On diagonal solutions of the reflection equation

Transcription

1 On diagonal solutions of the reflection equation Zengo Tsuboi Osaka city university advanced mathematical institute Sugimoto Sumiyoshi-ku Osaka Japan arxiv: v1 [math-ph] 26 Nov 2018 November Abstract We study solutions of the reflection equation associated with the quantum affine algebra U q ĝln and obtain diagonal K-operators in terms of the Cartan elements of a quotient of U q gln. We also consider intertwining relations for these K-operators and find an augmented q-onsager algebra like symmetry behind them. Key words: augmented q-onsager algebra; intertwining relation; K-matrix; L-operator; quantum algebra; reflection equation 1 Introduction The reflection equation [1] is a fundamental object in quantum integrable systems with open boundary conditions [2]. It has the following form y 1 x R 12 K 1 xr 21 xyk 2 y = K 2 yr 12 K 1 xr 21 xy C. 1.1 x xy y Here Rx is a solution R-matrix of the Yang-Baxter equation and Kx is a K- matrix. The indices 12 denote the space where the operators act non-trivially. In particular there is a N N diagonal matrix solution [3] of the reflection equation associated with the R-matrices [4] for the N-dimensional fundamental representation of U q ĝln. These R-matrices are specialization of more general operators called L- operators: L 12 xl 12 x U q gln EndC N. Namely they are given by R 12 x = π 1L 12 x R 21 x = π 1L 12 x where π is the fundamental representation of U q gln. In this context a natural problem is to investigate the solutions of the reflection equation associated with the L-operators: y 1 x L 12 K 1 xl 12 xyk 2 y = K 2 yl 12 K 1 xl x xy y 1

2 where the K-operator Kx is an operator in U q gln. In this paper we propose diagonal solutions of 1.2 namely solutions written in terms of only the Cartan elements of a quotient of U q gln. Evaluation of the K-operator for the fundamental representation of U q gln reproduces the N N diagonal K-matrix: Kx = πkx. In the context of Baxter Q-operators for integrable systems with open boundaries the essentially same K-operator for N = 2 case was previously proposed in [5]. This paper extends this to N 3 case in part. We also rewrite 1.2 as intertwining relations for the K-operator and speculate an explicit form of the underlying symmetry algebra. For N = 2 case it is known that the augmented q-onsager algebra [6 7] which is a co-ideal subalgebra of U q ŝl2 serves as this. We find an augmented q-onsager algebra like symmetry still exists for an oscillator representation even for N 3 case. At a little more abstract level the relevant algebras will be the reflection equation algebras [2] and some coideal subalgebras of quantum affine algebras [8]. In addition symmetries of the transfer matrix of open spin chains with diagonal K-matrices are discussed in [9]. Intertwining relations for K-matrices are studied for example in [10]. However an explicit expression for the higher rank analogue of the augmented q-onsager algebra seems to be missing in the literatures. We expect that our results can be a cue for further study on this. There are several versions of the reflection equation left right twisted untwisted. However their solutions K-matrices are considered to be related each other by some transformations. Then we concentrate on one of them. Throughout this paper we use the general gradation 1 of U q ĝln. This does not produce particularly new results since the difference of the gradation reduces to a simple similarity transformation and a rescaling of the spectral parameter on the level of the R-matrices. However there is a merit to use it since the parameters in the general gradation play a role as markers and make it easier to trace the actions of the generators of U q ĝln. 2 The quantum affine and finite algebras In this section we review the quantum algebras [ ] of type A and associated L-operators. We also refer the books [15 16] for review of this subject. 2.1 The quantum affine algebra U q ĝln We introduce a q-commutator [XY] q = XY qyx and set [XY] 1 = [XY]. The 2 quantum affine algebra U q ĝln is a Hopf algebra generated by the generators e i f i k i where i {1...N}. For ij {12...N} the defining relations of the 1 We emulate [11] and their subsequent papers for this. 2 Inthis paper wedonotusethe degreeoperatord. We willonlyconsiderlevelzerorepresentations. The notation e 0 f 0 conventionally used in literatures corresponds to e N f N. We assume that the deformation parameter q = e C is not a root of unity. 2

3 algebra are given by [k i k j ] = 0 [k i e j ] = δ ij δ ij+1 e j [k i f j ] = δ ij δ ij+1 f j [e i f j ] = δ ij q h i q h i q q 1 [e i e j ] = [f i f j ] = 0 for i j [e i [e i [e i e j ] q 2]] q 2 = [f i [f i [f i f j ] q 2]] q 2 = 0 for N = 2 i j [e i [e i e j ] q ] q 1 = [f i [f i f j ] q 1] q = 0 for N 3 i j = 1 whereh i = k i k i+1 andij shouldbeinterpreted modulon: N+1 1. ĉ = N i=1 k i is a central element of the algebra. The algebra has the co-product : U q ĝln U q ĝln UqĝlN defined by e i = e i 1+q h i e i f i = f i q h i +1 f i 2.2 k i = k i 1+1 k i. We will also use the opposite co-product defined by = σ σ X Y = Y X XY U q ĝln. 2.3 In addition to these there are anti-poide and co-unit which will not be used here. The Borel subalgebras B + resp. B is generated by the elements e i k i resp. f i k i where i {1...N}. There exists a unique element [12 17] R B + B called the universal R-matrix which satisfies the following relations a R = R a for a U q ĝln 1R = R 13 R R = R 13 R 12. where 3 R 12 = R 1 R 23 = 1 R R 13 = σ 1R 23. The Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R is a corollary of these relations 2.4. The universal R-matrix can be written in the form R = R q N i=1 k i k i. 2.6 Here R is the reduced universal R-matrix which is a series in e j 1 and 1 f j and does not contain Cartan elements. act. 3 We will use similar notation for the L-operators to indicate the space on which they non-trivially 3

4 2.2 The quantum finite algebra U q gln The finite quantum algebra U q gln is generated by the generators {e ii+1 e i+1i } N 1 i=1 and {e ii } N i=1 obeying the following defining relations: [e ii e jj ] = 0 [e ii e jj+1 ] = δ ij δ ij+1 e jj+1 [e ii e j+1j ] = δ ij δ ij+1 e j+1j [e ii+1 e j+1j ] = δ ij q e ii e i+1i+1 q e ii+e i+1i+1 q q 1 [e ii+1 e jj+1 ] = [e i+1i e j+1j ] = 0 for i j [e ii+1 [e ii+1 e jj+1 ] q ] q 1 = [e i+1i [e i+1i e j+1j ] q 1] q = 0 for i j = 1. Note that c = e 11 +e e NN is a central element of the algebra. We will also use elements defined recursively by e ij = [e ik e kj ] q for i > k > j e ij = [e ik e kj ] q 1 for i < k < j e ij = [e ik e kj ] q 1 for i > k > j e ij = [e ik e kj ] q for i < k < j where e ii+1 = e ii+1 e i+1i = e i+1i. We summarize relations among these elements in the appendix. There is an evaluation map ev x : U q ĝln U qgln: e N x s N q e 11 e N1 q e NN f N x s N q e NN e 1N q e 11 e i x s i e ii+1 f i x s i e i+1i for 1 i N 1 k i e ii for 1 i N 2.10 where x C is a spectral parameter; and the parameters s i Z fix the gradation of the algebra. We will also use another evaluation map ev x : U q ĝln U qgln which is defined by e N x s N q e 11 e N1 q e NN f N x s N q e NN e 1N q e 11 e i x s i e ii+1 f i x s i e i+1i for 1 i N 1 k i e ii for 1 i N Representations of quantum algebras Let π be the fundamental representation of U q gln and E ij be a N N matrix unit whose kl-element is δ ik δ jl. We chose the basis of the representation space so that πe ij = πe ij = E ij holds. The composition π x = π ev x = π ev x gives an evaluation representation 4 of U q ĝln. 4 Evaluation representations based the map ev x do not necessary coincide with the ones based on the map ev x for more general representations. 4

5 The q-oscillator algebra is generated by the generators c i c i n i i {12...N} obeying the defining relations: [c i c j ] q δ ij = δ ij q n i [c i c j ] q δ ij = δ ij q n i [n i c j ] = δ ij c j [n i c j ] = δ ijc j [n in j ] = [c i c j ] = [c i c j ] = A Fock space is given by the actions of the generators on the vacuum vector defined by c i 0 = n i 0 = 0. A highest weight representation of U q gln with the highest weight m0...0 and the highest weight vector 0 e ii 0 = mδ i1 0 e jj+1 0 = 0 1 i N 1 j N 1 can be realized in terms of the q-oscillator algebra cf. q-analogue of the Holstein-Primakoff realization [18 19] : e 11 = m n 2 n N e kk = n k for 2 k N e 1j = c j q j 1 k=2 n k e 1j = c j q j 1 k=2 n k for 2 j N [ ] [ ] e i1 = c i m n k q i 1 k=2 n k e i1 = c i m n k k=2 q e ij = c i c jq j 1 k=i+1 n k e ij = c i c jq j 1 k=i+1 n k for 2 i < j N e ij = c i c jq i 1 k=j+1 n k e ij = c i c jq i 1 k=j+1 n k for 2 j < i N where [x] q = q x q x /q q L-operators k=2 q q i 1 k=2 n k for 2 i N 2.13 The so-called L-operators are images of the universal R-matrix which are given by Lxy 1 = φxy 1 ev x π y R Lxy 1 = φxy 1 ev x π y R 21 where φxy 1 and φxy 1 are overall factors. They are solutions of the intertwining relations following from 2.4: ev x π y alxy 1 = Lxy 1 ev x π y a 2.14 ev x π y alxy 1 = Lxy 1 ev x π y a a U q ĝln One can solve 5 these to get cf. [14 20] Lxy 1 = Lxy 1 = j j { xy 1 ξ k +ξ j L + kj xy 1 } s ξ k +ξ j L kj E kj 2.16 { xy 1 s ξ k +ξ j L + kj + xy 1 } ξ k +ξ j L kj E kj Taking note on the expression 2.6 we adopt the solutions which satisfy L0 = L = N i=1 qeii E ii for the case s k > 0 for all k {12...N}. 5

6 where ξ k = s k +s k+1 + +s N s = ξ 1 and the coefficients are related to the generators of U q gln as L + ii = qe ii L ii = q e ii L + ij = q q 1 e ji q e jj L ij = 0 for i > j 2.18 L ij = q q 1 q e ii e ji L + ij = 0 for i < j L + ii = q e ii L ii = q e ii L + ij = q q 1 q e ii e ji L ij = 0 for i > j 2.19 L ij = q q 1 e ji q e jj L + ij = 0 for i < j. We remark that the second L-operator 2.17 follows from the first one 2.16 by the transformation e ij e N+1 in+1 j E ij E N+1 in+1 j for 1 ij N s l s N l for 1 l N s N s N which is composition of automorphisms of U q ĝln and U qgln. In fact the intertwining relation 2.15 follows from 2.14 by the same transformation Evaluating the L-operators for the fundamental representation 6 we obtain R- matrices [4] Rx = π 1Lx = q x s q 1 E ii E ii + 1 x s E ii E jj i j i=1 +q q 1 i<j x ξ i ξ j E ij E ji +q q 1 i>j x s+ξ i ξ j E ij E ji 2.21 Rx = π 1Lx = q x s q 1 E ii E ii + 1 x s E ii E jj i j i=1 +q q 1 x ξ i ξ j E ij E ji +q q 1 x s+ξ i ξ j E ij E ji i>j i<j 3 The reflection equation and its solutions In this section we will derive intertwining relations from the reflection equation associated with the L-operators and obtain K-operators in terms of the Cartan elements of a quotient of U q gln. 6 Up to an overall factor 2.22 coincides with π x π 1 R 21 since 3.27 and c = 1 hold true for the fundamental representation. 6

7 We start from the following form of the reflection equation for the R-matrices 2.21 and 2.22: y 1 x R 12 K 1 xr 12 xyk 2 y = K 2 yr 12 K 1 xr x xy y It is known that this equation allows a N N diagonal matrix solution K-matrix [3]. In our convention it reads Kx = a x 2s ξk ǫ +ǫ + x s E kk + x 2s ξk ǫ +ǫ + x s E kk 3.2 k=a+1 where a {01...N} ǫ ± C. Now we would like to consider the reflection equation for the L-operators 2.16 and 2.17: y 1 x L 12 K 1 xl 12 xyk 2 y = K 2 yl 12 K 1 xl x xy y The reflection equation 3.1 is the image of 3.3 for π 1. Expanding 3.3 with respect to y we obtain x 2ξ k x 2ξ k x 2ξ k L + ik KxL+ kj L ik KxL kj = 0 for ij a or ij > a 3.4 { } x s L + ik KxL kj +x s L ik KxL+ kj ǫ + +L ik KxL kj ǫ = 0 for i a < j 3.5 { } x s L + ik KxL kj +x s L ik KxL+ kj ǫ + +L + ik KxL+ kj ǫ = 0 One can rewrite 7 these in terms of e ij and e ij : for j a < i. 3.6 q 2e ii Kx = Kxq 2e ii 3.7 x 2ξ j e ji Kx q q 1 i 1 k=j+1 x 2ξ k e ki Kxe jk x 2ξ i Kxe ji = 0 for j < i a or a < j < i The commutation relations for the Cartan elements from 3.4 are of the form q eii Kxq eii = q eii Kxq eii from which e ii Kx = Kxe ii are duduced under expansion of e ii = logq eii /logq. We have used these to simplify

8 x 2ξ j e ji Kx+q q 1 j 1 k=i+1 x 2ξ k e ki Kxe jk x 2ξ i Kxe ji = 0 for i < j a or a < i < j 3.9 i 1 x 2ξ j e ji ǫ + x s q 2e jj +ǫ Kx q q 1 ǫ + x s x 2ξ k q e ii e ki q e kk Kxe jk N j 1 +ǫ + x s x 2ξ k e ki Kxq e kk e jk q e jj ǫ x 2ξ k e ki Kxe jk k=j+1 k=i+1 x 2ξ i Kxe ji ǫ + x s q 2e ii 2 +ǫ = 0 for i a < j 3.10 j 1 x 2ξ j e ji ǫ + x s q 2e jj +ǫ Kx+q q 1 ǫ + x s x 2ξ k e ki q e kk Kxe jk q e jj N i 1 +ǫ + x s x 2ξ k q e ii e ki Kxq e kk e jk ǫ x 2ξ k e ki Kxe jk k=i+1 k=j+1 x 2ξ i Kxe ji ǫ + x s q 2e ii+2 +ǫ = 0 for j a < i For i and j satisfying i j = 1 the relations 3.8 and 3.9 reduce to x 2ξ j e jj+1 Kx = x 2ξ j+1 Kxe jj+1 x 2ξ j+1 e j+1j Kx = x 2ξ j Kxe j+1j Taking note on 2.8 and 2.9 and 3.7 one can derive for j a x 2ξ j e ji Kx = x 2ξ i Kxe ji for ij a or a < ij 3.13 x 2ξ j e ji Kx = x 2ξ i Kxe ji for ij a or a < ij 3.14 from Then the relations 3.8 and 3.9 boil down to the relations e ji q q 1 e ji +q q 1 i 1 k=j+1 j 1 k=i+1 e ki e jk e ji = 0 for j < i 3.15 e ki e jk e ji = 0 for i < j 3.16 which are special cases of A2 and A3 under 3.13 and Thus it suffices to consider 3.12 instead of 3.8 and 3.9. We further rewrite 3.10 and 3.11 under 8

9 the relations 3.13 and 3.14 in the form of intertwining relations: x 2ξ j ǫ + x s A ji +ǫ B ji Kx = x 2ξ i Kxǫ + x s C ji +ǫ D ji A ji = e ji q 2e jj q q 1 B ji = e ji +q q 1 j 1 k=a+1 k=j+1 e ki e jk e ki e jk q e kk e jj +1 i 1 C ji = e ji q 2eii 2 +q q 1 e ki e jk q e kk+e ii 2 D ji = e ji q q 1 a k=i+1 e ki e jk for i a < j 3.17 x 2ξ j ǫ + x s A ji +ǫ B ji Kx = x 2ξ i Kxǫ + x s C ji +ǫ D ji j 1 A ji = e ji q 2e jj +q q 1 e ki e jk q e kk+e jj 1 B ji = e ji q q 1 a k=j+1 C ji = e ji q 2e ii+2 q q 1 D ji = e ji +q q 1 i 1 k=a+1 e ki e jk k=i+1 e ki e jk q e kk e ii +2 e ki e jk for j a < i We remark that the relation B ji = D ji holds due to A2 and A3. We find that the following operators 8 satisfy and 3.18 for the q-oscillator representation 2.13 and thus solve the reflection equation 3.3. k=a+1 ekk 2 ;q 2 Kx = q 2c 1 a e kk x 2 N = q 2c 1 a e kk x 2 N ǫ sθ1 k a ξ ke kk ǫ sθ1 k a ξ ke kk ǫ + x s q 2 N ǫ ǫ + x s q 2 a e kk ;q 2 for q > 1or 3.19 ekk+2 ;q 2 ǫ + x s q 2 a ǫ ǫ + x s q 2 N k=a+1 e kk ;q 2 8 In 2016 we were informed by S. Belliard that he found a non-diagonal solution for the rational case Ysl2. 9

10 = x 2 N ξ ke kk = x 2 N ξ ke kk for q < 1or 3.20 ǫ + ǫ x s q 2 a ekk 2 ;q 2 for q > 1or 3.21 ǫ + ǫ x s q 2 N k=a+1 e kk ;q 2 ǫ + ǫ x s q 2 N k=a+1 ekk+2 ;q 2 ǫ + ǫ x s q 2 a e kk ;q 2 for q < where c = N e kk θtrue = 1 θfalse = 0 and x;q = j=0 1 xqj. One can directly check that satisfy 3.7 and 3.12 for the generic generators of U q gln. One can also show that the operators satisfy the following relations KxE ji = x 2ξ i ξ j E ji Kx KxE ji = x 2ξ i ξ j E ji Kx ǫ + x s q 2 a e kk +ǫ ǫ + x s q 2 N k=a+1 ekk+2 +ǫ ǫ + x s q 2 N k=a+1 e kk +ǫ ǫ + x s q 2 a e kk 2 +ǫ for j a < i 3.23 for i a < j 3.24 where E ji is any operator obeying q e kk Eji q e kk = q δ kj δ kieji. Thus 3.17 and 3.18 reduce to B ji = A ji q 2 N k=a+1 e kk = C ji q 2 a e kk+2 B ji = A ji q 2 a e kk = C ji q 2 N k=a+1 e kk 2 for i a < j 3.25 for j a < i These relations 3.25 and 3.26 give a constraint on the class of representations of U q gln. In other wards we have to consider the quotient of U q gln by the relations 3.25 and Moreover we find that the first evaluation map 2.10 is almost 9 equivalent to the second one 2.11 under these: ev x e N = q 2c 1 ev x e N ev x f N = q 2c 1 ev x f N [under 3.25 and 3.26] One can check that 2.13 satisfies 3.25 and In addition the fundamental representation of U q gln which is a quotient of 2.13 at m = 1 also fulfills 3.25 and Then we recover 3.2 from 3.19: πkx = x s ǫ ǫ + x s q 2 ;q 2 Kx ǫ + ǫ ǫ + x s ;q 2 9 The factor q 2c 1 in the right hand side can be eliminated if one considers composition of the automorphism e i q 2ĉ 1δiN e i f i q 2ĉ 1δiN f i k i k i of U q ĝln and the evaluation map ev x. 10

11 It is important to note that the relations 3.25 and 3.26 become trivial for N = 2 case. This means that the K-operators written in terms of the generic Cartan elements solve the reflection equation 3.3 without any constraint on the representations for U q gl2 case. In fact 3.19 for N = 2 reproduces 10 the K-operator proposed in [5]. Let us consider a product of the L-operators Lxq 2c 1 s N j=1 ξ N je jj Lxq 2c 1 s q 2c 1 s j=1 ξ je jj = = i=1 x s x s +G i E ii +q q 1 i<j C ji A ji q 2c 1 q e jj e ii +1 E ij +q q 1 i>j A ji C ji q 2c 1 E ij 3.29 where G i = G + i +G i is defined by i 1 G + i = q 2e ii +q q 1 2 e ki e ik q e kk+e ii G i = q 2c 1 q 2e ii +q q 1 2 N k=i+1 e ki e ik q e kk e ii Here A ji and C ji for i < j resp. i > j are the ones in 3.17 resp [dependence on the parameter a is irrelevant]. In deriving 3.29 we have used A2 and A3. Note that 3.29 becomes diagonal under the relations A ji q 2c 1 = C ji for i < j and A ji = C ji q 2c 1 for j < i We remark that a part of the relations 3.25 and 3.26 automatically follows from For N = 2 the relation 3.32 becomes trivial and the diagonal elements G 1 and G 2 reduce to a Casimir element of U q gl2 [cf. eq. 3.5 in [5]]. On the other hand for N 3 case each diagonal element is not a Casimir element but rather twisted sums 11 N G+ k q 2k N G k q 2k and N G kq 2k are cf. eq. 41 in [24]. However there are cases where simplifications occur for particular representations. In fact one can check that the q-oscillator representation 2.13 satisfies the condition 3.32 and G i = q 2m +q 2 holds for any i. This is one of the desirable properties for construction of mutually commuting transfer matrices. 4 Augmented q-onsager algebra like symmetry In this section we will reconsider the intertwining relations in the previous section and point out their connection to the generators of U q ĝln. We will also mention our 10 x s2c q c 1s+s 2 c+s 1 H s Kxq c 1 s for N = 2 coincides with eq. 4.9 in [5]. 11 N G kq 2k is a part of the twisted trace of 3.29 by 1 πq 2 N ke kk. 11

12 observation on an underlying symmetry behind them. We find that the intertwining relations and 3.18 can be written in terms of the generators of U q ĝln: where Z ji is defined as follows: For 1 i = j N For 1 j < i a or a+1 j < i N For 1 i < j a or a+1 i < j N For j a < i ev x 1Z ji Kx = Kxev x Z ji 4.1 Z ii = k i. 4.2 Z ji = e [ji 1]. 4.3 Z ji = f [ij 1]. 4.4 Z ji = ǫ + Z ji + +ǫ Zji [[f [1j 1] f N ] q 1f [in 1] ] q q k j k i +1 for 2 j a < i N 1 Z ji + = [f N f [in 1] ] q q k 1 k i +1 for 1 = j a < i N 1 [f [1j 1] f N ] q 1q k j k N +1 for 2 j a < i = N f N q k 1 k N +1 for 1 = j a < i = N [e [ja] e [a+1i 1] ] q 1 for 1 j a < a+1 < i N Z ji = e [ja] for 1 j a < a+1 = i N e [ai 1] for 1 j = a < i N. For i a < j: Z ji = ǫ + Z + ji +ǫ Z ji [[e [jn 1] [e N e [1i 1] ] q 1] q for 2 i a < j N 1 Z + ji = [e [jn 1] e N ] q for 1 = i a < j N 1 [e N e [1i 1] ] q 1 for 2 i a < j = N e N for 1 = i a < j = N [f [a+1j 1] f [ia] ] q 1q k j k i +1 for 1 i a < a+1 < j N Z ji = f [ia] q k a+1 k i +1 for 1 i a < a+1 = j N f [aj 1] q k j k a+1 for 1 i = a < j N

13 Here we use the following notation for root vectors. For i < j we define e [ij] = [e i [e i+1...[e j 2 [e j 1 e j ] q 1] q 1...] q 1] q 1 e [ij] = [e i [e i+1...[e j 2 [e j 1 e j ] q ] q...] q ] q f [ij] = [f j [f j 1...[f i+2 [f i+1 f i ] q ] q...] q ] q 4.9 f [ij] = [f j [f j 1...[f i+2 [f i+1 f i ] q 1] q 1...] q 1] q 1 and e [ii] = e [ii] = e i f [ii] = f [ii] = f i. We remark that the third case in 4.6 resp. 4.8 is a special case of the first or the second ones. In deriving these we have used A2-A10. We introduce generators composed of Cartan elements k + ji = ǫ +q j l=1 k l N l=i k l k ji = ǫ q a l=j k l+ i l=a+1 k l for 1 j a < i N 4.10 and use notation Z aa+1 = e a Z N1 = e N Z a+1a = f a Z 1N = f N k + a = k + aa+1 k a = k aa+1 k+ N = k+ 1N k N = k 1N. We have observed 12 the following commutation relations among generators 13 for the q-oscillator representation of U q ĝln under the evaluation map 2.10 and 2.13 in addition to the relations 2.1 restricted to the generators {e i f i } i an and {k i } N i=1. [k + ji k rs] = [k ± ji k± rs] = [k ± ji k l] = 0 for 1 jr a < is N 1 l N; 4.11 [k l Z ji ] = δ lj δ il Z ji for 1 lij N; 4.12 k + ji Z rs = q θ1 r j θi r N θ1 s j+θi s N Z rs k + ji k ji Z rs = q θj r a+θa+1 r i+θj s a θa+1 s i Z rs k ji for 1 j a < i N 1 rs N; 4.13 [e a [e a [e a f a ] q 2]] q 2 = ρe a k + a 2 k a 2 e a [e N [e N [e N f N ] q 2]] q 2 = ρe N k N 2 k + N 2 e N [f a [f a [f a e a ] q 2]] q 2 = ρf a k a 2 k + a 2 f a 4.14 [f N [f N [f N e N ] q 2]] q 2 = ρf N k + N 2 k N 2 f N for 1 a N 1; [e a [e a [e a e N ] q 2]] q 2 = [f a [f a [f a f N ] q 2]] q 2 = 0 [e N [e N [e N e a ] q 2]] q 2 = [f N [f N [f N f a ] q 2]] q 2 = [f N e a ] q 2 = [e N f a ] q 2 = 0 for 2 a N 2; 12 We have checked these by Mathematica. 13 Note that the q-commutators [A[AB] q1 ] q2 and [A[A[AB] q1 ] q2 ] q3 are invariant under the permutation on {q 1 q 2 } and {q 1 q 2 q 3 } respectively. 13

14 [e a 1 [e a 1 e a ] q ] q 1 = [e a [e a e a 1 ] q ] q 1 = [f a 1 [f a 1 f a ]] q 2 = [f a [f a f a 1 ]] q 2 = 0 [e 1 [e 1 e N ] q ] q 1 = [e N [e N e 1 ] q ] q 1 = [f 1 [f 1 f N ]] q 2 = [f N [f N f 1 ]] q 2 = 0 [e a 1 f a ] q 1 = [e 1 f N ] q 1 = 0 for 2 a N 1; 4.16 [e a+1 [e a+1 e a ] q ] q 1 = [e a [e a e a+1 ] q ] q 1 = [f a+1 [f a+1 f a ]] q 2 = [f a [f a f a+1 ]] q 2 = 0 [e N 1 [e N 1 e N ] q ] q 1 = [e N [e N e N 1 ] q ] q 1 = 0 [f N 1 [f N 1 f N ]] q 2 = [f N [f N f N 1 ]] q 2 = 0 [e a+1 f a ] q 1 = [e N 1 f N ] q 1 = 0 for 1 a N 2; 4.17 [e i e a ] = [e i f a ] = [f i f a ] = 0 for 1 i a 2 or a+2 i N 1; 4.18 [e i e N ] = [e i f N ] = [f i f N ] = 0 for 2 i a 1 or a+1 i N 2; 4.19 [f i e a ] = [f i e N ] = 0 for 1 i a 1 or a+1 i N 1; 4.20 [e i 1 Z ij ] q = Z i 1j [Z ji f i 1 ] = Z ji 1 q 1+k i 1 k i for 2 i a < j N; 4.21 [Z ij e j ] q 1 = Z ij+1 [f j Z ji ] = Z j+1i q k j k j+1 for 1 i a < j N 1; 4.22 [f 1 [f 1 [f 1 f N ] q 3] q ] q 1 = q 1 ρf 1 k 1 2 f 1 Z 2N [f N [f N [f N f 1 ] q 3] q 1] q = q 4 ρf N k 1 2 f N Z 2N [e 1 [e 1 [e 1 e N ] q 3] q ] q 1 = q 5 ρe 1 k + N 2 e 1 Z N2 q k N k [e N [e N [e N e 1 ] q 3] q 1] q = q 2 ρe N k + N 2 e N Z N2 q k N k 2 for a = 1 N > 2; [f N 1 [f N 1 [f N 1 f N ] q 3] q ] q 1 = q 2 ρf N 1 k N 2 f N 1 Z 1N 1 q 2k 1 [f N [f N [f N f N 1 ] q 3] q 1] q = qρf N k N 2 f N Z 1N 1 q 2k 1 [e N 1 [e N 1 [e N 1 e N ] q 3] q ] q 1 = ρe N 1 k + N 1 2 e N 1 Z N 11 q k 1 k N [e N [e N [e N e N 1 ] q 3] q 1] q = q 1 ρe N k + N 1 2 e N Z N 11 q k 1 k N 1 for a = N 1 N > 2 where ρ = q 3 q 3 q 2 q 2 ρ = ρ/q q 1. We remark that the parameter a is a fixed parameter in the diagonal K-matrix 3.2. These commutation relations are valid without any restriction of the class of representations for N = 2 case and define the so-called augmented q-onsager algebra [6 7] a realization of which in terms of the generators of U q ŝl2 is known in [7]. In contrast not all of them hold true for 14

15 generic representations for N 3 case. Whether the elements Z ji for N 3 satisfy closed commutation relations on the level of the algebra remains to be clarified. 5 Rational limit In this section we consider the rational limit of some of the formulas in previous sections. Let ǫ ± = q 2p ± p = p + p and u C. The rational limit of the L-operators the K-matrix 3.2 the K-operators the intertwining relations and 3.18 are given respectively by Lu = lim q 1 q q 1 1 Lq 2u = ij Lu = lim q 1 q q 1 1 Lq 2u = ij Ku = lim q 1 q q 1 1 Kq 2u = suδ ij +e ji E ij 5.1 suδ ij +e ji E ij 5.2 a su+pe kk + Ku = lim Kq 2u 1 q 2 1 c+2su q 1±0 k=a+1 su+pe kk 5.3 Γ su p+ a = e kk Γ su p+1 from 3.19 or 3.22 or 5.4 N k=a+1 e kk Γ su+p+ N k=a+1 e kk = Γsu+p+1 a e from 3.20 or kk e ji Ku = Kue ji for ij a or ij a a su pe ji e ki e jk Ku = Ku su pe ji + e ki e jk su pe ji + k=a+1 for i a < j 5.7 a e ki e jk Ku = Ku su pe ji e ki e jk where e ij are generators of gln: k=a+1 for j a < i 5.8 [e ij e kl ] = δ jk e il δ li e kj

16 We find that 5.4 and 5.5 solve the relations and thus the reflection equation L 12 v uk 1 ul 12 u+vk 2 v = K 2 vl 12 u vk 1 ul 12 u v for uv C 5.10 which is the rational limit of 3.3 if the following conditions 14 are satisfied. N a a e ji e kk = ki e jk e ji e kk 1 = e ki e jk for i a < j 5.11 k=a+1 k=a+1e a a N e ji e kk = e ki e jk e ji e kk 1 = e ki e jk for j a < i k=a+1 k=a+1 The relations 5.11 and 5.12 correspond to the rational limit of 3.25 and 3.26 respectively. Then the rational limit of 2.13 satisfies 5.11 and Note that the above relations reproduce a part of a kind of generalized rectangular condition 15 [21]. e ki e jk = αe ji +βδ ij αβ C In fact sum of the first and the second relations in 5.11 or 5.12 gives N e ji e kk 1 = e ki e jk for i a < j or j a < i This corresponds to 5.13 for β = 0 since α := N e kk 1 is a central element of gln. 6 Concluding remarks We have derived intertwining relations and 3.18 from the reflection equation 3.3 for L-operators associated with U q ĝln and have obtained the diagonal K-operators in terms of the Cartan elements of a quotient of U q gln by the relations 3.25 and The intertwining relations can be expressed in terms of generators of U q ĝln as in 4.1. In particular the elements {Z ji } in the intertwining relations satisfy augmented q-onsager algebra like 14 These are sufficient conditions. If the parameter p in addition to u is interpreted as a free one these are necessary conditions as well. On the other hand 3.25 and 3.26 are necessary and sufficient conditions independent of ǫ ±. 15 In case the representations are finite dimensional this condition is satisfied by the representations labeled by rectangular Young diagrams. 16

17 commutation relations for the q-oscillator representation 2.13 under the evaluation map It will be an interesting problem to investigate the algebra generated by {Z ji } with some constraint on generators of U q ĝln if any which could be a version of higher rank generalization of the augmented q-onsager algebra. The reflection equation 3.1 defines [2] the reflection equation algebra if one adds the third component in the K-matrix: y 1 x R 12 K 13 xr 12 xyk 23 y = K 23 yr 12 K 13 xr 12 x xy y 6.1 K 13 x = E ij 1 k ij x K 23 x = 1 E ij k ij x ij=1 where the elements {k ij x} generate the algebra. In this context it will be desirable to clarify the direct connection between {Z ji } and a certain specialization of {k ij x} for the diagonal K-operators. In this paper we have defined the L-operator Lx based on the second evaluation map2.11. It will beworthwhile todefine itbased onthefirst one2.10asl xy 1 = φ xy 1 ev x π y R 21 and investigate solutions of the reflection equation. The reason why we avoided L x was that its expression based on the generators e ij or e ij might be involved for N 3 case. The conditions 3.25 and 3.26 for the solutions of the reflection equation are the ones that the L-operator Lx coincides with L x up to a fine tune by the central element see the relation The technical difficulty on L x may beresolved if we start fromthe FRTformulation ofthe algebra based Yang- Baxter relations [22] and use the matrix elements of the L-operators as the generators of the algebra instead of e ij or e ij. Another important problem is construction of Baxter Q-operators for open boundary conditions in the light of [23 5]. The K-operators obtained in this paper could be building blocks of Q-operators after taking limits on the parameter m of the q-oscillator representation as was already discussed in [5] for N = 2 case. ij=1 Acknowledgments The author would like to thank Pascal Baseilhac Sergey Khoroshkin and Masato Okado for correspondence. 17

18 Appendix A: Relations for U q gln Here we review relations among the generators of U q gln some of which are used in the main text. e ab = [e ac e cb ] q for a > c > b e ab = [e ac e cb ] q 1 for a < c < b [e ab e ba ] = qeaa e bb q e aa+e bb q q 1 for a < b [e dc e ba ] = q q 1 e da e bc for b < d < a < c or a < c < b < d [e dc e ba ] = 0 for d < c < b < a or d > c > b > a or d < b < a < c or d > b > a > c or d < c a < b or c < d b < a or d < a < b < c or c < b < a < d [e dc e ba ] = q q 1 q eaa ecc e da e bc for d < a < c < b [e dc e ba ] = q q 1 e da e bc q e bb e dd for a < d < b < c [e ba e ac ] = e bc q e bb e aa for a < b < c [e ba e ac ] = q eaa ecc e bc for a < c < b [e db e ba ] = e da q e bb e dd for a < d < b [e db e ba ] = q eaa e bb e da for d < a < b [e da e ba ] q 1 = 0 for a < b < d or b < d < a [e bc e ba ] q = 0 for c < a < b or b < c < a. A1 The relations for e ij can be obtained from the above relations through q q 1. In addition one can prove the following relations based on the above relations and induction. We remark that A3 for l = j 1 corresponds to eq. 10 in [24]. e ji q q 1 l k=j+1 e ki e jk = e ji +q q 1 i 1 k=l+1 e ki e jk = e ji for j l = i 1 = [e jl+1 e l+1i ] q 1 for j l < i 1 e ji for j = l < i A2 e ji q q 1 l k=i+1 e ki e jk = e ji +q q 1 j 1 k=l+1 e ki e jk = 18

19 e ji for i l = j 1 = [e jl+1 e l+1i ] q 1 for i l < j 1 e ji for i = l < j A3 j 1 e ji +q q 1 e ki e jk q e kk e jj 1 = [e jl e li ] q 2q e ll e jj for l < j < i A4 k=l i 1 e ji +q q 1 e ki e jk q e kk e ii = [e jl e li ] q 2q e ll e ii +1 for l < i < j A5 k=l l e ji q q 1 e ki e jk q e ii e kk = [e jl e li ] q 2q e ii e ll 1 k=i+1 for j < i < l A6 e ji q q 1 l k=j+1 e ki e jk q e jj e kk +1 = [e jl e li ] q 2q e jj e ll for i < j < l A7 [e jl e li ] q 2 = [e jl+1 e l+1i ]q e ll e l+1l+1 for i < j < l or j < i < l A8 [e jl e li ] q 2 = [e jl 1 e l 1i ]q e l 1l 1 e ll for l < i < j or l < j < i A9 [e jl 1 e l 1i ] q = [e jl e li ] q 1 for i < l 1 < l < j or j < l 1 < l < i. A10 We also remark that the third relation in the right hand side of A2 and A3 is a special case of the second one. References [1] I. V. Cherednik Factorizing particles on a half-line and root systems Theor. Math. Phys [2] E.K. Sklyanin Boundary conditions for integrable quantum systems J. Phys. A [3] H. J. de Vega A. González Ruiz Boundary K-Matrices for the Six Vertex and the n2n 1 A n 1 Vertex Models J.Phys. A L519-L524 [arxiv:hep-th/ ]. [4] I. V. Cherednik On a method of constructing factorized S matrices in elementary functions Theor. Math. Phys [5] P. Baseilhac Z. Tsuboi Asymptotic representations of augmented q-onsager algebra and boundary K-operators related to Baxter Q-operators Nucl. Phys. B [arxiv: ]. 19

20 [6] T. Ito P. Terwilliger The augmented tridiagonal algebra Kyushu J. Math [arxiv: [math.qa]]. [7] P. Baseilhac and S. Belliard The half-infinite XXZ chain in Onsager s approach Nucl. Phys. B [arxiv: [math-ph]]. [8] A. Molev E. Ragoucy P. Sorba Coideal subalgebras in quantum affine algebras Rev. Math. Phys [arxiv:math/ ]. [9] A.Doikou R.I.Nepomechie Dualityandquantum-algebrasymmetry ofthea 1 N 1 open spin chain with diagonal boundary fields Nucl. Phys. B [arxiv:hep-th/ ]. [10] G. W. Delius N. J. MacKay Quantum group symmetry in sine-gordon and affine Toda field theories on the half-line Commun.Math.Phys [arxiv:hep-th/ ]. [11] H. Boos F. Göhmann A. Klümper K.S. Nirov and A.V. Razumov Exercises with the universal R-matrix J. Phys. A: Math. Theor [arxiv: [math-ph]]. [12] V. Drinfeld Hopf algebras and the quantum Yang-Baxter equations Sov. Math. Dokl [13] M. Jimbo A q-difference analogue of Ug and the Yang-Baxter equation Lett. Math. Phys [14] M. Jimbo A q-analogue of UglN+1 Hecke algebra and the Yang-Baxter equation Lett. Math. Phys [15] V. Chari A. Pressley A guide to quantum groups Cambridge University Press [16] A. Klimyk K. Schmüdgen Quantum groups and their representations Springer- Verlag Berlin Heidelberg [17] S. M. Khoroshkin V. N. Tolstoy The uniqueness theorem for the universal R- matrix Lett. Math. Phys [18] T. D. Palev A Holstein-Primakoff and a Dyson realization for the quantum algebra U q [sln+1] J. Phys. A [arxiv:math/ [math.qa]]. [19] T. Hayashi Q-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras Commun. Math. Phys [20] S. Khoroshkin S. Pakuliak A computation of an universal weight function for the quantum affine algebra U q ĝl N J. Math. Kyoto Univ [arxiv: [math.qa]]. 20

21 [21] R. Frassek T. Lukowski C. Meneghelli M. Staudacher Baxter operators and Hamiltonians for gnearly allh integrable closed gln spin chains Nucl. Phys. B [arxiv: [math-ph]]. [22] L. D. Faddeev N. Y. Reshetikhin and L. A. Takhtajan Quantization of Lie Groups and Lie Algebras Leningrad Math. J [Alg. Anal ]. [23] R. Frassek I. M. Szecsenyi Q-operators for the open Heisenberg spin chain Nucl. Phys. B [arxiv: [math-ph]]. [24] R. B. Zhang Universal L operator and invariants of the quantum supergroup U q glm/n J. Math. Phys