasi R-matrix possesses a remarkable factorization property expressed in terms of the braid group action on U q (g) of the Weyl group of g, see [KR90,LS90]. Recently it has become clear that many of these properties extend to the universal K-matrix K. It has a distinguished factor called quasi K-matrix, introduced in [BW18a] for certain coideal subalgebras of U q (sl N ) and in a more general setting in [BK15]. This object plays a prominent role in the theory of canonical bases for quantum symmetric pairs [BW18b]; for a historical note we refer the reader to [BW18b,Rmk. 4.9]. In [DK19] a factorization property is established for the quasi K-matrix using a braid group action of the restricted Weyl group. As a consequence of the present work, this factorization property naturally extends to quasi K-matrices defined in terms of generalized Satake diagrams.
The Kac-Moody generalization of this approach will be addressed in a future work. Another outstanding issue is a Lie-theoretic motivation of the subalgebra k, which we define in an ad hoc manner directly in terms of the combinatorial data (X, τ ), see Definition 2. Therefore we now provide a further motivation for the study of the subalgebra k and its quantization B.
1.1. Some remarks on the representation theory of (U q (g), B). Consider the completion U of U q (g) with respect to the category of integrable U q (g)-modules, so that objects in them have well-defined images under any finite-dimensional representation, see e.g. [Lu94,Jan96]. Then U ⊗ U can be embedded in a completion U (2) of U q (g) ⊗2 and one can construct an invertible R ∈ U (2) satisfying R∆(a) = ∆ op (a)R for all a ∈ U q (g), (∆ ⊗ id)(R) = R 13 R 23 , (id ⊗ ∆)(R) = R 13 R 12 , where ∆ is the coproduct and ∆ op the opposite coproduct (these can be viewed as maps from U to U (2) ).
Analogously, according to [BK19,Ko20], one can construct an invertible K ∈ U and an involutive Hopf algebra automorphism φ of U such that (φ ⊗ φ)(R) = R and where R φ = (φ ⊗ id)(R), the subscript 21 denotes the simple transposition of tensor factors in U (2) and B (2) ⊆ U (2) is a particular completion of B ⊗ U q (g), see [Ko20,Eq. (3.31)]. As a consequence, the (universal) φ-twisted reflection equation is satisfied: The automorphism φ is given by τ τ 0 where τ 0 is the diagram automorphism corresponding to the longest element of the Weyl group of g. The expression for K is given in [BK19, Cor. 7.7]. One could argue in favour of making the automorphism φ inner: adjoin to U a group-like element c φ such that φ(u) = c φ uc −1 φ for all u ∈ U. Then the object K φ := c −1 φ K satisfies (1.1-1.3) with φ replaced by id. However, for certain nontrivial diagram automorphisms φ, c φ cannot be chosen inside U so that K φ cannot be evaluated in all finite-dimensional representations. This relates to the fact that the weights defining certain fundamental representations are not fixed by φ. For instance, if ρ is the vector representation of U q (sl N ) with N > 2 one checks that the matrices ρ(φ(u)) and ρ(u) are not simultaneously similar for all u ∈ U q (g). Now let ρ the vector representation of U q (g); if g is of exceptional type by this we mean the smallest fundamental representation (for E 6 one has a choice of two representations).
Applying ρ ⊗ ρ to (1.4) one obtains the matrix reflection equation where the subscript 21 indicates conjugation by the permutation operator in GL(V ⊗ V ). Starting with g of classical Lie type and a coideal subalgebra B = B(X, τ ) where (X, τ ) is a Satake diagram, the matrices ρ(K) correspond to the solutions of (1.5) used in [NDS95,NS95] to define quantum symmetric pairs. Treating the matrix R as given, one can of course solve (1.5) for K ∈ GL(V ). For U q (sl N ) and V = C N this was done by A. Mudrov [Mu02]. From this result and computations for U q (g) whose vector representation is of dimension at most 9 (i.e. with g of types B n , C n , D n (n ≤ 4) and G 2 ) one obtains a classification of solutions K of (1.5) for those pairs (U q (g), ρ). One can match this list of solutions K one-to-one with a list of generalized Satake diagrams (X, τ ) by checking which K satisfies Kρ(b) = ρ(φ(b))K for all b ∈ B = B(X, τ ), i.e. the image of (1.1) under ρ. Although this intertwining equation does not determine K uniquely, it turns out that, provided K / ∈ C Id, each K intertwines ρ| B for a unique B = B(X, τ ) with X not equal to the whole Dynkin diagram I. In the case X = I we must have τ = τ 0 and B = U q (g) so that the excluded case K ∈ C Id can be matched to it. It leads to the following conjecture.
In the Letzter-Kolb approach, the generators of the coideal subalgebra B associated to a node i ∈ I\X carry extra parameters: scalars γ i = 0 and σ i , see Definition 4 and we can sharpen Conjecture 1 (i). Namely, let d i denote the squared length of root α i and write q i = q di . Consider the set I ns = {i ∈ I\X | i does not neighbour X, τ (i) = i}, see (3.24), and the sets Γ q and Σ q , see (4.5); these definition go back to [Le03,Ko14]. Conjecturally, any invertible matrix solution K of (1.5) is proportional to ρ(K) for some B(X, τ ) with (X, τ ) a generalized Satake diagram whose parameters satisfy (γ i ) i∈I\X ∈ Γ q , σ i = 0 if i / ∈ I ns and for all (i, j) ∈ I ns × I ns such that i = j one of three conditions must hold: the Cartan integer a ij is even, The set Σ q does not cover the third possibility, which appeared in [BB10] for a ij ∈ {−1, −3}. Conjecture 1 (ii) can be made more precise in an analogous way.
The approach in [BK19] requires also certain constraints on γ i and σ i under the transformation q → q −1 which are given in (4.22) and (4.23) in the present notation and generality.
1.2. Outline. The paper is organized as follows. In Section 2 we define the basic objects associated to a finite-dimensional semisimple complex Lie algebra g and its Cartan subalgebra h. We introduce generalized Satake diagrams and explain how they emerge in the work of A. Heck.
In Section 3 we define the Lie subalgebra k ⊆ g in terms of (X, τ ). In Theorem 3.1, the main result of this section, we show that k satisfies k ∩ h = h θ precisely if (X, τ ) is a generalized Satake diagram. In Propositions 3.2 and 3.3 we describe the derived subalgebra of k and establish that when k is not reductive it is a semidirect product of a reductive subalgebra and a nilpotent ideal of class 2. We end this section discussing the universal enveloping algebra U (k).
In Section 4 we indicate the necessary modifications to the papers [Ko14, BK15, BK19, Ko20, DK19] so that they apply to the quantum pair algebras B = U q (k) associated to generalized Satake diagrams.
Appendix A contains three technical lemmas in aid of Section 3. We use the symbol to indicate the end of definitions, examples and remarks.

Finite-dimensional semisimple Lie algebras and root system involutions
Let I be a finite set and A = (a ij ) i,j∈I a Cartan matrix. In particular, there exist positive rationals d i (i ∈ I) such that d i a ij = d j a ji . Let g = g(A) be the corresponding finite-dimensional semisimple Lie algebra over C. It is generated by The simple roots α i ∈ h * (i ∈ I) satisfy α j (h i ) = a ij for i, j ∈ I. Let Q = i∈I Zα i denote the root lattice and write Q + = i∈I Z ≥0 α i . For all α, β ∈ Q, we write α > β if α − β ∈ Q + \{0}. The Lie algebra g is Q-graded in terms of the root spaces g α = {x ∈ g | [h, x] = α(h) x for all h ∈ h} and we have the following identities for h-modules: Hence the root system Φ : Weyl group W is the (finite) subgroup of GL(h * ) generated by the simple reflections s i (i ∈ I) acting via Then Aut(Φ) = W ⋊ Aut(A), with Aut(A) acting by relabelling.
2.1. Compatible decorations and involutions of Φ. Given a subset X ⊆ I denote the corresponding Cartan submatrix by A X = (a ij ) i,j∈X and consider the semisimple Lie algebra g X := e i , f i , h i | i ∈ X ⊆ g with Cartan subalgebra h X = h ∩ g X and dual Weyl vector ρ ∨ X ∈ h X . The unique longest element w X of the Weyl group W X := s i | i ∈ X is an involution and there exists τ 0,X ∈ Aut(A X ) which satisfies We recall here the basic fact that both w X and τ 0,X naturally factorize with respect to the decomposition of X into connected components. Furthermore, if X is connected then τ 0,X is trivial unless X is of type A n with n > 1, D n with n > 4 odd or E 6 (in each case of which there is a unique nontrivial diagram automorphism). Note that Ad(w X )| gX = τ 0,X ω| gX and Ad(w X ) 2 = Ad(ζ), where ζ ∈ H is defined by ζ(α) = (−1) α(2ρ ∨ X ) for all α ∈ Q.
In the associated Dynkin diagram one marks a compatible decoration by filling the nodes corresponding to X and drawing bidirectional arrows for the nontrivial orbits of τ .
As explained above, the map dual to θ can be extended to an element of Aut inv (g, h), also called θ and given by θ = Ad(w X ) τ ω. Owing to aforementioned properties of Ad(w X ) we have Hence it vanishes so that We fix a subset I * ⊆ I\X containing precisely one element from each τ -orbit in I\X. As a consequence of (2.13) we have (2.14) 2.2. Generalized Satake diagrams and the restricted Weyl group. For i ∈ I\X denote byX(i) the union of connected components of X neighbouring {i, τ (i)}. Remark 1. Generalized Satake diagrams were first considered by A. Heck in [He84]. He uses the term "Satake diagrams" in a more general setting, see [He84, §1 - §2]: he starts with σ = −θ ∈ Aut inv (Φ) and calls a base Π of Φ σ-fundamental if for all α ∈ Π either θ(α) = α or θ(α) ∈ Z ≤0 Π. Letting X consist of the nodes corresponding to Π θ in the Dynkin diagram corresponding to Π, it follows that τ := σw X is an involutive diagram automorphism restricting on X to τ 0,X . He calls (X, τ ) the Satake diagram of σ, which we call a compatible decoration; what he calls an "admissible Satake diagram" is in our case a generalized Satake diagram. Since the term "Satake diagram" has come to be associated to involutions of the complex Lie algebra g, we prefer the nomenclature "compatible decoration" and "generalized Satake diagram".
Note that (X, τ ) ∈ CDec(A) is a generalized Satake diagram precisely if which is the condition needed in [Ko14, Proof of Lemma 5.11, Step 1] and [BK19, Proof of Lemma 6.4]. Straightforwardly one checks that it is equivalent to any of the following conditions: Satake diagrams can be defined as the following subset of compatible decorations of A: Satake diagrams classify involutive Lie algebra automorphisms up to conjugacy, see e.g. [Ar62]. In our notation, for (X, τ ) ∈ GSat(A) and γ ∈ (C × ) I * define χ γ ∈ H and θ γ ∈ Aut(g) by . We refer the reader to the classification of generalized Satake diagrams in [He84, Table I]. Since this does not distinguish between elements of Sat(A) and GSat(A)\Sat(A), for convenience we list the elements of GSat(A)\Sat(A), see Table 1; note that outside type A n we have GSat(A) = Sat(A). Table 1. The set GSat(A)\Sat(A) for indecomposable Cartan matrices A. By a case-bycase analysis there is a unique i ∈ I such that i = τ (i) and α i (ρ ∨ X ) / ∈ Z and we have indicated that node in the diagrams. The classical families of diagrams are labelled in the standard way. For types C n and D n upper bounds on i are imposed to avoid the cases when θ is an involution whose fixed-point subalgebra is isomorphic to gl n .
Now consider the restricted Weyl group and the set of restricted roots Theorem 2.1 ( [He84] and [Lu76]). Let (X, τ ) ∈ CDec(A). The following are equivalent: and w X do not commute so that s 2 i = id V . Finally, to show (i) ⇔ (iv), note that by factorizability of τ 0,X[i] over connected components of X[i], without loss of generality we may restrict to the case where X[i] is connected and equals I. Since there is nothing to prove if τ 0,X[i] = id, it remains to check the cases where X[i] is of type A n with n > 1, D n with n > 4 odd or E 6 . Classifying all (X, τ ) ∈ CDec(A) such that I = X[i] for some i ∈ I * , I is connected and τ 0,X[i] = id we obtain the following diagrams in the top row: The bottom row shows the corresponding compatible decoration (X[i], τ 0,X[i] ). The first case is not in GSat(A) and for the remaining six cases the subset X is preserved by τ 0,X[i] .
Remark 2. Note that Φ is not always a root system. By [He84, Thm. 6.1], Φ is a (possibly non-reduced or empty) root system precisely if τ 0,X[i] preserves X for all i ∈ I * or (X, τ ) = .
3. The subalgebra k For (X, τ ) ∈ Sat(A) the subalgebra g θ can be described in terms of generators; see e.g. [Ko14, Lemma 2.8]. This motivates the following more general definition.
It is convenient to suppress the dependence on γ and simply write b i and k if there is no cause for confusion.
It follows that k is generated by n + X := {e i | i ∈ X}, h θ and b i for i ∈ I. Owing to (2.1-2.2) and (2.12), these satisfy In Appendix A we study the repeated adjoint action of b i on b j for i, j ∈ I such that i = j. By setting m = M ij in Lemmas (A.1-A.3) one obtains the following Serre-type relations for the generators b i : By induction with respect to m, it can be shown that these integers satisfy Indeed, (3.9) is true for m ∈ {0, 1}. Suppose 0 ≤ 2r ≤ m and 1 < m ≤ M ij and assume (3.9) holds with m replaced by m − 1 and by m − 2. If p is nonzero and the observation that (m − 1)(M ij + 1 − m) > 0 completes the induction step.
(i) Definition 2 can be used in the general Kac-Moody case, so that (3.2-3.6) still hold. Also the results of Appendix A are valid in this general setting and hence so is (3.7). We will discuss the subalgebra k(X, τ ) in the Kac-Moody setting in future work. 3.1. Basic structure of k. From now on we assume that the γ i are nonzero. In order to state the main result of this section, we need some notation. For all i, j ∈ I such that i = j denote λ ij := M ij α i + α j ∈ Q + \Φ and consider the sets Then {f i } i∈J is a basis of n − and {e i } i∈JX is a basis of n + X . Theorem 3.1. Let (X, τ ) ∈ CDec(A) and γ ∈ (C × ) I\X . The following statements are equivalent: (iii) We have the following identity for h θ -modules: (3.14) Proof.
(ii) =⇒ (iii): Owing to (3.3-3.5) it is sufficient to prove (3.14) as an identity for vector spaces. First we prove that k = n + as vector spaces. Hence it suffices to prove that for all j ∈ ∪ ℓ I ℓ we have We will prove this by induction with respect to the height ℓ. Since for all j ∈ I we have dim(g −αj ) = 1 and hence (j) ∈ J , the case ℓ = 1 is trivial. Now fix ℓ ∈ Z >1 and assume that (3.17) holds true for all smaller positive integers. Fix j ∈ I ℓ and repeatedly apply the Serre relations (2.2) to obtain that for all i ∈ J ℓ there exist a i ∈ C such that f j = i∈J ℓ a i f i . Hence, by virtue of (ii) and equations Using the induction hypothesis for the elements b i in the last summation one obtains (3.17). It remains to show that the sum in (3.17) is direct. Let j ∈ J . Then f j is nonzero. Because of the explicit formula (3.1) we have where π α is the projection on g α for α ∈ Φ, see (2.3). Thus the linear independence of {f j } j∈J together with (2.3) implies that the sum is direct. (iii) =⇒ (iv): By definition, h θ ⊆ k ∩ h so it suffices to show that k ∩ h ⊆ h θ . Suppose h ∈ k ∩ h θ . By π −α j (b j ) = f j and the triangular decomposition (2.3), part (iii) implies h ∈ n + X ⊕ h θ ⊆ g θ so h ∈ h θ . (iv) =⇒ (ii): We prove the contrapositive. If (3.13) fails then (3.7) and (2.14) imply In either case (3.15) does not hold.
Conjecture 2. Let (X, τ ) ∈ GSat(A) and γ ∈ Γ. The Lie algebra k generated by symbols h i , e i (i ∈ X), h i − h τ (i) (i ∈ I * , i = τ (i)), b i (i ∈ I) and the relations obtained from (3.2-3.7) by adding tildes appropriately is isomorphic to k.
The only obstacle to promoting this to a theorem, and thereby settling the question posed in [Ko14, Rmk. 2.10], is the lack of a general explicit formula for the right-hand sides in (3.7) in terms of the e k with k ∈ X, instead of θ(f i ); for individual choices of (X, τ ) ∈ GSat(A) these explicit expressions can be found, or such relations do not occur at all, and in those cases one could prove the statement in the conjecture as follows. Because the generators of the Lie subalgebra k satisfy (3.2-3.7), one has a surjective Lie algebra homomorphism φ : k → k defined on generators by removing the tilde. On the other hand, in k there are no relations involving the b i other than (3.2-3.7), as otherwise applying the appropriate projection π −α onto g −α with α ∈ Φ + maximal would yield a relation involving the f i other than those given in (2.1-2.2). From this one can deduce that φ is injective and obtain the statement in Conjecture 2.
3.2. Semidirect product decompositions of k. In this section we assume that A is indecomposable, so that g is simple. In order to describe the derived subalgebra of k recall the set I diff ∈ I * and define (3.24) Proposition 3.2. Let (X, τ ) ∈ GSat(A) and γ ∈ Γ. The set is a basis for the derived subalgebra k ′ and we have Proof. Fix (X, τ ) ∈ GSat(A). Note that in (3.2-3.7) neither h i −h τ (i) (i ∈ I diff ) nor b j (j ∈ I nsf ) appears in the right-hand side. From the decomposition (3.14) it follows that these elements are not linear combinations of Lie brackets in k. It suffices to show that the remaining basis elements specified in (3.21) are linear combinations of Lie brackets in k.
• For b i with i ∈ J ℓ and e i with i ∈ J X,ℓ with ℓ > 1, this holds by definition.
• For e i , f i , h i with i ∈ X, this follows from (3.2-3.4).
• For h i − h τ (i) with i ∈ I * \I diff and i = τ (i), the constraint on i is equivalent to w X (α i ) = α i and • For b j with j ∈ I ns \I nsf there exists i ∈ I ns such that a ij is odd. From (3.7) we deduce From Proposition 3.2 it follows that the codimension of k ′ in k equals |I diff | + |I nsf |. For (X, τ ) ∈ Sat(A), in [Le02, Sec. 7, Variation 1] it was noted that |I diff | ≤ 1 if A is of finite type. In light of the above it is natural to generalize this by involving the set I nsf and allowing (X, τ ) ∈ GSat(A). Namely we have |I diff | + |I nsf | ≤ 1 for all (X, τ ) ∈ GSat(A) and the upper bound is sharp unless A is of type E 8 , F 4 or G 2 . This extends the known result for involutive θ that g θ is reductive with abelian summand at most 1-dimensional, see [Ar62]. For (X, τ ) ∈ WSat(A) we will obtain a semidirect product decomposition in terms of a reductive Lie subalgebra and a nilpotent ideal. For any r ∈ Z ≥0 and any i ∈ I denote by k(i) r the span of all b j with j ∈ J such that the coefficient of α i in α j is at least r. Set k(i) := k(i) 1 ⊆ k and recall the γ-modified automorphism θ γ defined in (2.20).
(i) The subalgebra g I\{i} is θ γ -stable, θ γ | g I\{i} is an involution and kî := k ∩ g I\{i} is its fixed-point subalgebra in g I\{i} ; (ii) We have ad(b i )(k(i) r ) ⊆ k(i) r+1 for all r ∈ Z ≥0 and the subspaces k(i) r are ad(kî)-modules; (iii) The subspace k(i) is an ideal of k, k = k(i) ⋊ kî and we have the lower central series (iv) The subalgebra k γ is isomorphic to the subalgebra of g generated by g X , h θ , b j;γj for j ∈ I\(X ∪ {i}) and b i;0 = f i .
(i) Since θ γ (g α ) = g θ(α) for all α ∈ Φ and applying θ = −w X τ to any root of g I\{i} does not modify the coefficient of α i , it follows that g I\{i} is θ γ -stable. Note that kî is fixed pointwise by θ γ . Furthermore a dimension count in each simple summand of g I\{i} combined with (3.22) implies that kî is the fixed-point subalgebra of θ γ . (ii) The first statement is immediate. From the adjoint action of e j (j ∈ X), h ∈ h θ and b j (j ∈ I\{i}) on elements of k(i) r , subject to (3.2-3.7), we obtain that ad(kî)(k(i) r ) ⊆ k(i) r . (iii) From part (ii) it follows that [k(i), k] ⊂ k(i) and combining this with k = k(i) ⊕ kî we obtain the semidirect product decomposition. A case-by-case analysis using Table 1 yields that the coefficient in front of α i in the highest root of Φ is always 2. This implies that the lower central series becomes trivial after 2 steps. (iv) This follows from the facts that the relations involving b i;γi in (3.2-3.7) do not depend on γ i and that the derivation of (3.7) did not require γ i = 0.
(ii) Proposition 3.3 excludes the case (X, τ ) = 1 2 . We will now see that is the only element of GSat(A)\Sat(A) such that k is a reductive Lie algebra. By definition, k is the subalgebra of g = Lie(G 2 ) generated by e 1 , h 1 , b 1 = f 1 and b 2 = f 2 + γ 2 θ(f 2 ) for some γ 2 ∈ C × . The relations (3.2-3.7) give The standard basis of k is given by {e 1 , h 1 , b 1 , b 2 , b (2,1) , b (2,2,1) , b (2,2,2,1) , b (1,2,2,2,1) }. Proposition 3.2 yields k = k ′ . Moreover, using (3.29), the adjoint action of e 1 , b 1 and b 2 on k implies that any ideal of k equals k if it contains any of the above standard basis elements. Then some straightforward computations show that k is in fact a simple Lie algebra and hence isomorphic to sl 3 .
Proposition 3.4. Let (X, τ ) ∈ GSat(A)\Sat(A) and γ ∈ Γ. Then k is not the fixed-point subalgebra of any φ ∈ Aut(g) such that 1 is a simple root of the minimal polynomial of φ.
Proof. We first show this in the case (X, τ ) = 1 2 . Suppose there exists φ ∈ Aut(g) such that k = g φ .
As a consequence of Proposition 3.4, k is not the fixed-point subalgebra of any semisimple (in particular, finite-order) automorphism of g.
Finally we comment on the centre z of k for (X, τ ) ∈ WSat(A). In Example 2 (i) we saw that it is one-dimensional if (X, τ ) = . Let c ∈ z and as before denote by i the unique element of I\X such that i = τ (i) and α i (ρ ∨ X ) / ∈ Z. Proposition 3.3 implies that c = c ′ + c ′′ with c ′ ∈ kî, c ′′ ∈ k(i). Moreover, since c ∈ z we have [x, c ′ ] = 0 for all x ∈ kî. We claim that c ′ = 0. If kî is semisimple, we are done. By a case-by-case analysis using Table 1, note that kî is semisimple unless (X, τ ) = 1 2 n with n > 2, in which case kî has a one-dimensional centre spanned by b 1 . Since [b 1 , b 2 ] = 0, it follows that also in this case c ′ = 0. Hence c ∈ k(i). Since the centre of k(i) is k(i) 2 we must have z ⊆ k(i) 2 . Define J even := {j ∈ J | the coefficient of α k in α j is even for all k ∈ I\X}. Remark 4. This should be compared with the situation for Satake diagrams and the associated fixed-point subalgebras, where the centre of k = g θ has dimension |I diff | + |I nsf | ∈ {0, 1}. Casework suggests that it is generated by a linear combination of either h i − h τ (i) (for i ∈ I diff ) or b i (for i ∈ I nsf ) and at least one other standard basis element of k.
3.3. The universal enveloping algebra U (k). Let (X, τ ) ∈ GSat(A) and γ ∈ Γ. We identify k with its image in U (k) under the canonical Lie algebra embedding. The generators of U (k) corresponding to b i (i ∈ I\X) can be modified by scalar terms, which is a straightforward generalization of [Ko14, Cor. 2.9].
Proposition 3.5. For (X, τ ) ∈ GSat(A), γ ∈ Γ and σ ∈ C I\X , the universal enveloping algebra U (k γ ) σ is generated by e i , f i (i ∈ X), h ∈ h θ and Again, if there is no cause for confusion, we will suppress γ and σ from the notation. From Conjecture 2 we obtain the following conditional result for U (k), which would address the question raised in [Ko14, Rmk. 2.10]: for (X, τ ) ∈ GSat(A), γ ∈ Γ and σ ∈ C I\X , U (k) is isomorphic to the algebra with generators h i , e i (i ∈ X), h i − h τ (i) (i ∈ I * , i = τ (i)), b i (i ∈ I) and relations (3.2-3.7).
We may view U (k) as a Hopf subalgebra of U (g) so that Lie algebra automorphisms of g lift to Hopf algebra automorphisms of U (g). Call two Hopf subalgebras B, B ′ of U (g) equivalent if there exists φ ∈ Aut Hopf (U (g)) such that B ′ = φ(B). Define Proposition 3.6. Let (X, τ ) ∈ GSat(A), γ ∈ Γ and σ ∈ C I\X . There exist γ ∈ Γ and σ ′ ∈ Σ such that Proof. The existence of γ follows from an argument analogous to the proof of [Ko14, Prop. 9.2 (i)]. Hence U (k γ ) σ is equivalent to U (k γ ) σ for some σ ∈ C I\X . Regarding the existence of σ ′ ∈ Σ, note that b i; γi ∈ (k γ ) ′ if i / ∈ I nsf , by Proposition 3.2. Hence U (k γ ) σ is already generated by e i , f i (i ∈ X), h ∈ h θ , b i; γi,0 for i ∈ (I\X)\I nsf and b i; γi, si for i ∈ I nsf . Hence we may take σ ′ i = σ i if i ∈ I nsf and σ ′ i = 0 otherwise.

Quantum pair algebras and the universal K-matrix revisited
Assume the d i are dyadic rationals and let K be a quadratic closure of C(q) where q is an indeterminate, so that q i := q di ∈ K for all i ∈ I. The Drinfeld-Jimbo quantum group U q (g) is an associative unital algebra over K which quantizes the universal enveloping algebra U (g). It is generated by i } where i ∈ I, satisfying the relations given in e.g. [Lu94, 3.1.1]. The Hopf algebra structure is the one defined in [Lu94, 3.1.3, 3.1.11, 3.3]. We write U q (h) for the Hopf subalgebra generated by K ±1 i for i ∈ I. We also write U q (n ± ) for the coideal subalgebras generated by the E i and F i (i ∈ I), respectively. The algebra U q (g) is Q-graded in terms of the root spaces We discuss some automorphisms of U q (g). Diagram automorphisms act (by relabelling) as Hopf algebra automorphisms on U q (g). Moreover, we have Lusztig's automorphisms T i for i ∈ I, given as T ′′ i,1 in [Lu94, 37.1.3], which generate an image of the braid group in Aut alg (U q (g)) and reproduce Ad(s i ) as q → 1. They satisfy T i (U q (g) α ) ⊆ U q (g) si(α) for all α ∈ Q and T i (K j ) = K j K −aij i for all j ∈ I. For X ⊆ I with w X = s i1 · · · s i ℓ a reduced decomposition we write T wX = T i1 · · · T i ℓ . If τ ∈ Aut(A) stabilizes X then [τ, T wX ] = 0. Finally, the assignments define ω q ∈ Aut alg (U q (g)) which is a particular quantum analogue of the Chevalley involution commuting with each T i , see [BK19, Lemma 7.1], and with Aut(A). We now discuss the changes to definitions and statements in the papers [Ko14, BK15, BK19, Ko20, DK19] needed to generalize these works to all generalized Satake diagrams.
In the remainder of this section we assume (X, τ ) ∈ CDec(A). The quantum analogon of the map θ = Ad(w X )τ ω is the map The quantization of the fixed-point subalgebra in the formalism by [Ko14] relies on the description of g θ in terms of generators given in [Ko14, Lemma 2.8]. Our k(X, τ ) by definition can be quantized to a right coideal subalgebra in the same way.
Definition 4. Let γ ∈ (K × ) I\X and σ ∈ K I\X . The quantum pair algebra B = B γ,σ (X, τ ) is the coideal subalgebra generated by U q (g X ), U q (h θ ) and the elements Remark 5.
(i) Note that U q (h θ ) is denoted U 0 Θ ′ in [Ko14]. In Moreover, if (X, τ ) ∈ GSat(A) and the tuples γ, σ lie in the sets 4.1. The bar involution for the subalgebra B. The bar involution of U q (g) is the algebra automorphism of U q (g) fixing E i , F i and inverting K ±1 i and q; it plays a crucial role in Lusztig's construction of the quasi R-matrix, see [Lu94]. In order to have a natural modification of Lusztig's theory of bar involutions and quasi R-matrices to the setting of quantum symmetric pair algebras, the paper [BK15] deals with the existence of an analogous map of B. This follows earlier work by [ES13] and [BW18a] in the case of certain quantum symmetric pairs of type AIII. More precisely, for suitable parameters γ, there exists an involutive algebra automorphism B : B → B which coincides with on U q (g X )U q (h θ ) and satisfies B i  Table 1, which we will now explain. We also note that the result [BK15, Prop. 2.3] was generalized in [BW18c,Thm. 4.1] to the Kac-Moody setting, but this did not explicitly include the cases where B is defined in terms of (X, τ ) ∈ GSat(A)\Sat(A). Here we provide an elementary proof for g of finite type which works for all compatible decorations. Let σ be the unique algebra anti-automorphism of U q (g) which fixes E i and F i and inverts K i . For i ∈ I, denote by r i Lusztig's right skew derivation, see [Lu94, 1.2.13]; it is the unique linear map on U q (n + ) such that for all x, y ∈ U q (n + ) with y ∈ U q (g) µ (µ ∈ Q + ) we have (4.6) r i (xy) = q µ(hi) i r i (x)y + xr i (y).
We denote [x, y] p := xy − pyx for x, y ∈ U q (g) and p ∈ K; note that σ( We start with a lemma that simplifies the proof drastically. Call a connected component of X simple if it is of the form {j} for some j ∈ I such that a ij = a ji ∈ {0, −1} for all i ∈ I\X. Lemma 4.1. Let i ∈ I\X and suppose ∅ = X(i) = X 1 ∪ · · · ∪ X ℓ is a decomposition into connected components. If i = τ (i) then ℓ ≤ 1 and if i = τ (i), all X s are simple except at most one. Denote by Y the non-simple connected component of X(i) if present and otherwise let Y be any simple connected component.
Proof. The first part of the Lemma follows from the classification of Satake diagrams in [Ar62] and an inspection of Table 1. Since adding simple components does not change the statement, it is true for all compatible decorations.
The second part is proven by induction with respect to the number of simple components. If there are none, then X(i) = Y and the statement is true. Otherwise, by the induction hypothesis we may suppose Since τ (j) = j, applying στ yields the desired result.
Proof. The proof is essentially casework, but first we make some observations.
(i) Since T wX (E i ) = T X(i) (E i ) we may assume that {i, τ (i)} is the only τ -orbit outside X.
(ii) We may assume X is nonempty as otherwise (r i T wX )(E i ) = 1. (iii) By Lemma 4.1 it suffices to prove the statement in the case that X is connected. Hence it suffices to prove the statement for those diagrams in Table 1 where the node i is the only node outside X, X is connected and |X| > 1. The only infinite family satisfying this condition is given by . In this case the proof is identical to the proof for the type BII case in (4.8) From the first expression we readily obtain (4.9) Now note that (s 3 s 4 s 2 s 3 s 2 s 4 s 3 )(α 2 ) = (s 3 s 2 s 3 )(α 2 ) = α 2 and s 3 s 4 s 2 s 3 s 2 s 4 s 3 and s 3 s 2 s 3 are reduced elements in W . Appealing to [Jan96,Prop. 8.20] we arrive at (4.10) By the q-Serre relation E 2 2 E 3 − (q 2 + q −2 )E 2 E 3 E 2 + E 3 E 2 2 = 0 we have (4.13) 3 ) so that for the first term of (4.12) we have (4.14) [ . The reduced elements s 3 s 2 s 3 and s 3 s 4 map α 2 to itself and α 3 to α 4 , respectively, so that ( where we have used the q-Serre relation E 2 E 4 − E 4 E 2 = 0. As a consequence, (4.12) yields (4.16) where we have used T 3 T 4 T 3 = T 4 T 3 T 4 . Because r i and T j commute if a ij = 0 we have (στ )((r 1 T wX )(E 1 )) = (r 1 T 4 T 3 T 4 T wX )(E 1 ) and by virtue of (4.8) the proof is complete.

4.2.
Quantum Serre relations for the B i . We now introduce some notation in order to discuss q-Serre relations for the B i . For x, y ∈ U q (g) and i, j ∈ I such that i = j recall the shorthand M ij = 1 − a ij and define Here Mij n qi is the q i -deformed binomial coefficient defined in [Lu94,1.3.3]. Note that the q-Serre relations for U q (g) are the relations According to [BK15,Thm. 3.7 (Case 4) and Thm. 3.9], for all i, j ∈ I such that i = j the elements These theorems cover all generalized Satake diagrams in Table 1 except 1 2 which we discuss now; without loss of generality we may assume d 1 = 3 and d 2 = 1. Note that Y 2 = −r 2 (T 1 (E 2 )) = −q −3 (q 3 − q −3 )E 1 . In this case we have, by a direct computation, qγ 2 Y 2 r 1 (qγ 2 Y 2 )q 9 K 1 + (σr 1 σ)(qγ 2 Y 2 )q −9 K −1 1 .
where ρ X ∈ (h X ) * is the Weyl vector of g X ; this specializes to [BK15,Prop. 3.5] if we set γ i = s(α τ (i) )c i . Note that Proposition 4.2 is crucial in this context; in the notation of [ for all i ∈ I\X, which specializes to [BK15, Eq. (3.28)] if γ i = s(α τ (i) )c i . Combining (4.20-4.21), the existence of the bar involution for B is equivalent to where we have used the analogue of (2.13) for roots, which can be obtained in the same way.
4.3. The universal K-matrix. Building upon the work in [BK15] on the bar involution, in [BK19] the universal K-matrix K = K(X, τ ) for B = B(X, τ ) is constructed for (X, τ ) ∈ Sat(A) and the relations (1.1) and (1.3) are derived. Again we comment on those places in this text with a nontrivial generalization to the setting of quantum pair algebras defined in terms of generalized Satake diagrams. In addition to (4.22) we also assume (4.23)  [Ar62]. Denoting by τ 0 the diagram automorphism corresponding to the longest element of the Weyl group of g, it is derived that τ 0 preserves X and commutes with τ ; furthermore one can choose γ ∈ Γ and σ ∈ Σ such that γ τ (i) = γ τ0(i) and σ τ (i) = σ τ0(i) for all i ∈ I\X and the above transformation properties with respect to the bar involution hold. Extending this analysis to Table 1, one checks that the remark is valid for all generalized Satake diagrams. In [Ko20] it is shown that K satisfies the axiom (1.2) for a universal K-matrix and the centre of B is described in terms of K, without using the defining condition of Satake diagrams or a case-by-case analysis; it follows that the results remain valid for (X, τ ) ∈ GSat(A). This is also the case for [DK19, Section 2] which entails an analysis of the restricted Weyl group and restricted root system following [Lu76] in order to establish a factorization of the quasi K-matrix (subject to a condition on Satake diagrams of restricted rank 2). In particular, from the statement τ 0 (X) = X for any (X, τ ) ∈ Sat(A) it is inferred that τ 0,X[i] (X) = X for all i ∈ I * and hence {w X w X[i] | i ∈ I * } is a Coxeter system for the group it generates. For all generalized Satake diagrams these conclusions follow directly from Theorem 2.1.
Acknowledgments. The authors thank A. Appel, M. Balagović, S. Kolb, J. Stokman, W. Wang and an anonymous referee for useful comments and discussions. V.R. was supported by the European Social Fund, grant number 09.3.3-LMT-K-712-02-0017. B.V. was supported by the Engineering and Physical Sciences Research Council, grant numbers EP/N023919/1 and EP/R009465/1. The authors gratefully acknowledge the financial support.
Appendix A. Deriving Serre relations for k The following three technical lemmas are used to derive the key equation (3.7). It is convenient to introduce the notation Q + X := Q + ∩ Q X = i∈X Z ≥0 α i .
First we consider the case w X (α i ) − (m − 1) α i − α j ∈ Φ + . Then m = 2 and since w X (α i ) − α i − α j ∈ Q + X it follows that [θ(f i ), [f i , f j ]] ∈ n + X . The claimed expression for L 2 follows immediately from (A.6) and those for L m with m > 2 from (3.2).
By virtue of the induction hypothesis, (A.14) simplifies to