Symmetry, Integrability and Geometry: Methods and Applications A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions ⋆

For a finite-dimensional simple Lie algebra $\mathfrak{g}$, let $U^+_q(\mathfrak{g})$ be the positive part of the quantized universal enveloping algebra, and $A_q(\mathfrak{g})$ be the quantized algebra of functions. We show that the transition matrix of the PBW bases of $U^+_q(\mathfrak{g})$ coincides with the intertwiner between the irreducible $A_q(\mathfrak{g})$-modules labeled by two different reduced expressions of the longest element of the Weyl group of $\mathfrak{g}$. This generalizes the earlier result by Sergeev on $A_2$ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for $C_2$. Our proof is based on a realization of $U^+_q(\mathfrak{g})$ in a quotient ring of $A_q(\mathfrak{g})$.


Introduction
Let g be a finite-dimensional simple Lie algebra and U q (g) be the Drinfeld-Jimbo quantized enveloping algebra. U q (g) has the subalgebra U + q (g) generated by the Chevalley generators e 1 , . . . , e n (n = rank g) corresponding to the simple roots. Denote by W = s 1 , . . . , s n the Weyl group of g generated by the simple reflections s 1 , . . . , s n . It is well known (see for example [15]) that for each reduced expression w 0 = s i 1 · · · s i l of the longest element w 0 ∈ W , one can associate the Poincaré-Birkhoff-Witt (PBW) basis of U + q (g) having the form

be another PBW basis
This paper is a contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa. The full collection is available at http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html associated with a yet different reduced expression w 0 = s j 1 · · · s j l . Following Lusztig [14], one expands a basis in terms of another as and obtains the transition coefficient γ A B uniquely. We have suppressed its dependence on i, j in the notation. Many remarkable properties are known for γ A B including the fact γ A B ∈ Z[q]. See [14,Proposition 2.3] for example.
In this paper we show that the transition coefficients γ = (γ A B ) coincide with the matrix elements of the intertwiner between the irreducible A q (g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. Here A q (g) denotes the quantized algebra of functions associated with g. It is a Hopf subalgebra of the dual U q (g) * which has been studied from a variety of aspects. See [5,11,17,18,21,22,23] for example. Let us briefly recall the most relevant result to the present paper due to Vaksman and Soibelman [21,22,23]. To each reduced expression of a (not necessarily longest) element w = s i 1 · · · s ir ∈ W , one can associate an irreducible representation π i labeled by i = (i 1 , . . . , i r ) having the form π i = π i 1 ⊗ · · · ⊗ π ir : A q (g) → End F q i 1 ⊗ · · · ⊗ F q ir , where each component π i : A q (g) → End(F q i ) is the fundamental representation of A q (g) on the q-oscillator Fock space F q i = m≥0 C(q)|m . See Section 4.1. The two irreducible representations π i and π j with j = (j 1 , . . . , j r ) are isomorphic if s i 1 · · · s ir = s j 1 · · · s jr ∈ W are reduced expressions (Theorem 4). Thus one has the intertwiner Φ = Φ i,j : F q i 1 ⊗· · ·⊗F q ir → F q j 1 ⊗· · ·⊗F q jr characterized by π j (g) • Φ = Φ • π i (g) ∀ g ∈ A q (g) up to an overall constant. Writing the basis of the Fock space F q i 1 ⊗ · · · ⊗ F q ir as |A = |a 1 ⊗ · · · ⊗ |a r with A = (a 1 , . . . , a r ) ∈ (Z ≥0 ) r , we define the matrix elements of Φ = (Φ A B ) by Φ|B = A Φ A B |A and the normalization Φ 0,...,0 0,...,0 = 1. Our main result (Theorem 5) is concerned with the longest element case r = l and is stated for each pair (i, j) as γ A B = Φ A B , i.e., For a convenience we also introduce the "checked" intertwiner Φ ∨ = Φ • σ, where σ(|a 1 ⊗ · · · ⊗ |a l ) = |a l ⊗ · · · ⊗ |a 1 is the reversal of the components. Our work is inspired by recent developments in 3-dimensional (3D) integrable systems related to rank 2 cases. Recall the Zamolodchikov tetrahedron equation [27] and the Isaev-Kulish 3D reflection equation [8]: They are equalities among the linear operators acting on the tensor product of 6 and 9 vector spaces, respectively. The indices specify the components in the tensor product on which the operators R and K act nontrivially. They serve as 3D analogue of the Yang-Baxter and reflection equations postulating certain factorization conditions of straight strings which undergo the scattering R and the reflection K by a boundary plane. For g = A 2 , Kapranov and Voevodsky [10] showed that R = Φ ∨ ∈ End(F ⊗3 q ) provides a solution to the tetrahedron equation (2). Moreover it was discovered by Sergeev [20] that the solution of the tetrahedron equation R in [2] (given also in [10] with misprint) is related with the transition matrix as γ = R • σ. Thus the equality (1) for g = A 2 is a corollary of their results. Apart from the A 2 case, it has been shown more recently [13] that K = Φ ∨ for g = C 2 yields the first nontrivial solution to the 3D reflection equation (3). See also [12] for g = B 2 . These results motivated us to investigate the general g case and have led to (1). It is our hope that it provides a useful insight into higher-dimensional integrable systems from the representation theory of quantum groups.
The layout of the paper is as follows. In Section 2, we summarize the definitions of U q (g) and PBW bases. In Section 3, we recall the basic facts on A q (g) following Kashiwara [11]. A fundamental role is played by the Peter-Weyl type Theorem 1. The relation with the Reshetikhin-Takhtadzhyan-Faddeev realization by generators and relations [18] is explained and its concrete forms are quoted for A n , C n and G 2 [19] which will be of use in later sections. The construction of a certain quotient ring A q (g) S of A q (g) and the special elements σ i ∈ A q (g) (Definition 1) and ξ i ∈ A q (g) S (36) will play a key role in our proof of (1). In Section 4, we briefly review the representation theory of A q (g) in [21,22,23] and sketch the intertwiners for the rank 2 cases. Section 5 is devoted to the proof of the main theorem γ = Φ. It reduces to the rank 2 cases and is done without recourse to explicit formulae of γ or Φ. Our method is to identify their characterizations under the correspondence e i → ξ i . Actually, this map extends to an algebra homomorphism U + q (g) → A q (g) S for general g as shown by Yakimov [26]. We give a direct proof of a part of his results in Section 6.

Def inition
In this paper g stands for a finite-dimensional simple Lie algebra. Its weight lattice, simple roots, simple coroots, fundamental weights are denoted by P , {α i } i∈I , {h i } i∈I , { i } i∈I where I is the index set of the Dynkin diagram of g. The Cartan matrix (a ij ) i,j∈I is given by The quantized enveloping algebra U q (g) is an associative algebra over Q(q) generated by i | i ∈ I} satisfying the relations: Here we use the following notations: We normalize the simple roots so that q i = q when α i is a short root. U q (g) is a Hopf algebra. As its comultiplication we adopt the following one

PBW basis
Let W be the Weyl group of g. It is generated by simple reflections {s i | i ∈ I} obeying the relations: s 2 i = 1, (s i s j ) m ij = 1 (i = j) where m ij = 2, 3, 4, 6 for h i , α j h j , α i = 0, 1, 2, 3, respectively. Let w 0 be the longest element of W and fix a reduced expression w 0 = s i 1 s i 2 · · · s i l .
Then every positive root occurs exactly once in Correspondingly, define elements e βr ∈ U q (g) (r = 1, . . . , l) by Here T i is the action of the braid group on U q (g) introduced by Lusztig [15]. It is an algebra automorphism and is given on the generators {e j } by U q (g) has a subalgebra generated by {e i | i ∈ I}, denoted by U + q (g). It is known that e βr ∈ U + q (g) holds for any r. U + q (g) has the so-called Poincaré-Birkhoff-Witt (PBW) basis. It depends on the reduced expression s i 1 s i 2 · · · s i l of w 0 . Set i = (i 1 , i 2 , . . . , i l ) and define for A = (a 1 , a 2 , . . . , a l ) ∈ (Z ≥0 ) l Then We hope that the notations e ir with i r ∈ I and e βr with a positive root β r can be distinguished properly from the context. In particular 3 Quantized algebra of functions A q (g)

Def inition
Following [11] we give the definition of the quantized algebra of functions A q (g). It is valid for any symmetrizable Kac-Moody algebra g. Let O int (g) be the category of integrable left U q (g)-modules M such that, for any u ∈ M , there exists l ≥ 0 satisfying e i 1 · · · e i l u = 0 for any i 1 , . . . , i l ∈ I. Then O int (g) is semisimple and any simple object is isomorphic to the irreducible module V (λ) with dominant integral highest weight λ. Similarly, we can consider the category O int (g opp ) of integrable right U q (g)-modules M r such that, for any v ∈ M r , there exists l ≥ 0 satisfying vf i 1 · · · f i l = 0 for any i 1 , . . . , i l ∈ I. O int (g opp ) is also semisimple and any simple object is isomorphic to the irreducible module V r (λ) with dominant integral highest weight λ. Let u λ (resp. v λ ) be a highest-weight vector of V (λ) (resp. V r (λ)). Then there exists a unique bilinear form ( , ) satisfying (v λ , u λ ) = 1 and (vP, u) = (v, P u) for v ∈ V r (λ), u ∈ V (λ), P ∈ U q (g).
Let U q (g) * be Hom Q(q) (U q (g), Q(q)) and , be the canonical pairing between U q (g) * and U q (g). The comultiplication ∆ of U q (g) induces a multiplication of U q (g) * by thereby giving U q (g) * the structure of Q(q)-algebra. It also has a U q (g)-bimodule structure by xϕy, P = ϕ, yP x for x, y, P ∈ U q (g).
We define the subalgebra A q (g) of U q (g) * by A q (g) = ϕ ∈ U q (g) * ; U q (g)ϕ belongs to O int (g) and ϕU q (g) belongs to O int (g opp ) , and call it the quantized algebra of functions.
The following theorem is the q-analogue of the Peter-Weyl theorem. See e.g. [11] for a proof.
where λ runs over all dominant integral weights, by the homomorphisms given by for v ∈ V r (λ), u ∈ V (λ), and P ∈ U q (g).
Let us now assume that g is a finite-dimensional simple Lie algebra. Then A q (g) turns out a Hopf algebra. See e.g. [9,Chapter 9]. Its comultiplication is also denoted by ∆.
Let R be the universal R matrix for U q (g). For its explicit formula see e.g. [4, p. 273]. For our purpose it is enough to know that where q (wt ·,wt ·) is an operator acting on the tenor product u λ ⊗ u µ of weight vectors u λ , u µ of weight λ, µ by q (wt ·,wt ·) (u λ ⊗ u µ ) = q (λ,µ) u λ ⊗ u µ , Q + = i Z ≥0 α i , and (U ± q ) ±β is the subspace of U ± q (g) spanned by root vectors corresponding to ±β. Fix λ, let {u λ j } and {v λ i } be bases of V (λ) and V r (λ) such that (v λ i , u λ j ) = δ ij , and ϕ λ ij = Ψ λ (v λ i ⊗ u λ j ). Let R be the so-called constant R matrix for V (λ) ⊗ V (µ). Denoting the homomorphism U q (g) → End(V (λ)) by π λ , it is given as where σ stands for the exchange of the first and second components. The scalar multiple is determined appropriately depending on g. The reason we apply σ is that it agrees to the convention of [18]. R satisfies We call such a relation RT T relation.

Right quotient ring
For later use we require a certain right quotient ring of A q (g) by a suitable multiplicatively closed subset S. We first review the general construction from [16,Chapter 2]. Let R be a noncommutative ring with 1 and S a multiplicatively closed subset of R. The following condition is called the right Ore condition: Then under the right Ore condition ass S turns out a two-sided ideal. Let : R → R/ass S denote the canonical projection. Suppose By passing to the images by , it suffices to consider the case when ass S = 0, and then elements of R S are of the form r/s. For r i /s i ∈ R/S (i = 1, 2) the addition and multiplication formulae are given by where u, u , v, v are so chosen that Let us return to our case where R = A q (g).
The following proposition is proven in [9]. However, we dare to prove again, since conventions might be different.
Proposition 1 (Corollary 9.1.4 of [9]). Let ϕ λµ be an element of A q (g) such that k i ϕ λµ = q ϕ λµ for any i ∈ I. Then the following commutation relation holds: Proof . Without loss of generality one can assume In view of (9), (10) we have Then (11) implies the commutation relation. The second relation follows from the first one, Let n be the rank of g and define which is obviously multiplicatively closed subset of A q (g).
In particular σ i is characterized as the unique element (up to an overall constant) in Im Ψ i such that f j σ i = σ i f j = 0 for all j ∈ I. We remark that Theorem 1 implies that if a nonzero element ϕ λ,µ ∈ A q (g) satisfies the assumption of Proposition 1 and f j ϕ λ,µ = ϕ λ,µ f j = 0 for all j, then λ = w 0 µ must hold.
In [9] it is shown that A q (g) is an integral domain (Lemma 9.1.9), hence (reg) is satisfied, and that (Ore) is also satisfied (Lemma 9.1.10). Therefore we have the following theorem. (A proof is attached for self-containedness.) Theorem 3. The right quotient ring A q (g) S exists.
Let us prove (1). Let ϕ = j ϕ j be the two-sided weight decomposition. If ϕ j s = 0 for some j, ϕs = 0 since the weights of ϕ j s are distinct. Hence we can reduce the claim when ϕ is a weight vector. Suppose ϕ = µ ϕ µ , ϕ µ ∈ Im Ψ µ and let λ be a maximal weight, with respect to the standard ordering on weights, such that ϕ λ = 0. Choose sequences i 1 , . . . , i k and j 1 , . . . , j l such that f i k · · · f i 1 ϕ λ f j 1 · · · f j l turns out a left-lowest and right-highest weight vector. Then by Lemma 1 it coincides with cs with some c ∈ Q(q) × , s ∈ S. Then with another c ∈ Q(q) × . By the maximality of λ the remaining part + · · · in the right-hand side does not contain the terms with the same two-sided weight. Hence · · · = 0. Therefore, the left-hand side is not 0 and we conclude ϕs = 0.
(1 ) is similar. For (2) we can reduce the claim when ϕ is a weight vector, and in this case the claim is clear from Proposition 1.

Realization by generators and relations
We consider the fundamental representation V ( 1 ) of U q (g) for g = A n−1 , C n , G 2 . Set N = dim V ( 1 ). It is known [5,18] that A q (g) for g = A n−1 , C n , G 2 is realized as an associative algebra with appropriate generators (t ij ) 1≤i,j≤N corresponding to and additional ones depending on g. See below for each g under consideration. In all cases, there exists a comultiplication ∆ : 3.3.1 A n−1 case We present formulae for A q (A n−1 ). In this case N = n. Let u 1 and v 1 be the highest-weight In A n−1 case we need another condition that the quantum determinant is 1, i.e., where S n = W (A n−1 ) is the symmetric group of degree n and (σ) is the length of σ. According to Definition 1, we have σ 1 = t 13 and σ 2 = t 12 t 23 − qt 22 t 13 . As an exposition, we note that σ i e i in (39) is derived from . See e.g. [17] for an extensive treatment.
Define t ij = Ψ 1 (v i ⊗u j ). The RT T relations are given by (13) with the above R ij,kl . Additional relations are given by

G 2 case
We have N = 7 in this case. We adopt the basis {u i } of V ( 1 ) that has the representation matrices given as in [19, equation (29)], and let {v i } the dual basis in is generated by (t ij ) 1≤i,j≤7 satisfying (i) and (ii) given below.
(ii) Additional relations where g ij and f ij k are given by [19, equations (30), (31)]. The relations [19, equations (20), (22)] are equivalent to (15) if the RT T relations are imposed. See the explanation after [19,Definition 7]. Note also that we use the opposite indices of the Dynkin diagram to [19].
4 Representations of A q (g) 4

.1 General remarks
Let us recall the results in [22,23] on the representations of A q (g) necessary in this paper. Consider the simplest example A q (A 1 ) generated by t 11 , t 12 , t 21 , t 22 with the relations It has a representation on the Fock space F q = m≥0 C(q)|m : In what follows, the symbols k, a + , a − shall also be regarded as the elements from End(F q ). It is easy to check that the following map π defines an irreducible representation of A q (A 1 ) on F q : where α, µ are nonzero parameters.
(1) For each vertex i of the Dynkin diagram of g, A q (g) has an irreducible representation π i factoring through (18) via A q (g) A q i (sl 2,i ). (sl 2,i denotes the sl 2 -subalgebra of g associated to i.) (2) Irreducible representations of A q (g) are in one to one correspondence with the elements of the Weyl group W of g.
(3) Let w = s i 1 · · · s i l ∈ W be an reduced expression in terms of the simple reflections. Then the irreducible representation corresponding to w is isomorphic to π i 1 ⊗ · · · ⊗ π i l .
A crucial corollary of Theorem 4 is the following: If s i 1 · · · s i l = s j 1 · · · s j l ∈ W are reduced expressions, then π i 1 ,...,i l π j 1 ,...,j l .
In particular, there exists the isomorphism Φ : F q i 1 ⊗ · · · ⊗ F q i l → F q j 1 ⊗ · · · ⊗ F q j l characterized (up to an overall constant) by Here π i 1 ,...,i l (g = t ij ) for example means the tensor product representation r 1 ,...,r l−1 π i 1 (t ir 1 ) ⊗ · · · ⊗ π i l (t r l−1 ,j ) obtained by the (l − 1)-fold application of the coproduct (14). Elements of the Fock space |m 1 ⊗ · · · ⊗ |m l ∈ F q j 1 ⊗ · · · ⊗ F q j l will simply be denoted by |m 1 , . . . , m l . We will always normalize the intertwiner by the condition Φ|0, 0, . . . , 0 = |0, 0, . . . , 0 . The exchange of the ith and the jth tensor components from the left will be denoted by P ij . In the remainder of this section we concentrate on A q (g) of rank 2 cases g = A 2 , C 2 and G 2 , and present the concrete forms of the fundamental representations, definition of the intertwiners with a few examples of their matrix elements.

A 2 case
Let T = (t ij ) 1≤i,j≤3 be the 3 × 3 matrix of the generators of A q (A 2 ). The fundamental representations π i : A q (A 2 ) → End(F q ) (i = 1, 2) are given by where α i , µ i are nonzero parameters. The Weyl group W = s 1 , s 2 is the Coxeter system with the relations s 2 1 = s 2 2 = 1, s 1 s 2 s 1 = s 2 s 1 s 2 .
Thus we have the isomorphism π 2121 π 1212 . Let Φ be the corresponding intertwiner and denote by K the checked intertwiner Φ ∨ π 2121 Φ = Φπ 1212 , π 2121 K = Kπ 2121 , π 2121 = P 14 P 23 π 1212 P 14 P 23 , Define the matrix elements of K and its parameter-free part K by Then the following properties are valid for K = (K abcd ijkl ) [13]: Due to (24), K is the infinite direct sum of finite-dimensional matrices. An explicit formula of K abcd ijkl is available in [13, equations (3.27), (3.28)]. This K and R in Section 4.2 satisfy [13] the 3D reflection equation (3).
The Weyl group W = s 1 , s 2 is the Coxeter system with the relations s 2 1 = s 2 2 = 1, s 2 s 1 s 2 s 1 s 2 s 1 = s 1 s 2 s 1 s 2 s 1 s 2 .
Thus we have the isomorphism π 212121 π 121212 . Let Φ be the corresponding intertwiner and denote by F the checked intertwiner Φ ∨ π 212121 Φ = Φπ 121212 , π 212121 F = F π 212121 , π 212121 = P 16 P 25 P 34 π 121212 P 16 P 25 P 34 , Define the matrix elements of F and its parameter-free part F by Then the following properties are valid for F = (F abcdef ijklmn ): Due to (29), F is the infinite direct sum of finite-dimensional matrices. The formula for F abcdef ijklmn | q=0 can be deduced by the ultradiscretization (tropical form) of [3, Theorem 3.1(c)]. Although a tedious algorithm can be formulated for calculating any given F abcdef ijklmn by using (28), an explicit formula for it is yet to be constructed. Example 3. The following is the list of all the nonzero F abcdef 010101 :

Main theorem
In this section we fix two reduced words i = (i 1 , . . . , i l ), j = (j 1 , . . . , j l ) of the longest element w 0 ∈ W .

Def initions of γ A B and Φ A B
In the U q (g) side, we defined the PBW bases E A i , E B j of U + q (g) in Section 2.2. We define their transition coefficient γ A B by While, in the A q (g) side, we have the intertwiner Φ : F q i 1 ⊗· · ·⊗F q i l → F q j 1 ⊗· · ·⊗F q j l satisfying We take the parameters µ, α in (18) to be 1. This in particular means for rank 2 cases that µ i , α i entering π i (T ) in (19), (23) and (27) are all 1. The intertwiner Φ is normalized by Φ|0, 0, . . . , 0 = |0, 0, . . . , 0 . Under these conditions a matrix element Φ A B of Φ is uniquely specified by where A = (a 1 , . . . , a l ) ∈ (Z ≥0 ) l and |A = |a 1 ⊗ · · · ⊗ |a l ∈ F q j 1 ⊗ · · · ⊗ F q j l and similarly for |B ∈ F q i 1 ⊗ · · · ⊗ F q i l . Then our main result is For any pair (i, j), from i one can reach j by applying Coxeter relations. In view of the uniqueness of γ and Φ and the fact that the braid group action T i is an algebra homomorphism, the proof of this theorem reduces to establishing the same equality for all g of rank 2. This will be done in the rest of this section.
See Definition 1 for σ i and (39), (41), (42) for the concrete forms in rank 2 cases. In Section 5.3 we will check the following statement case by case.
Proposition 2. For g of rank 2, π i (σ i ) is invertible and the following equality is valid: where the right-hand side means λ i π i (σ i e i )π i (σ i ) −1 .
Proof of Theorem 5 for rank 2 case. We write the both sides of (37) as M i AB and the one for i instead of i as M i AB . From On the other hand, the action of the two sides of (31) with g = ξ i and j = i are calculated as A satisfy the same relation. Moreover the maps π i and ρ i are both homomorphism, i.e., π i (gh) = π i (g)π i (h) and ρ i (xy) = ρ i (x)ρ i (y). We know that Φ is the intertwiner of the irreducible A q (g) modules and (33) obviously holds as 1 = 1 at A = B = (0, . . . , 0). Thus it is valid for arbitrary A and B. Conjecture 1. The equality (37) is valid for any g.

Explicit formulae for rank 2 cases: Proof of Proposition 2
Here we present the explicit formulae of (34) with x = e i and (35) with g = σ i , σ i e i that allow one to check Proposition 2. We use the notation i = q i − q −i . In each case, there are two i-sequences, 1 and 2 = 1 corresponding to the two reduced words. Let χ be the anti-algebra involution such that χ(e i ) = e i . Then the relation χ(Ẽ A i ) =ẼĀ i holds, whereĀ = (a l , . . . , a 2 , a 1 ) denotes the reversal of A = (a 1 , a 2 , . . . , a l ). Applying χ to (34) with x = e i yields the right multiplication formulaẼĀ i · e i = BẼB i ρ i (e i ) BA for i -sequence. In view of this fact, we shall present the left and right multiplication formulae for i = 2 only.
As for (35) with g = ξ i in (36), explicit formulae for σ i , σ i e i ∈ A q (g) and their image by the both representations π 1 and π 2 will be given. We include an exposition on how to use these data to check (37) along the simplest A 2 case. The C 2 and G 2 cases are similar.

A 2 case
The q-Serre relations are  Proof . By induction, we have The lemma is a direct consequence of these formulae.
By applying χ to the first two relations in Lemma 2, we get Thus we find ρ i (e i ) = ρ i (e 3−i ). This property is only valid for A 2 and not in C 2 and G 2 .
See the exposition at the end of Section 3.3.1 and the remark after Lemma 1.
In terms of the checked intertwiner R in Section 4.2, Theorem 5 implies This is valid either for i = 1 or 2 thanks to the middle property in (21). This relation connecting the PBW bases with the solution of the tetrahedron equation is due to [20].

C 2 case
The q-Serre relations are Their commutation relations are as follows: Proof . By induction, we have

Discussion
In view of Proposition 2 it is natural to expect that the map defined on generators of U + q (g) as e i → η i := σ i e i /σ i extends to an algebra homomorphism from U + q (g) to A q (g) S , namely, η i satisfies q-Serre relations. In fact, it is true not only for rank 2 cases but also for any g.
Here we have set s = 1 − a 12 − r. Recalling that σ 1 and σ 2 commute with each other, we can reduce the claim to showing Note that the right (resp. left) weight of Z is (1 − a 12 )( 1 − α 1 ) + ( 2 − α 2 ) (resp. w 0 ((1 − a 12 ) 1 + 2 )). The two weights are not related by the longest element w 0 ∈ W . Hence if we show f i Z = Zf i = 0 for any i, we can conclude Z = 0 by the remark after Lemma 1. The properties f i Z = 0 for any i and Zf i = 0 for i = 1, 2 are trivial.
First we show Zf 2 = 0. We have Next, we show Zf 1 = 0. Remark 1. The special case w = w 0 of [26,Theorem 3.7] gives Theorem 6 here. Moreover [26,Theorem 3.7] also shows that U + q (g) is isomorphic to an explicit subalgebra of A q (g) S . We would like to thank the referee for pointing this out and for giving helpful comments.
It will be interesting to investigate it further in the light of the quantum cluster algebra which has been recognized as a fundamental structure in the quantized algebra of functions [6]. The representations via multiplication on PBW bases also play a fundamental role in the study of the positive principal series representations and modular double [7].
In this paper we have not discussed the analogue of the tetrahedron and 3D reflection equations for general g. However, from our proof of Theorem 5, we expect that the basic constituents are R and K only, and their compatibility condition is reduced to the rank 2 cases (2) and (3).