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 g, let U⁺q(g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U⁺q(g) coincides with the intertwiner between the irreducible Aq(g)-mo...

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Datum:	2013
Hauptverfasser:	Kuniba, A., Okado, M., Yamada, Y.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2013
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1493422019-02-22T01:24:08Z A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions Kuniba, A. Okado, M. Yamada, Y. For a finite-dimensional simple Lie algebra g, let U⁺q(g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U⁺q(g) coincides with the intertwiner between the irreducible Aq(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A₂ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C₂. Our proof is based on a realization of U⁺q(g) in a quotient ring of Aq(g). 2013 Article A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Function / A. Kuniba, M. Okado, Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 20G42; 81R50; 17B80 DOI: http://dx.doi.org/10.3842/SIGMA.2013.049 http://dspace.nbuv.gov.ua/handle/123456789/149342 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For a finite-dimensional simple Lie algebra g, let U⁺q(g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U⁺q(g) coincides with the intertwiner between the irreducible Aq(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A₂ related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C₂. Our proof is based on a realization of U⁺q(g) in a quotient ring of Aq(g).
format	Article
author	Kuniba, A. Okado, M. Yamada, Y.
spellingShingle	Kuniba, A. Okado, M. Yamada, Y. A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Kuniba, A. Okado, M. Yamada, Y.
author_sort	Kuniba, A.
title	A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions
title_short	A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions
title_full	A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions
title_fullStr	A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions
title_full_unstemmed	A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions
title_sort	common structure in pbw bases of the nilpotent subalgebra of uq(g) and quantized algebra of functions
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149342
citation_txt	A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Function / A. Kuniba, M. Okado, Y. Yamada // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 049, 23 pages A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions? Atsuo KUNIBA †, Masato OKADO ‡ and Yasuhiko YAMADA § † Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan E-mail: atsuo@gokutan.c.u-tokyo.ac.jp ‡ Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan E-mail: okado@sigmath.es.osaka-u.ac.jp § Department of Mathematics, Faculty of Science, Kobe University, Hyogo 657-8501, Japan E-mail: yamaday@math.kobe-u.ac.jp Received March 19, 2013, in final form July 10, 2013; Published online July 19, 2013 http://dx.doi.org/10.3842/SIGMA.2013.049 Abstract. For a finite-dimensional simple Lie algebra g, let U+ q (g) be the positive part of the quantized universal enveloping algebra, and Aq(g) be the quantized algebra of functions. We show that the transition matrix of the PBW bases of U+ q (g) coincides with the intertwiner between the irreducible Aq(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. This generalizes the earlier result by Sergeev on A2 related to the tetrahedron equation and endows a new representation theoretical interpretation with the recent solution to the 3D reflection equation for C2. Our proof is based on a realization of U+ q (g) in a quotient ring of Aq(g). Key words: quantized enveloping algebra; PBW bases; quantized algebra of functions; tetra- hedron equation 2010 Mathematics Subject Classification: 17B37; 20G42; 81R50; 17B80 Dedicated to Professors Anatol N. Kirillov and Tetsuji Miwa who taught us the joy of doing mathematics. 1 Introduction Let g be a finite-dimensional simple Lie algebra and Uq(g) be the Drinfeld–Jimbo quantized enveloping algebra. Uq(g) has the subalgebra U+ q (g) generated by the Chevalley generators e1, . . . , en (n = rank g) corresponding to the simple roots. Denote by W = 〈s1, . . . , sn〉 the Weyl group of g generated by the simple reflections s1, . . . , sn. It is well known (see for example [15]) that for each reduced expression w0 = si1 · · · sil of the longest element w0 ∈W , one can associate the Poincaré–Birkhoff–Witt (PBW) basis of U+ q (g) having the form EAi = e (a1) β1 e (a2) β2 · · · e(al) βl , A = (a1, . . . , al) ∈ (Z≥0)l, where e (ai) βi ’s are the divided powers of the positive root vectors determined by the choice i = (i1, . . . , il). See Section 2.2. Let { EAj \| A ∈ (Z≥0)l } with j = (j1, . . . , jl) 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 mailto:atsuo@gokutan.c.u-tokyo.ac.jp mailto:okado@sigmath.es.osaka-u.ac.jp mailto:yamaday@math.kobe-u.ac.jp http://dx.doi.org/10.3842/SIGMA.2013.049 http://www.emis.de/journals/SIGMA/InfiniteAnalysis2013.html 2 A. Kuniba, M. Okado and Y. Yamada associated with a yet different reduced expression w0 = sj1 · · · sjl . Following Lusztig [14], one expands a basis in terms of another as EAi = ∑ B∈(Z≥0)l γABE B j and obtains the transition coefficient γAB uniquely. We have suppressed its dependence on i, j in the notation. Many remarkable properties are known for γAB including the fact γAB ∈ Z[q]. See [14, Proposition 2.3] for example. In this paper we show that the transition coefficients γ = (γAB) coincide with the matrix ele- ments of the intertwiner between the irreducible Aq(g)-modules labeled by two different reduced expressions of the longest element of the Weyl group of g. Here Aq(g) denotes the quantized algebra of functions associated with g. It is a Hopf subalgebra of the dual Uq(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 = si1 · · · sir ∈ W , one can associate an irreducible representation πi labeled by i = (i1, . . . , ir) having the form πi = πi1 ⊗ · · · ⊗ πir : Aq(g)→ End ( Fqi1 ⊗ · · · ⊗ Fqir ) , where each component πi : Aq(g)→ End(Fqi) is the fundamental representation of Aq(g) on the q-oscillator Fock space Fqi = ⊕ m≥0 C(q)\|m〉. See Section 4.1. The two irreducible representa- tions πi and πj with j = (j1, . . . , jr) are isomorphic if si1 · · · sir = sj1 · · · sjr ∈W are reduced ex- pressions (Theorem 4). Thus one has the intertwiner Φ = Φi,j : Fqi1⊗· · ·⊗Fqir → Fqj1⊗· · ·⊗Fqjr characterized by πj(g) ◦ Φ = Φ ◦ πi(g) ∀ g ∈ Aq(g) up to an overall constant. Writing the basis of the Fock space Fqi1 ⊗ · · · ⊗ Fqir as \|A〉 = \|a1〉 ⊗ · · · ⊗ \|ar〉 with A = (a1, . . . , ar) ∈ (Z≥0)r, we define the matrix elements of Φ = (ΦA B) by Φ\|B〉 = ∑ A ΦA B\|A〉 and the normalization Φ0,...,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 γAB = ΦA B, i.e., γ = Φ. (1) For a convenience we also introduce the “checked” intertwiner Φ∨ = Φ ◦ σ, where σ(\|a1〉 ⊗ · · · ⊗ \|al〉) = \|al〉 ⊗ · · · ⊗ \|a1〉 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]: R356R246R145R123 = R123R145R246R356, (2) R456R489K3579R269R258K1678K1234 = K1234K1678R258R269K3579R489R456. (3) They are equalities among the linear operators acting on the tensor product of 6 and 9 vec- tor 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 = A2, 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 PBW Bases and Quantized Algebra of Functions 3 transition matrix as γ = R ◦ σ. Thus the equality (1) for g = A2 is a corollary of their results. Apart from the A2 case, it has been shown more recently [13] that K = Φ∨ for g = C2 yields the first nontrivial solution to the 3D reflection equation (3). See also [12] for g = B2. 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 Uq(g) and PBW bases. In Section 3, we recall the basic facts on Aq(g) following Kashiwara [11]. A funda- mental 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 An, Cn and G2 [19] which will be of use in later sections. The construction of a certain quotient ring Aq(g)S of Aq(g) and the special elements σi ∈ Aq(g) (Definition 1) and ξi ∈ Aq(g)S (36) will play a key role in our proof of (1). In Section 4, we briefly review the representation theory of Aq(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 ei 7→ ξi. Actually, this map extends to an algebra homomorphism U+ q (g)→ Aq(g)S for general g as shown by Yakimov [26]. We give a direct proof of a part of his results in Section 6. 2 Quantized enveloping algebra Uq(g) 2.1 Definition 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 , {αi}i∈I , {hi}i∈I , {$i}i∈I where I is the index set of the Dynkin diagram of g. The Cartan matrix (aij)i,j∈I is given by aij = 〈hi, αj〉 = 2(αi, αj)/(αi, αi). The quantized enveloping algebra Uq(g) is an associative algebra over Q(q) generated by {ei, fi, k±1 i \| i ∈ I} satisfying the relations: kikj = kjki, kik −1 i = k−1 i ki = 1, kiejk −1 i = q 〈hi,αj〉 i ej , kifjk −1 i = q −〈hi,αj〉 i fj , [ei, fj ] = δij ki − k−1 i qi − q−1 i , 1−aij∑ r=0 (−1)re (r) i eje (1−aij−r) i = 1−aij∑ r=0 (−1)rf (r) i fjf (1−aij−r) i = 0, i 6= j. (4) Here we use the following notations: qi = q(αi,αi)/2, [m]i = (qmi −q −m i )/(qi−q−1 i ), [n]i! = n∏ m=1 [m]i, e (n) i = eni /[n]i!, f (n) i = fni /[n]i!. We normalize the simple roots so that qi = q when αi is a short root. Uq(g) is a Hopf algebra. As its comultiplication we adopt the following one ∆(ki) = ki ⊗ ki, ∆(ei) = ei ⊗ 1 + ki ⊗ ei, ∆(fi) = fi ⊗ k−1 i + 1⊗ fi. 2.2 PBW basis Let W be the Weyl group of g. It is generated by simple reflections {si \| i ∈ I} obeying the relations: s2 i = 1, (sisj) mij = 1 (i 6= j) where mij = 2, 3, 4, 6 for 〈hi, αj〉〈hj , αi〉 = 0, 1, 2, 3, respectively. Let w0 be the longest element of W and fix a reduced expression w0 = si1si2 · · · sil . 4 A. Kuniba, M. Okado and Y. Yamada Then every positive root occurs exactly once in β1 = αi1 , β2 = si1(αi2), . . . , βl = si1si2 · · · sil−1 (αil). Correspondingly, define elements eβr ∈ Uq(g) (r = 1, . . . , l) by eβr = Ti1Ti2 · · ·Tir−1(eir). (5) Here Ti is the action of the braid group on Uq(g) introduced by Lusztig [15]. It is an algebra automorphism and is given on the generators {ej} by Ti(ei) = −kifi, Ti(ej) = −aij∑ r=0 (−1)rqri e (r) i eje (−aij−r) i , i 6= j. Uq(g) has a subalgebra generated by {ei \| i ∈ I}, denoted by U+ q (g). It is known that eβr ∈ 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 si1si2 · · · sil of w0. Set i = (i1, i2, . . . , il) and define for A = (a1, a2, . . . , al) ∈ (Z≥0)l EAi = e (a1) β1 e (a2) β2 · · · e(al) βl . (6) Then {EAi \| A ∈ (Z≥0)l} forms a basis of U+ q (g). We hope that the notations eir with ir ∈ I and eβr with a positive root βr can be distinguished properly from the context. In particular e (ar) βr = (eβr)ar/ ar∏ m=1 pmr −p −m r pr−p−1 r with pr = q(βr,βr)/2. 3 Quantized algebra of functions Aq(g) 3.1 Definition Following [11] we give the definition of the quantized algebra of functions Aq(g). It is valid for any symmetrizable Kac–Moody algebra g. Let Oint(g) be the category of integrable left Uq(g)-modules M such that, for any u ∈M , there exists l ≥ 0 satisfying ei1 · · · eilu = 0 for any i1, . . . , il ∈ I. Then Oint(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 Oint(g opp) of integrable right Uq(g)-modules M r such that, for any v ∈ M r, there exists l ≥ 0 satisfying vfi1 · · · fil = 0 for any i1, . . . , il ∈ I. Oint(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 uλ (resp. vλ) be a highest-weight vector of V (λ) (resp. V r(λ)). Then there exists a unique bilinear form ( , ) V r(λ)⊗ V (λ)→ Q(q) satisfying (vλ, uλ) = 1 and (vP, u) = (v, Pu) for v ∈ V r(λ), u ∈ V (λ), P ∈ Uq(g). Let Uq(g)∗ be HomQ(q)(Uq(g),Q(q)) and 〈 , 〉 be the canonical pairing between Uq(g)∗ and Uq(g). The comultiplication ∆ of Uq(g) induces a multiplication of Uq(g)∗ by 〈ϕϕ′, P 〉 = 〈ϕ⊗ ϕ′,∆(P )〉 for P ∈ Uq(g), (7) PBW Bases and Quantized Algebra of Functions 5 thereby giving Uq(g)∗ the structure of Q(q)-algebra. It also has a Uq(g)-bimodule structure by 〈xϕy, P 〉 = 〈ϕ, yPx〉 for x, y, P ∈ Uq(g). (8) We define the subalgebra Aq(g) of Uq(g)∗ by Aq(g) = { ϕ ∈ Uq(g)∗;Uq(g)ϕ belongs to Oint(g) and ϕUq(g) belongs to Oint(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. Theorem 1. As a Uq(g)-bimodule Aq(g) is isomorphic to ⊕ λ V r(λ)⊗V (λ), where λ runs over all dominant integral weights, by the homomorphisms Ψλ : V r(λ)⊗ V (λ)→ Aq(g) given by 〈Ψλ(v ⊗ u), P 〉 = (v, Pu) for v ∈ V r(λ), u ∈ V (λ), and P ∈ Uq(g). Let us now assume that g is a finite-dimensional simple Lie algebra. Then Aq(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 Uq(g). For its explicit formula see e.g. [4, p. 273]. For our purpose it is enough to know that R ∈ q(wt ·,wt ·) ⊕ β∈Q+ (U+ q )β ⊗ (U−q )−β, (9) where q(wt ·,wt ·) is an operator acting on the tenor product uλ ⊗ uµ of weight vectors uλ, uµ of weight λ, µ by q(wt ·,wt ·)(uλ⊗ uµ) = q(λ,µ)uλ⊗ uµ, Q+ = ⊕ i Z≥0αi, and (U±q )±β is the subspace of U±q (g) spanned by root vectors corresponding to ±β. Fix λ, let {uλj } and {vλi } be bases of V (λ) and V r(λ) such that (vλi , u λ j ) = δij , and ϕλij = Ψλ(vλi ⊗ uλj ). Let R be the so-called constant R matrix for V (λ) ⊗ V (µ). Denoting the homo- morphism Uq(g)→ End(V (λ)) by πλ, it is given as R ∝ (πλ ⊗ πµ)(σR), (10) 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 R∆(x) = ∆′(x)R for any x ∈ Uq(g), where ∆′ = σ ◦∆. Define matrix elements Rij,kl by R(uλk ⊗u µ l ) = ∑ i,j Rij,klu λ i ⊗u µ j . Define the right action of R on V r(λ)⊗V r(µ) in such a way that ((vλi ⊗v µ j )R, uλk⊗u µ l ) = (vλi ⊗v µ j , R(uλk⊗u µ l )) holds. Then we have (vλi ⊗ v µ j )R = ∑ k,lRij,klv λ k ⊗ v µ l . From∑ m,p Rij,mp〈ϕλmkϕ µ pl, x〉 = ∑ m,p Rij,mp〈ϕλmk ⊗ ϕ µ pl,∆(x)〉 = ∑ m,p Rij,mp(v λ m ⊗ vµp ,∆(x)(uλk ⊗ u µ l )) = ((vλi ⊗ v µ j )R,∆(x)(uλk ⊗ u µ l )) = (vλi ⊗ v µ j , R∆(x)(uλk ⊗ u µ l )) = ∑ m,p (vλi ⊗ v µ j ,∆ ′(x)(uλm ⊗ uµp ))Rmp,kl 6 A. Kuniba, M. Okado and Y. Yamada = ∑ m,p (vµj ⊗ v λ i ,∆(x)(uµp ⊗ uλm))Rmp,kl = ∑ m,p 〈ϕµjp ⊗ ϕ λ im,∆(x)〉Rmp,kl = ∑ m,p 〈ϕµjpϕ λ im, x〉Rmp,kl for any x ∈ Uq(g), we have∑ m,p Rij,mpϕ λ mkϕ µ pl = ∑ m,p ϕµjpϕ λ imRmp,kl. (11) We call such a relation RTT relation. 3.2 Right quotient ring Aq(g)S For later use we require a certain right quotient ring of Aq(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: (Ore) For any r ∈ R, s ∈ S, rS ∩ sR 6= ∅. Set assS = {r ∈ R \| rs = 0 for some s ∈ S}. Then under the right Ore condition assS turns out a two-sided ideal. Let : R → R/assS denote the canonical projection. Suppose (reg) S consists of regular elements, namely, elements x such that both xr = 0 and rx = 0 imply r = 0. Then a theorem in [16, Chapter 2] states Theorem 2 (Theorem 2.1.12 of [16]). The right quotient ring RS exists, if and only if (Ore) and (reg) are satisfied. By passing to the images by , it suffices to consider the case when assS = 0, and then elements of RS are of the form r/s. For ri/si ∈ R/S (i = 1, 2) the addition and multiplication formulae are given by r1/s1 + r2/s2 = (r1u+ r2u ′)/(s1u), (r1/s1)(r2/s2) = (r1v ′)/(s2v), (12) where u, u′, v, v′ are so chosen that s1u = s2u ′ (u ∈ S, u′ ∈ R), r2v = s1v ′ (v ∈ S, v′ ∈ R). Let us return to our case where R = Aq(g). Definition 1. For any i ∈ I, let uw0$i (resp. v$i) be a lowest (resp. highest) weight vector of V ($i) (resp. V r($i)). Set σi = Ψ$i(v$i ⊗ uw0$i). 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 ϕλµ be an element of Aq(g) such that kiϕλµ = q 〈hi,µ〉 i ϕλµ, ϕλµki = q 〈hi,λ〉 i ϕλµ for any i ∈ I. Then the following commutation relation holds: q($i,λ)σiϕλµ = q(w0$i,µ)ϕλµσi. In particular, σiσj = σjσi for any i, j. PBW Bases and Quantized Algebra of Functions 7 Proof. Without loss of generality one can assume ϕλµ = Ψν(vλ ⊗ uµ) for some ν, vλ ∈ V r(ν), uµ ∈ V (ν) such that kiuµ = q 〈hi,µ〉 i uµ, vλki = q 〈hi,λ〉 i vλ. In view of (9), (10) we have R(uw0$i ⊗ uµ) = q(w0$i,µ)uw0$i ⊗ uµ, (v$i ⊗ vλ)R = q($i,λ)v$i ⊗ vλ. Then (11) implies the commutation relation. The second relation follows from the first one, since ($i, $i) = (w0$i, w0$i). � Let n be the rank of g and define S = { σm1 1 · · ·σmn n \| m1, . . . ,mn ∈ Z≥0 } , which is obviously multiplicatively closed subset of Aq(g). Lemma 1. Let s be a nonzero element in Im Ψλ satisfying fis = sfi = 0 for any i ∈ I. Then s ∈ Q(q)×σλ11 · · ·σλnn where Q(q)× = Q(q) \ {0} and λi = 〈hi, λ〉. Proof. By (7), (8) fiσ λ1 1 · · ·σλnn = σλ11 · · ·σλnn fi = 0 for any i and σλ11 · · ·σλnn belongs to Im Ψλ. By Theorem 1 such an element is unique up to an element of Q(q)×. � In particular σi is characterized as the unique element (up to an overall constant) in Im Ψ$i such that fjσi = σifj = 0 for all j ∈ I. We remark that Theorem 1 implies that if a nonzero element ϕλ,µ ∈ Aq(g) satisfies the assumption of Proposition 1 and fjϕλ,µ = ϕλ,µfj = 0 for all j, then λ = w0µ must hold. In [9] it is shown that Aq(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 Aq(g)S exists. Proof. In view of Theorem 2 it is enough to show that (1) if ϕ 6= 0, then ϕs 6= 0 for any s ∈ S, (1′) if ϕ 6= 0, then sϕ 6= 0 for any s ∈ S, and (2) the right Ore condition is satisfied, since (1) implies assS = 0, then (1) and (1′) imply S = S consists of regular elements. Let us prove (1). Let ϕ = ∑ j ϕj be the two-sided weight decomposition. If ϕjs 6= 0 for some j, ϕs 6= 0 since the weights of ϕjs are distinct. Hence we can reduce the claim when ϕ is a weight vector. Suppose ϕ = ∑ µ ϕµ, ϕµ ∈ Im Ψµ and let λ be a maximal weight, with respect to the standard ordering on weights, such that ϕλ 6= 0. Choose sequences i1, . . . , ik and j1, . . . , jl such that fik · · · fi1ϕλfj1 · · · fjl 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 fik · · · fi1(ϕs)fj1 · · · fjl = c′s′s+ · · · 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 6= 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. � 8 A. Kuniba, M. Okado and Y. Yamada 3.3 Realization by generators and relations We consider the fundamental representation V ($1) of Uq(g) for g = An−1, Cn, G2. Set N = dimV ($1). It is known [5, 18] that Aq(g) for g = An−1, Cn, G2 is realized as an associative algebra with appropriate generators (tij)1≤i,j≤N corresponding to V r($1) ⊗ V ($1) satisfying RTT relations∑ m,p Rij,mptmktpl = ∑ m,p tjptimRmp,kl, (13) and additional ones depending on g. See below for each g under consideration. In all cases, there exists a comultiplication ∆ : Aq → Aq ⊗Aq given by ∆(tij) = ∑ k tik ⊗ tkj . (14) 3.3.1 An−1 case We present formulae for Aq(An−1). In this case N = n. Let u1 and v1 be the highest-weight vectors of V ($1) and V r($1) such that (v1, u1) = 1 and set uj = fj−1fj−2 · · · f1u1, vj = v1e1e2 · · · ej−1 for 2 ≤ j ≤ n. Then the constant R matrix is given by∑ i,j,k,l Rij,klEik ⊗ Ejl = q ∑ i Eii ⊗ Eii + ∑ i 6=j Eii ⊗ Ejj + ( q − q−1 )∑ i>j Eij ⊗ Eji, where Eij is the matrix unit. Define tij = Ψ$1(vi ⊗ uj). Then the RTT relations among (tij)1≤i,j≤N read explicitly as follows [tik, tjl] = { 0, i < j, k > l,( q − q−1 ) tjktil, i < j, k < l, tiktjk = qtjktik, i < j, tkitkj = qtkjtki, i < j. In An−1 case we need another condition that the quantum determinant is 1, i.e.,∑ σ∈Sn (−q)`(σ)t1σ1 · · · tnσn = 1, where Sn = W (An−1) is the symmetric group of degree n and `(σ) is the length of σ. According to Definition 1, we have σ1 = t13 and σ2 = t12t23 − qt22t13. As an exposition, we note that σiei in (39) is derived from 〈σ1e1, P 〉 = 〈t13e1, P 〉 = (v1e1, Pu3) = (v2, Pu3) = 〈t23, P 〉, 〈σ2e2, P 〉 = 〈(t12 ⊗ t23 − qt22 ⊗ t13)∆(e2),∆(P )〉 = 〈t12k2 ⊗ t23e2 − qt22e2 ⊗ t13,∆(P )〉 = 〈t12 ⊗ t33 − qt32 ⊗ t13,∆(P )〉 = 〈t12t33 − qt32t13, P 〉 for any P ∈ Uq(A2). See e.g. [17] for an extensive treatment. 3.3.2 Cn case We present formulae for Aq(Cn). In this case N = 2n. Let u1 be the highest-weight vector of V ($1) and define uj for 2 ≤ j ≤ 2n recursively by uj+1 = fjuj (j ≤ n), −f2n−juj (j > n). Let {vi} be the dual basis to {ui} in V r($1), namely, {vi} are determined by (vi, uj) = δij . Then the constant R matrix is given by∑ i,j,k,l Rij,klEik ⊗ Ejl = q ∑ i Eii ⊗ Eii + ∑ i 6=j,j′ Eii ⊗ Ejj + q−1 ∑ i Eii ⊗ Ei′i′ PBW Bases and Quantized Algebra of Functions 9 + ( q − q−1 )∑ i>j Eij ⊗ Eji − ( q − q−1 )∑ i>j εiεjq %i−%jEij ⊗ Ei′j′ , i′ = 2n+ 1− i, εi = 1, 1 ≤ i ≤ n, εi = −1, n < i ≤ 2n, (%1, . . . , %2n) = (n− 1, n− 2, . . . , 1, 0, 0,−1, . . . ,−n+ 1). Define tij = Ψ$1(vi⊗uj). The RTT relations are given by (13) with the above Rij,kl. Additional relations are given by∑ j,k,l CjkClmtijtlk = ∑ j,k,l CijCkltkjtlm = −δim, Cij = δi,j′εiq %j . 3.3.3 G2 case We have N = 7 in this case. We adopt the basis {ui} of V ($1) that has the representation matrices given as in [19, equation (29)], and let {vi} the dual basis in V r($1). Define tij = Ψ$1(vi ⊗ uj). Then Aq(G2) is generated by (tij)1≤i,j≤7 satisfying (i) and (ii) given below. (i) RTT relations (13) with the structure constants specified by Rij,kl = Rijkl in [19, equa- tion (33)]. (ii) Additional relations gij = ∑ k,l tjltikg kl, ∑ k f ijktkm = ∑ k,l tjltikf kl m, (15) where gij and f ijk are given by [19, equations (30), (31)]. The relations [19, equations (20), (22)] are equivalent to (15) if the RTT 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 Aq(g) 4.1 General remarks Let us recall the results in [22, 23] on the representations of Aq(g) necessary in this paper. Consider the simplest example Aq(A1) generated by t11, t12, t21, t22 with the relations t11t21 = qt21t11, t12t22 = qt22t12, t11t12 = qt12t11, t21t22 = qt22t21, [t12, t21] = 0, [t11, t22] = (q − q−1)t21t12, t11t22 − qt12t21 = 1. Let Oscq = 〈a+,a−,k〉 be the q-oscillator algebra, i.e., an associative algebra with the relations ka+ = qa+k, k a− = q−1a−k, a−a+ = 1− q2k2, a+a− = 1− k2. (16) It has a representation on the Fock space Fq = ⊕ m≥0 C(q)\|m〉: k\|m〉 = qm\|m〉, a+\|m〉 = \|m+ 1〉, a−\|m〉 = (1− q2m)\|m− 1〉. (17) In what follows, the symbols k, a+, a− shall also be regarded as the elements from End(Fq). It is easy to check that the following map π defines an irreducible representation of Aq(A1) on Fq: π : ( t11 t12 t21 t22 ) 7→ ( µa− αk −qα−1k µ−1a+ ) , (18) where α, µ are nonzero parameters. 10 A. Kuniba, M. Okado and Y. Yamada Theorem 4 ([22, 23]). (1) For each vertex i of the Dynkin diagram of g, Aq(g) has an irreducible representation πi factoring through (18) via Aq(g) � Aqi(sl2,i). (sl2,i denotes the sl2-subalgebra of g asso- ciated to i.) (2) Irreducible representations of Aq(g) are in one to one correspondence with the elements of the Weyl group W of g. (3) Let w = si1 · · · sil ∈ W be an reduced expression in terms of the simple reflections. Then the irreducible representation corresponding to w is isomorphic to πi1 ⊗ · · · ⊗ πil. Actually the assertions (2) and (3) hold up to the degrees of freedom of the parameters α, µ in (18). See [22] for the detail. We call πi (i = 1, . . . , rank g) the fundamental representations. For simplicity we denote πi1 ⊗ · · · ⊗ πil by πi1,...,il . A crucial corollary of Theorem 4 is the following: If si1 · · · sil = sj1 · · · sjl ∈W are reduced expressions, then πi1,...,il ' πj1,...,jl . In particular, there exists the isomorphism Φ : Fqi1 ⊗· · ·⊗Fqil → Fqj1 ⊗· · ·⊗Fqjl characterized (up to an overall constant) by πj1,...,jl(g) ◦ Φ = Φ ◦ πi1,...,il(g) ∀ g ∈ Aq(g). Here πi1,...,il(g = tij) for example means the tensor product representation ∑ r1,...,rl−1 πi1(tir1)⊗ · · · ⊗ πil(trl−1,j) obtained by the (l − 1)-fold application of the coproduct (14). Elements of the Fock space \|m1〉 ⊗ · · · ⊗ \|ml〉 ∈ Fqj1 ⊗ · · · ⊗ Fqjl will simply be denoted by \|m1, . . . ,ml〉. 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 Pij . In the remainder of this section we concentrate on Aq(g) of rank 2 cases g = A2, C2 and G2, and present the concrete forms of the fundamental representations, definition of the intertwiners with a few examples of their matrix elements. 4.2 A2 case Let T = (tij)1≤i,j≤3 be the 3 × 3 matrix of the generators of Aq(A2). The fundamental repre- sentations πi : Aq(A2)→ End(Fq) (i = 1, 2) are given by π1(T ) =  µ1a − α1k 0 −qα−1 1 k µ−1 1 a+ 0 0 0 1  , π2(T ) = 1 0 0 0 µ2a − α2k 0 −qα−1 2 k µ−1 2 a+  , (19) where αi, µi are nonzero parameters. The Weyl group W = 〈s1, s2〉 is the Coxeter system with the relations s2 1 = s2 2 = 1, s1s2s1 = s2s1s2. Thus we have the isomorphism π121 ' π212. Let Φ be the corresponding intertwiner and denote by R the checked intertwiner Φ∨ explained after (1) π121Φ = Φπ212, π121R = Rπ′212, π′212 = P13π212P13, R = ΦP13 ∈ End ( F⊗3 q ) . For example π′212(tij) = ∑ k,l π2(tl,j) ⊗ π1(tk,l) ⊗ π2(tik). Define the matrix elements of R and its parameter-free part R by R\|i, j, k〉 = ∑ a,b,c Rabcijk \|a, b, c〉, Rabcijk = µa−j+k1 µb−a−k2 Rabcijk . PBW Bases and Quantized Algebra of Functions 11 Then the following properties are valid for R = (Rabcijk) [13]: Rabcijk ∈ Z[q], Rabcijk = 0 unless (a+ b, b+ c) = (i+ j, j + k), (20) R−1 = R, Rabcijk = Rcbakji, Rabcijk = (q2)i(q 2)j(q 2)k (q2)a(q2)b(q2)c R ijk abc, (21) Rabcijk \|q=0 = δi,b+(a−c)+δj,min(a,c)δk,b+(c−a)+ . (22) Here (q2)a = a∏ m=1 (1 − q2m) and (y)+ = max(0, y). Due to (20), R is the infinite direct sum of finite-dimensional matrices. An explicit formula of Rabcijk was obtained in [10] (unfortunately with misprint) and in [1, equation (59)] (in a different context and gauge including square roots). The formula exactly matching the present convention is [13, equation (2.20)]. The R satisfies [10] the tetrahedron equation (2). Example 1. The following is the list of all the nonzero Rabc314: R041 314 = −q2 ( 1− q4 )( 1− q6 )( 1− q8 ) , R132 314 = ( 1− q6 )( 1− q8 )( 1− q4 − q6 − q8 − q10 ) , R223 314 = q2 ( 1 + q2 )( 1 + q4 )( 1− q6 )( 1− q6 − q10 ) , R314 314 = q6 ( 1 + q2 + q4 − q8 − q10 − q12 − q14 ) , R405 314 = q12. Thus Rabc314\|q=0 = δa,1δb,3δc,2 in agreement with (22). 4.3 C2 case We have (q1, q2) = (q, q2). Let T = (tij)1≤i,j≤4 be the 4× 4 matrix of the generators of Aq(C2). We use Oscq2 = 〈A+,A−,K〉 in addition to Oscq = 〈a+,a−,k〉 (16). The fundamental repre- sentations πi : Aq(C2)→ End(Fqi) (i = 1, 2) are given by π1(T ) =  µ1a − α1k 0 0 −qα−1 1 k µ−1 1 a+ 0 0 0 0 µ1a − −α1k 0 0 qα−1 1 k µ−1 1 a+  , π2(T ) =  1 0 0 0 0 µ2A − α2K 0 0 −q2α−1 2 K µ−1 2 A+ 0 0 0 0 1  , (23) where αi, µi are nonzero parameters. The Weyl group W = 〈s1, s2〉 is the Coxeter system with the relations s2 1 = s2 2 = 1, s2s1s2s1 = s1s2s1s2. Thus we have the isomorphism π2121 ' π1212. Let Φ be the corresponding intertwiner and denote by K the checked intertwiner Φ∨ π2121Φ = Φπ1212, π2121K = Kπ′2121, π′2121 = P14P23π1212P14P23, K = ΦP14P23 ∈ End(Fq2 ⊗Fq ⊗Fq2 ⊗Fq). 12 A. Kuniba, M. Okado and Y. Yamada Define the matrix elements of K and its parameter-free part K by K\|i, j, k, l〉 = ∑ a,b,c,d Kabcd ijkl \|a, b, c, d〉, Kabcd ijkl = µ 2(c−k) 1 µb−j2 Kabcd ijkl . Then the following properties are valid for K = (Kabcd ijkl ) [13]: Kabcd ijkl ∈ Z[q], Kabcd ijkl = 0 unless (a+ b+ c, b+ 2c+ d) = (i+ j + k, j + 2k + l), (24) K−1 = K, Kabcd ijkl = (q4)i(q 2)j(q 4)k(q 2)l (q4)a(q2)b(q4)c(q2)d K ijkl abcd, (25) Kabcd ijkl \|q=0 = δi,a′δj,b′δk,c′δl,d′ , (26) a′ = x+ a+ b− d, b′ = c+ d− x−min(a, c+ x), c′ = min(a, c+ x), d′ = b+ (c− a+ x)+, x = (c− a+ (d− b)+)+. Due to (24), K is the infinite direct sum of finite-dimensional matrices. An explicit formula of Kabcd 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). Example 2. The following is the list of all the nonzero Kabcd 2110: K1300 2110 = q8 ( 1− q8 ) , K2110 2110 = −q4 ( 1− q8 + q14 ) , K2201 2110 = −q6 ( 1 + q2 )( 1− q2 + q4 − q6 − q10 ) , K3011 2110 = 1− q8 + q14, K3102 2110 = −q10 ( 1− q + q2 )( 1 + q + q2 ) , K4003 2110 = q4. Thus Kabcd 2110\|q=0 = δa,3δb,0δc,1δd,1 in agreement with (26). 4.4 G2 case We have (q1, q2) = (q, q3). Let T = (tij)1≤i,j≤7 be the 7× 7 matrix of the generators of Aq(G2). We use Oscq3 = 〈A+,A−,K〉 in addition to Oscq = 〈a+,a−,k〉 (16). The fundamental repre- sentations πi : Aq(G2)→ End(Fqi) (i = 1, 2) are given by π1(T ) =  µ1a − α1k 0 0 0 0 0 −qα−1 1 kµ−1 1 a+ 0 0 0 0 0 0 0 (µ1a −)2 [2]1α1µ1k a− (α1k)2 0 0 0 0 −qα−1 1 µ1a − k a−a+ − k2 α1µ −1 1 k a+ 0 0 0 0 (qα−1 1 k)2 −[2]1(α1µ1)−1k a+ (µ−1 1 a+)2 0 0 0 0 0 0 0 µ1a − α1k 0 0 0 0 0 −qα−1 1 kµ−1 1 a+  , π2(T ) =  1 0 0 0 0 0 0 0 µ2A − α2K 0 0 0 0 0 −q3α−1 2 K µ−1 2 A+ 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 µ2A − α2K 0 0 0 0 0 −q3α−1 2 K µ−1 2 A+ 0 0 0 0 0 0 0 1  , (27) where αi, µi are nonzero parameters and [2]1 = q + q−1 as defined after (4). PBW Bases and Quantized Algebra of Functions 13 The Weyl group W = 〈s1, s2〉 is the Coxeter system with the relations s2 1 = s2 2 = 1, s2s1s2s1s2s1 = s1s2s1s2s1s2. Thus we have the isomorphism π212121 ' π121212. Let Φ be the corresponding intertwiner and denote by F the checked intertwiner Φ∨ π212121Φ = Φπ121212, π212121F = Fπ′212121, π′212121 = P16P25P34π121212P16P25P34, F = ΦP16P25P34 ∈ End(Fq3 ⊗Fq ⊗Fq3 ⊗Fq ⊗Fq3 ⊗Fq). (28) Define the matrix elements of F and its parameter-free part F by F \|i, j, k, l,m, n〉 = ∑ a,b,c,d,e,f F abcdefijklmn\|a, b, c, d, e, f〉, F abcdefijklmn = µ3c−3k+d−l+3e−3m 1 µ2k−2c+l−d+3m−3e+n−f 2 F abcdef ijklmn. Then the following properties are valid for F = (Fabcdefijklmn): F abcdef ijklmn ∈ Z[q], F abcdef ijklmn = 0 unless ( a+ b+ 2c+ d+ e b+ 3c+ 2d+ 3e+ f ) = ( i+ j + 2k + l +m j + 3k + 2l + 3m+ n ) , (29) F−1 = F, F abcdef ijklmn = (q6)i(q 2)j(q 6)k(q 2)l(q 6)m(q2)n (q6)a(q2)b(q6)c(q2)d(q6)e(q2)f F ijklmn abcdef . (30) 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: F000200 010101 = q4 ( 1− q2 )( 1− q2 − q4 − q6 ) , F001001 010101 = −q ( 1− q2 )( 1− q2 − q4 + q8 + q10 ) , F010010 010101 = −q ( 1− q2 )( 1− q2 − q4 + q8 + q10 ) , F010101 010101 = 1− 2q2 + 2q6 + 3q8 − 2q12 − 2q14 − q16, F020002 010101 = q4 ( −2 + 2q6 + q8 + q10 ) , F100011 010101 = −q3 ( 1− q2 )( 1− q6 − q8 ) , F100102 010101 = q ( 1− q2 − q4 − q6 + q10 + q12 + q14 ) , F200004 010101 = q4, F110003 010101 = q ( 1− q + q2 )( 1 + q + q2 )( 1− q2 − q8 ) . 5 Main theorem In this section we fix two reduced words i = (i1, . . . , il), j = (j1, . . . , jl) of the longest element w0 ∈W . 14 A. Kuniba, M. Okado and Y. Yamada 5.1 Definitions of γA B and ΦA B In the Uq(g) side, we defined the PBW bases EAi , EBj of U+ q (g) in Section 2.2. We define their transition coefficient γAB by EAi = ∑ B γABE B j . While, in the Aq(g) side, we have the intertwiner Φ : Fqi1⊗· · ·⊗Fqil → Fqj1⊗· · ·⊗Fqjl satisfying πj(g) ◦ Φ = Φ ◦ πi(g) ∀ g ∈ Aq(g). (31) We take the parameters µ, α in (18) to be 1. This in particular means for rank 2 cases that µi, αi entering πi(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 ΦA B of Φ is uniquely specified by Φ\|B〉 = ∑ A ΦA B\|A〉, where A = (a1, . . . , al) ∈ (Z≥0)l and \|A〉 = \|a1〉 ⊗ · · · ⊗ \|al〉 ∈ Fqj1 ⊗ · · · ⊗ Fqjl and similarly for \|B〉 ∈ Fqi1 ⊗ · · · ⊗ Fqil . Then our main result is Theorem 5. γAB = ΦA B. 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 Ti 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. 5.2 Proof of Theorem 5 for rank 2 cases In the rank 2 cases, there are two reduced expressions si1 · · · sil for the longest element of the Weyl group. Denote the associated sequences i = (i1, . . . , il) by 1, 2 and set 1′ = 2, 2′ = 1. Concretely, we take them as A2 : 1 = (1, 2, 1), 2 = (2, 1, 2), (q1, q2) = (q, q), C2 : 1 = (1, 2, 1, 2), 2 = (2, 1, 2, 1), (q1, q2) = ( q, q2 ) , G2 : 1 = (1, 2, 1, 2, 1, 2), 2 = (2, 1, 2, 1, 2, 1), (q1, q2) = ( q, q3 ) , where qi defined after (4) is also recalled. In order to simplify the formulae in Section 5.3, we use the PBW bases and the Fock states in yet another normalization as follows: ẼAi := ([a1]i1 ! · · · [al]il !)E A i = ea1β1 · · · e al βl , \|A〉〉 := di1,a1 · · · dil,al \|A〉, di,a = q −a(a−1)/2 i λai , λi = ( 1− q2 i )−1 , (32) where A = (a1, . . . , al). See after (4) for the symbol [a]i!. eβr is defined in (5). Accordingly we introduce the matrix elements γ̃AB and Φ̃A B by ẼAi = ∑ B γ̃ABẼ B i′ , Φ\|B〉〉 = ∑ A Φ̃A B\|A〉〉, i = 1,2. PBW Bases and Quantized Algebra of Functions 15 It follows that γAB = γ̃AB l∏ k=1 ([bk]ik !/[ak]ik !) and ΦA B = Φ̃A B l∏ k=1 (dik,ak/dik,bk) for B = (b1, . . . , bl). On the other hand, we know ΦA B = ΦB A l∏ k=1 ((q2 ik )bk/(q 2 ik )ak) from (21), (25) and (30). Due to the identity (q2 i )mdi,m = [m]i!, the assertion γAB = ΦA B of Theorem 5 is equivalent to γ̃AB = Φ̃B A . (33) Let ρi(x) = (ρi(x)AB) be the matrix for the left multiplication of x ∈ U+ q (g): x · ẼAi = ∑ B ẼBi ρi(x)BA. (34) Let further πi(g) = (πi(g)AB) be the representation matrix of g ∈ Aq(g): πi(g)\|A〉〉 = ∑ B \|B〉〉πi(g)BA. (35) The following element in the right quotient ring Aq(g)S will play a key role in our proof. ξi = λi(σiei)/σi, i = 1, 2. (36) See Definition 1 for σi 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, πi(σi) is invertible and the following equality is valid: ρi(ei)AB = πi(ξi)AB, i = 1, 2, (37) where the right-hand side means λiπi(σiei)πi(σi) −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 ′iAB. From∑ B,C ẼCi′M ′i CB γ̃ A B = ei ∑ B ẼBi′ γ̃ A B = eiẼ A i = ∑ B ẼBi M i BA = ∑ B,C ẼCi′ γ̃ B CM i BA we have ∑ BM ′i CB γ̃ A B = ∑ B γ̃ B CM i BA. On the other hand, the action of the two sides of (31) with g = ξi and j = i′ are calculated as πi′(ξi) ◦ Φ\|A〉〉 = πi′(ξi) ∑ B \|B〉〉Φ̃B A = ∑ B,C \|C〉〉M ′iCBΦ̃B A and Φ ◦ πi(ξi)\|A〉〉 = Φ ∑ B \|B〉〉M i BA = ∑ B,C \|C〉〉Φ̃C BM i BA. Hence ∑ BM ′i CBΦ̃B A = ∑ B Φ̃C BM i BA. Thus γ̃AB and Φ̃B A satisfy the same relation. Moreover the maps πi and ρi are both homomorphism, i.e., πi(gh) = πi(g)πi(h) and ρi(xy) = ρi(x)ρi(y). We know that Φ is the intertwiner of the irreducible Aq(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. 16 A. Kuniba, M. Okado and Y. Yamada 5.3 Explicit formulae for rank 2 cases: Proof of Proposition 2 Here we present the explicit formulae of (34) with x = ei and (35) with g = σi, σiei that allow one to check Proposition 2. We use the notation 〈i〉 = qi − 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(ei) = ei. Then the relation χ(ẼAi ) = ẼĀi′ holds, where Ā = (al, . . . , a2, a1) denotes the reversal of A = (a1, a2, . . . , al). Applying χ to (34) with x = ei yields the right multiplication formula ẼĀi′ · ei = ∑ B Ẽ B̄ i′ ρi(ei)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 = ξi in (36), explicit formulae for σi, σiei ∈ Aq(g) and their image by the both representations π1 and π2 will be given. We include an exposition on how to use these data to check (37) along the simplest A2 case. The C2 and G2 cases are similar. 5.3.1 A2 case The q-Serre relations are e2 1e2 − [2]1e1e2e1 + e2e 2 1 = 0, e2 2e1 − [2]1e2e1e2 + e1e 2 2 = 0, where [m]1 = 〈m〉/〈1〉. Let b1, b2, b3 be the generator for positive roots: b1 = e2, b2 = e1e2−qe2e1 and b3 = e1. In the notation of Section 2.2, they are the root vectors bi = eβi associated with the reduced expression w0 = s2s1s2 for 2 = (2, 1, 2). The corresponding positive roots are (β1, β2, β3) = (α2, α1 + α2, α1). In particular, b2 = T2(e1). Their commutation relations are b2b1 = q−1b1b2, b3b1 = b2 + qb1b3, b3b2 = q−1b2b3. Lemma 2. For Ẽa,b,c2 = ba1b b 2b c 3, we have Ẽa,b,c2 · e1 = Ẽa,b,c+1 2 , Ẽa,b,c2 · e2 = qc−bẼa+1,b,c 2 + [c]1Ẽ a,b+1,c−1 2 , e1 · Ẽa,b,c2 = qa−bẼa,b,c+1 2 + [a]1Ẽ a−1,b+1,c 2 , e2 · Ẽa,b,c2 = Ẽa+1,b,c 2 . Proof. By induction, we have b3b n 1 = qnbn1b3 + [n]1b n−1 1 b2, b3b n 2 = q−nbn2b3, bn3b1 = qnb1b n 3 + [n]1b2b n−1 3 , bn2b1 = q−nb1b n 2 . The lemma is a direct consequence of these formulae. � Set Ẽa,b,c1 = χ(Ẽc,b,a2 ) = χ(ba3)χ(bb2)χ(bc1) = ba3b ′b 2 b c 1, where b′2 := χ(b2) = e2e1 − qe1e2. By applying χ to the first two relations in Lemma 2, we get e1 · Ẽa,b,c1 = Ẽa+1,b,c 1 , e2 · Ẽa,b,c1 = qa−bEa,b,c+1 1 + [a]1Ẽ a−1,b+1,c 1 . (38) Thus we find ρi′(ei) = ρi(e3−i). This property is only valid for A2 and not in C2 and G2. Let us turn to the representations πi of Aq(A2). The elements σi in Definition 1 and σiei are given by σ1 = t13, σ2 = t12t23 − qt22t13, σ1e1 = t23, σ2e2 = t12t33 − qt32t13. (39) See the exposition at the end of Section 3.3.1 and the remark after Lemma 1. PBW Bases and Quantized Algebra of Functions 17 From (14) and (19) with αi = µi = 1, we find π1(σ1) = k1k2, π1(σ1e1) = a+ 1 k2, π1(σ2) = k2k3, π1(σ2e2) = a−1 a+ 2 k3 + k1a + 3 , where the notation like k1a + 3 = k ⊗ 1 ⊗ a+ has been used. Since k ∈ End(Fq) is invertible, so is πi(σi) and we may write π1(ξ1) = λ1a + 1 k−1 1 , π1(ξ2) = λ2(a−1 a+ 2 k−1 2 + k1k −1 2 a+ 3 k−1 3 ), where λ1 = λ2 = (1− q2)−1. The action of each component on the ket vector \|m〉〉 := di,m\|m〉 ∈ Fqi (cf. (32)) takes the form a+\|m〉〉 = λ−1 i qmi \|m+ 1〉〉, a−\|m〉〉 = [m]i\|m− 1〉〉, k\|m〉〉 = qmi \|m〉〉, (40) due to (17). (The formula (40) is valid also for C2 and G2 provided that a+, a−, k are interpreted as A+, A−, K for i = 2.) Thus one has π1(ξ1)\|a, b, c〉〉 = \|a+ 1, b, c〉〉, π1(ξ2)\|a, b, c〉〉 = [a]1\|a− 1, b+ 1, c〉〉+ qa−b\|a, b, c+ 1〉〉. This agrees with (38) thereby proving (37) for i = 1. The other case i = 2 also holds due to the symmetry π2(ξi) = π1(ξ3−i). Thus Proposition 2 is established for A2. In terms of the checked intertwiner R in Section 4.2, Theorem 5 implies Ea,b,ci = ∑ i,j,k RabcijkE k,j,i i′ . 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]. 5.3.2 C2 case The q-Serre relations are e3 1e2 − [3]1e 2 1e2e1 + [3]1e1e2e 2 1 − e2e 3 1 = 0, e2 2e1 − [2]2e2e1e2 + e1e 2 2 = 0, where [m]1 = 〈m〉/〈1〉 and [m]2 = 〈2m〉/〈2〉. Let b1, . . . , b4 be the generator for positive roots: b1 = e2, b2 = e1e2− q2e2e1, b3 = 1 [2]1 (e1b2− b2e1) and b4 = e1. Their commutation relations are b2b1 = q−2b1b2, b3b1 = −q−1〈1〉[2]−1 1 b22+b1b3, b4b1 = b2 + q2b1b4, b3b2 = q−2b2b3, b4b2 = [2]1b3 + b2b4, b4b3 = q−2b3b4. Lemma 3. For Ẽa,b,c,d2 = ba1b b 2b c 3b d 4, we have Ẽa,b,c,d2 · e1 = Ẽa,b,c,d+1 2 , Ẽa,b,c,d2 · e2 = [d]1q d−2c−1Ẽa,b+1,c,d−1 2 + q2(d−b)Ẽa+1,b,c,d 2 − 〈1〉q2d−2c+1[c]2[2]−1 1 Ẽa,b+2,c−1,d 2 + [d− 1]1[d]1Ẽ a,b,c+1,d−2 2 , e1 · Ẽa,b,c,d2 = [2]1[b]1q 2a−b+1Ẽa,b−1,c+1,d 2 + q2a−2cẼa,b,c,d+1 2 + [a]2Ẽ a−1,b+1,c,d 2 , e2 · Ẽa,b,c,d2 = Ẽa+1,b,c,d 2 . 18 A. Kuniba, M. Okado and Y. Yamada Proof. By induction, we have b4b n 1 = bn1b4q 2n + [n]2b n−1 1 b2, b4b n 2 = [2]1[n]1b n−1 2 b3q −n+1 + bn2b4, b4b n 3 = q−2nbn3b4, bn4b1 = [n]1b2b n−1 4 qn−1 + b1b n 4q 2n + [n− 1]1[n]1b3b n−2 4 , bn3b1 = −q1−2n〈1〉[n]2[2]−1 1 b22b n−1 3 + b1b n 3 , bn3b2 = q−2nb2b n 3 , bn2b1 = q−2nb1b n 2 . The lemma is a direct consequence of these formulae. � Set Ẽa,b,c,d1 = χ ( Ẽd,c,b,a2 ) . The left multiplication formula for this basis is deduced from the above lemma by applying χ. One can adjust the definition of EAi in (6) with that in [24] by setting v = q−1. Let us turn to the representations πi of Aq(C2). The elements σi in Definition 1 and σiei are given by σ1 = t14, σ2 = t13t24 − qt23t14, σ1e1 = t24, σ2e2 = t13t34 − qt33t14. (41) From (14) and (23) with αi = µi = 1, we have π1(σ1) = −k1K2k3, π1(σ1e1) = −a+ 1 K2k3, π1(σ2) = −K2k3 2K4, π1(σ2e2) = −a−1 2 A+ 2 k3 2K4 − [2]1a − 1 k1a + 3 k3K4 − k1 2A−2 a+ 3 2 K4 −A+ 4 k1 2K2, λ−1 1 π1(ξ1) = a+ 1 k1 −1, λ−1 2 π1(ξ2) = a−1 2 A+ 2 K2 −1 + k1 2A−2 K2 −1a+ 3 2 k3 −2 + [2]1a − 1 k1K2 −1a+ 3 k3 −1 + k1 2k3 −2A+ 4 K4 −1, π2(σ1) = −k2K3k4, π2(σ1e1) = −K1k2a + 4 −K1a − 2 A+ 3 k4 −A−1 a+ 2 K3k4, π2(σ2) = −K1k 2 2K3, π2(σ2e2) = −A+ 1 k2 2K3, λ−1 1 π2(ξ1) = A−1 a+ 2 k−1 2 + K1a − 2 k−1 2 A+ 3 K−1 3 + K1K −1 3 a+ 4 k−1 4 , λ−1 2 π2(ξ2) = A+ 1 K1 −1. We find that πi(σi) is invertible. Comparing these formulae with Lemma 3 by using (40), the equality (37) is directly checked. Thus Proposition 2 is established for C2. In terms of the checked intertwiner K in Section 4.3, Theorem 5 implies Ea,b,c,d2 = ∑ i,j,k,l Kabcd ijkl E l,k,j,i 1 . Thus the solution to the 3D reflection equation [13] is identified with the transition coefficient of the PBW bases for U+ q (C2). 5.4 G2 case PBW Bases and Quantized Algebra of Functions 19 The q-Serre relations are e4 1e2 − [4]1e 3 1e2e1 + [4]1[3]1/[2]−1 1 e2 1e2e 2 1 − [4]1e1e2e 3 1 + e2e 4 1 = 0, e2 2e1 − [2]2e2e1e2 + e1e 2 2 = 0, where we remind that [m]1 = 〈m〉/〈1〉 and [m]2 = 〈3m〉/〈3〉. Let b1, . . . , b6 be the generator for positive roots: b1 = e2, b2 = e1e2− q3e2e1, b4 = 1 [2]1 (e1b2− qb2e1), b5 = 1 [3]1 (e1b4 − q−1b4e1), b3 = 1 [3]1 (b4b2 − q−1b2b4) and b6 = e1. Their commutation relations are as follows: b2b1 = b1b2q −3, b3b1 = 〈1〉2b32q−3[3]−1 1 + b1b3q −3, b4b1 = b1b4− b22〈1〉q−1, b5b1 = b1b5q 3−b2b4〈1〉q−1−(q4+q2−1)b3q −3, b6b1 = b1b6q 3+b2, b3b2 = b2b3q −3, b4b2 = b2b4q −1+ b3[3]1, b5b2 = b2b5 − b24〈1〉q−1, b6b2 = qb2b6 + b4[2]1, b4b3 = b3b4q −3, b5b3 = 〈1〉2b34q−3[3]−1 1 + b3b5q −3, b6b3 = b3b6 − b24〈1〉q−1, b5b4 = b4b5q −3, b6b4 = [3]1b5 + b4b6q −1, b6b5 = b5b6q −3. Lemma 4. For Ẽa,b,c,d,e,f2 = ba1b b 2 · · · b f 6 , we have Ẽa,b,c,d,e,f2 · e1 = Ẽa,b,c,d,e,f+1 2 , Ẽa,b,c,d,e,f2 · e2 = −〈1〉[e]2q−3c−d+3f−1Ẽa,b+1,c,d+1,e−1,f 2 + 〈1〉2[e− 1]2[e]2[3]−1 1 q−3e+3f+3Ẽa,b,c,d+3,e−2,f 2 − 〈3〉[d− 1]1[d]1q −3c−2d+3e+3f+1Ẽa,b+1,c+1,d−2,e,f 2 − 〈1〉[d]1q −6c−d+3(e+f)Ẽa,b+2,c,d−1,e,f 2 + [f−1]1[f ]1q −3e+f−2Ẽa,b,c,d+1,e,f−2 2 + [3]1[d]1[f ]1q 2f−2dẼa,b,c+1,d−1,e,f−1 2 + [f ]1q −3c−d+2f−2Ẽa,b+1,c,d,e,f−1 2 + q−3(b+c−e−f)Ẽa+1,b,c,d,e,f 2 + 〈1〉2[c]2[3]−1 1 q3(−2c+e+f+1)Ẽa,b+3,c−1,d,e,f 2 − 〈3〉[d− 2]1[d− 1]1[d]1q 3(−d+e+f+2)Ẽa,b,c+2,d−3,e,f 2 − 〈1〉[e]2[f ]1q −3e+2f Ẽa,b,c,d+2,e−1,f−1 2 − [e]2q −3d+3f (q2d+1[3]1 − [2]2)Ẽa,b,c+1,d,e−1,f 2 + [f − 2]1[f − 1]1[f ]1Ẽ a,b,c,d,e+1,f−3 2 , e1 · Ẽa,b,c,d,e,f2 = −〈1〉[c]2q3a+b−3c+2Ẽa,b,c−1,d+2,e,f 2 + [3]1[b− 1]1[b]1q 3a−b+2Ẽa,b−2,c+1,d,e,f 2 + [3]1[d]1q 3a+b−2d+2Ẽa,b,c,d−1,e+1,f 2 + q3a+b−d−3eẼa,b,c,d,e,f+1 2 + [2]1[b]1q 3(a−c)Ẽa,b−1,c,d+1,e,f 2 + [a]2Ẽ a−1,b+1,c,d,e,f 2 , e2 · Ẽa,b,c,d,e,f2 = Ẽa+1,b,c,d,e,f 2 . Proof. By induction, we have b6b n 1 = q3nbn1b6 + [n]2b n−1 1 b2, b6b n 2 = [3]1q 2−n[n− 1]1[n]1b n−2 2 b3 + qnbn2b6 + [2]1[n]1b n−1 2 b4, b4b n 3 = q−3nbn3b4, b6b n 3 = bn3b6 − 〈1〉q2−3n[n]2b n−1 3 b24, b6b n 4 = [3]1q 2−2n[n]1b n−1 4 b5 + q−nbn4b6, b6b n 5 = q−3nbn5b6, and bn6b1 = qn−2[n− 1]1[n]1b4b n−2 6 + q3nb1b n 6 + q2(n−1)[n]1b2b n−1 6 + [n− 2]1[n− 1]1[n]1b5b n−3 6 , bn5b1 = 〈1〉2q−3(n−1)[n− 1]2[n]2[3]−1 1 b34b n−2 5 + q3nb1b n 5 − q−3(q4 + q2 − 1)[n]2b3b n−1 5 − q−1〈1〉[n]2b2b4b n−1 5 , 20 A. Kuniba, M. Okado and Y. Yamada bn5b2 = b2b n 5 − 〈1〉q2−3n[n]2b 2 4b n−1 5 , bn5b4 = q−3nb4b n 5 , bn4b1 = −〈3〉q6−3n[n− 2]1[n− 1]1[n]1b 2 3b n−3 4 − 〈1〉q−n[n]1b 2 2b n−1 4 − 〈3〉q1−2n[n− 1]1[n]1b2b3b n−2 4 + b1b n 4 , bn4b2 = [3]1q 2−2n[n]1b3b n−1 4 + q−nb2b n 4 , bn4b3 = q−3nb3b n 4 , bn3b1 = q−3nb1b n 3 + 〈1〉2q3−6n[n]2[3]−1 1 b32b n−1 3 , bn3b2 = q−3nb2b n 3 , bn2b1 = q−3nb1b n 2 . The lemma is a direct consequence of these formulae. � A part of the above results have also been obtained in [25]. Let us turn to the representations πi of Aq(G2). The elements σi in Definition 1 and σiei are given by σ1 = t17, σ2 = t26t17 − qt27t16, σ1e1 = t27, σ2e2 = t36t17 − qt37t16. (42) From (14) and (27) with αi = µi = 1, we have π1(σ1) = k1K2k 2 3K4k5, π1(σ2) = K2k 3 3K 2 4k 3 5K6, π1(σ1e1) = a+ 1 K2k 2 3K4k5, π1(σ2e2) = k3 1K 2 2k 3 3K4A + 6 + [2]2k 3 1A − 2 K2A + 4 K4k 3 5K6 + a− 3 1A + 2 k3 3K 2 4k 3 5K6 + [3]1a −2 1 k1a + 3 k2 3K 2 4k 3 5K6 + [3]1a − 1 k2 1K2k 2 3K4a + 5 k2 5K6 − q[3]1k 3 1A − 2 K2k 2 3A + 4 K4k 3 5K6 + [3]1k 3 1K 2 2a − 3 k2 3a +2 5 k5K6 + k3 1K 2 2k 3 3A − 4 a+3 5 K6 + [3]1a − 1 k2 1A − 2 a+2 3 k3K 2 4k 3 5K6 + [3]1a − 1 k2 1K2a − 3 k3A + 4 K4k 3 5K6 + k3 1A −2 2 a+3 3 K2 4k 3 5K6 + [3]1k 3 1A − 2 K2a + 3 k3K4a + 5 k2 5K6 + k3 1K 2 2a −3 3 A+2 4 k3 5K6 + [3]1k 3 1K 2 2a −2 3 k3A + 4 a+ 5 k2 5K6, λ−1 1 π1(ξ1) = a+ 1 k−1 1 , λ−1 2 π1(ξ2) = a−3 1 A+ 2 K−1 2 + [2]2k 3 1A − 2 k−3 3 A+ 4 K−1 4 − q[3]1k 3 1A − 2 k−1 3 A+ 4 K−1 4 + [3]1a −2 1 k1K −1 2 a+ 3 k−1 3 + [3]1a − 1 k2 1a − 3 k−2 3 A+ 4 K−1 4 + [3]1a − 1 k2 1k −1 3 K−1 4 a+ 5 k−1 5 + k3 1K2K −1 4 k−3 5 A+ 6 K−1 6 + [3]1a − 1 k2 1A − 2 K−1 2 a+2 3 k−2 3 + [3]1k 3 1A − 2 a+ 3 k−2 3 K−1 4 a+ 5 k−1 5 + k3 1A −2 2 K−1 2 a+3 3 k−3 3 + [3]1k 3 1K2a − 3 k−1 3 K−2 4 a+2 5 k−2 5 + k3 1K2A − 4 K−2 4 a+3 5 k−3 5 + k3 1K2a −3 3 k−3 3 A+2 4 K−2 4 + [3]1k 3 1K2a −2 3 k−2 3 A+ 4 K−2 4 a+ 5 k−1 5 , π2(σ1) = k2K3k 2 4K5k6, π2(σ2) = K1k 3 2K 2 3k 3 4K5, π2(σ1e1) = K1k 2 2K3k4a + 6 + A−1 a+ 2 K3k 2 4K5k6 + K1k 2 2K3a − 4 A+ 5 k6 + K1a −2 2 A+ 3 k2 4K5k6 + [2]1K1a − 2 k2a + 4 k4K5k6 + K1k 2 2A − 3 a+2 4 K5k6, π2(σ2e2) = A+ 1 k3 2K 2 3k 3 4K5, λ−1 1 π2(ξ1) = A−1 a+ 2 k−1 2 + [2]1K1a − 2 K−1 3 a+ 4 k−1 4 + K1a −2 2 k−1 2 A+ 3 K−1 3 + K1k2a − 4 k−2 4 A+ 5 K−1 5 + K1k2k −1 4 K−1 5 a+ 6 k−1 6 + K1k2A − 3 K−1 3 a+2 4 k−2 4 , λ−1 2 π2(ξ2) = A+ 1 K−1 1 . We find that πi(σi) is invertible. Comparing these formulae with Lemma 4 by using (40), the equality (37) is directly checked. Thus Proposition 2 is established for G2. PBW Bases and Quantized Algebra of Functions 21 In terms of the checked intertwiner F in Section 4.4, Theorem 5 implies Ea,b,c,d,e,f2 = ∑ i,j,k,l,m,n F abcdef ijklmnE n,m,l,k,j,i 1 . 6 Discussion In view of Proposition 2 it is natural to expect that the map defined on generators of U+ q (g) as ei 7→ ηi := σiei/σi extends to an algebra homomorphism from U+ q (g) to Aq(g)S , namely, ηi satisfies q-Serre relations. In fact, it is true not only for rank 2 cases but also for any g. Theorem 6. In Aq(g)S the following relation holds for any i, j (i 6= j): 1−aij∑ r=0 (−1)rη (r) i ηjη (1−aij−r) i = 0. Proof. By relabeling of Dynkin indices we can assume i = 1, j = 2. Set τi = σiei for i = 1, 2. Then from Proposition 1 we have σiτi = qiτiσi, i = 1, 2, σiτj = τjσi, i, j = 1, 2; i 6= j. (43) Using (12) with these relations one verifies ηr1η2η s 1 = q (r+s)(r+s−1)/2 1 ( τ r1 τ2τ s 1 ) / ( σr1σ2σ s 1 ) . Here we have set s = 1 − a12 − r. Recalling that σ1 and σ2 commute with each other, we can reduce the claim to showing Z := 1−a12∑ r=0 (−1)rτ (r) 1 τ2τ (s) 1 = 0. Note that the right (resp. left) weight of Z is (1− a12)($1 − α1) + ($2 − α2) (resp. w0((1− a12)$1 + $2)). The two weights are not related by the longest element w0 ∈ W . Hence if we show fiZ = Zfi = 0 for any i, we can conclude Z = 0 by the remark after Lemma 1. The properties fiZ = 0 for any i and Zfi = 0 for i 6= 1, 2 are trivial. First we show Zf2 = 0. We have( τ r1 τ2τ s 1 ) f2 = τ r1 (τ2f2) ( τ1k −1 2 )s = τ r1σ2(βτ1)s = βsτ r+s1 σ2, where β = q −〈h2,$1−α1〉 2 = qa212 = qa121 and we have used (43). Hence, Zf2 = ( ∑ r+s=1−a12 (−q−a121 )s [r]1![s]1! ) (−τ1)1−a12σ2 = 0. In the last equality we have used the following formula: m∑ i=0 (−z)i [ m i ] = m∏ j=1 ( 1− zq2j−m−1 ) , where [ m i ] = [m]!/([i]![m− i]!). 22 A. Kuniba, M. Okado and Y. Yamada Next, we show Zf1 = 0. ( τ r1 τ2τ s 1 ) f1 = r∑ i=1 τ r−i1 σ1 ( τ1k −1 1 )i−1( τ2k −1 1 )( τ1k −1 1 )s + τ r1 τ2 s∑ i=1 τ s−i1 σ1 ( τ1k −1 1 )i−1 = r∑ i=1 δγi−1−sτ r−1 1 τ2τ s 1σ1 + s∑ i=1 γi−1τ r1 τ2τ s−1 1 σ1, where constants γ, δ are determined by σ1 ( τ1k −1 1 ) = γτ1σ1, σ1 ( τ2k −1 1 ) = δτ2σ1 and hence we have γ = q1q −〈h1,$1−α1〉 1 = q2 1, δ = q −〈h1,$2−α2〉 1 = qa121 . Then, we obtain Zf1 = ∑ r+s=1−a12 (−1)r [r]1![s]1! ( r∑ i=1 δγi−1+sτ r−1 1 τ2τ s 1σ1 + s∑ i=1 γi−1τ r1 τ2τ s−1 1 σ1 ) = ∑ r+s=1−a12 (−1)r [r]1![s]1! ( δγs 1− γr 1− γ τ r−1 1 τ2τ s 1σ1 + 1− γs 1− γ τ r1 τ2τ s−1 1 σ1 ) = ∑ r+s=1−a12 ( −(−1)r−1qs1τ (r−1) 1 τ2τ (s) 1 + (−1)rqs−1 1 τ (r) 1 τ2τ (s−1) 1 ) σ1 = 0 as desired. � Remark 1. The special case w = w0 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 Aq(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 equa- tions 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). Acknowledgments The authors thank Ivan C.H. Ip, Anatol N. Kirillov, Toshiki Nakashima and Masatoshi Noumi for communications. 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JETP 52 (1980), 325–336. http://dx.doi.org/10.1007/s00029-012-0099-x http://arxiv.org/abs/1104.0531 http://arxiv.org/abs/1205.2940 http://dx.doi.org/10.1142/S0217732397000443 http://arxiv.org/abs/hep-th/9702013 http://dx.doi.org/10.1215/S0012-7094-93-06920-7 http://arxiv.org/abs/1210.6430 http://dx.doi.org/10.1088/1751-8113/45/46/465206 http://arxiv.org/abs/1208.1586 http://dx.doi.org/10.2307/1990961 http://dx.doi.org/10.1063/1.531350 http://dx.doi.org/10.1007/s11005-008-0219-x http://arxiv.org/abs/0707.4029 http://dx.doi.org/10.1142/S0217751X92004087 http://dx.doi.org/10.1007/BF01077623 http://dx.doi.org/10.1006/jabr.1998.7688 http://dx.doi.org/10.1112/plms/pdq006 http://arxiv.org/abs/0905.0852 1 Introduction 2 Quantized enveloping algebra Uq(g) 2.1 Definition 2.2 PBW basis 3 Quantized algebra of functions Aq(g) 3.1 Definition 3.2 Right quotient ring Aq(g)S 3.3 Realization by generators and relations 3.3.1 An-1 case 3.3.2 Cn case 3.3.3 G2 case 4 Representations of Aq(g) 4.1 General remarks 4.2 A2 case 4.3 C2 case 4.4 G2 case 5 Main theorem 5.1 Definitions of AB and AB 5.2 Proof of Theorem 5 for rank 2 cases 5.3 Explicit formulae for rank 2 cases: Proof of Proposition 2 5.3.1 A2 case 5.3.2 C2 case 5.4 G2 case 6 Discussion References

A Common Structure in PBW Bases of the Nilpotent Subalgebra of Uq(g) and Quantized Algebra of Functions

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