JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.1 (1-13) Available online at www.sciencedirect.com ScienceDirect 1 2 Nuclear Physics B ••• (••••) •••–••• www.elsevier.com/locate/nuclphysb 4 5 6 10 Non-Abelian symmetries of the half-infinite XXZ spin chain 11 10 11 a 12 Pascal Baseilhac , Samuel Belliard 13 b 12 13 14 a Laboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350, Fédération Denis Poisson FR2964, 14 15 Fédération Denis Poisson, Université de Tours, Parc de Grammont, 37200 Tours, France b Institut de Physique Théorique, Orme des Merisiers, batiment 774, Point courrier 136, CEA/DSM/IPhT, CEA/Saclay, Gif-sur-Yvette Cedex, 91191 Saint-Aubin, France 15 16 17 Received 14 December 2016; received in revised form 12 January 2017; accepted 13 January 2017 18 19 16 17 18 19 Editor: Hubert Saleur 20 20 21 21 22 22 23 Abstract 24 25 26 27 28 29 30 23 24 The non-Abelian symmetries of the half-infinite XXZ spin chain for all possible types of integrable boundary conditions are classified For each type of boundary conditions, an analog of the Chevalleytype presentation is given for the corresponding symmetry algebra In particular, two new algebras arise that are, respectively, generated by the symmetry operators of the model with triangular and special Uq (gl2 )-invariant integrable boundary conditions © 2017 Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 25 26 27 28 29 30 31 31 32 32 33 33 34 Introduction 35 36 37 38 39 40 41 35 Identifying non-Abelian infinite dimensional symmetries of quantum integrable systems provides a very efficient starting point for transferring the Hamiltonian and eigenstates formulation into the language of operator algebra, representation theory and related special functions, on which the non-perturbative analysis is thus based Besides the restricted class of models with conformal symmetry for which the Virasoro algebra and its representation theory play a central role, in the thermodynamic limit of lattice systems it is known that quantum groups, related 42 45 46 47 36 37 38 39 40 41 42 43 44 34 43 E-mail addresses: baseilha@lmpt.univ-tours.fr (P Baseilhac), samuel.belliard@cea.fr (S Belliard) http://dx.doi.org/10.1016/j.nuclphysb.2017.01.012 0550-3213/© 2017 Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Funded by SCOAP3 44 45 46 47 JID:NUPHB AID:13972 /FLA 10 11 12 13 14 15 16 17 current algebras and infinite dimensional q-vertex operators representations provide an appropriate mathematical framework As an important example, in the thermodynamic limit N → ∞ of the finite XXZ spin chain with N sites and periodic boundary conditions, the Uq (sl2 ) algebra emerges as a hidden non-Abelian symmetry of the Hamiltonian [15,18,13] Based on the representation theory of the Uq (sl2 ) quantum affine algebra at level one and its current algebra, an explicit characterization of the Hamiltonian’s spectrum, corresponding eigenstates, as well as multiple integral representations of correlation functions and form factors of local operators has been given [13] This approach has been later on extended to lattice systems with periodic boundary conditions associated with higher spins [16] or higher rank affine Lie algebras [22], as well as for certain class of boundary conditions, see for instance [19,5,8,9,20] For lattice models with general integrable boundary conditions, identifying the non-Abelian symmetries in the thermodynamic limits has remained essentially unexplored although coideal subalgebras of quantum affine Lie algebras are natural candidates For instance, in the thermodynamic limit N → ∞ of the finite open XXZ spin chain it was expected that non-Abelian infinite dimensional symmetries emerge, that are associated with certain coideal subalgebras of Uq (sl2 ) Recall that the Hamiltonian of the half-infinite XXZ spin chain is formally defined as (see also [19,5,8,9]): 18 H XXZ = − 2 19 20 21 22 27 28 29 31 32 33 34 35 36 37 38 39 40 41 + σ2k+1 σ2k + σ3k+1 σ3k 10 11 12 13 14 15 16 17 19 20 22 − ++ + σ31 − ) ( + − + −) −) 23 k+ σ+1 + k− σ−1 24 (1.2) Given the Hamiltonian of the half-infinite XXZ spin chain (1.1), until recently the hidden nonAbelian symmetry for q generic and any type of boundary conditions2 k± , ± has remained unknown However, at the end of the article [5], it was pointed out that for the generic case (k± = and ± = 0) and the diagonal case (k± = and ± = 0) the respective hidden non-Abelian symmetries are associated with two different coideal subalgebras of Uq (sl2 ): the q-Onsager algebra [27,1] and the augmented q-Onsager algebra [11,5] (see also [17]) In this letter, we present a unified picture that complete the preliminary observations of [5] Namely, for each type of boundary conditions, the non-Abelian symmetry algebra of the half-infinite XXZ spin chain is identified and characterized through generators and relations In particular, two new coideal subalgebras are obtained 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 σ+ = 0 , σ− = 0 , 46 47 21 V = · · · ⊗ C ⊗ C2 ⊗ C2 44 45 (1.1) k=1 42 43 + hB Here σ1,2,3 and σ± = (σ1 ± iσ2 )/2 are usual Pauli matrices,1 = (q + q −1 )/2 denotes the anisotropy parameter and ± , k± are scalar parameters associated with the right boundary field Formally, the Hamiltonian acts on an infinite tensor product of 2-dimensional vector spaces Note that the ordering of the tensor components in (1.1) is such that: 30 18 σ1k+1 σ1k (q − q −1 ) ( hB = − ( 24 26 ∞ with boundary interaction 23 25 [m1+; v1.242; Prn:18/01/2017; 10:20] P.2 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• 2 In (1.1), we implicitly assume that + + − = σ3 = 0 −1 44 45 46 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.3 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• 10 11 12 The text is organized as follows In Section 2, four different types of coideal subalgebras of Uq (sl2 ) are defined though generators and relations in a Chevalley-type presentation The coaction and counit maps are given in each case Then, it is shown that these algebras are symmetry algebras of the Hamiltonian (1.1) for generic ( ± = 0, k± = 0), triangular ( ± = 0, k+ = 0, k− = 0), diagonal ( ± = 0, k± = 0) and special3 boundary conditions ( + = 1, − = 0, k± = 0), respectively Note that since the case of upper triangular (k− = 0, k+ = 0) and lower triangular (k+ = 0, k− = 0) boundary conditions are related through conjugation of the Hamiltonian (1.1) x by the spin-reversal operator νˆ = ∞ j =1 σj , it is sufficient for our purpose to restrict our attention to the case of lower boundary conditions In Section 3, we propose an alternative and simpler derivation of the symmetry operators which is based on the remarkable connection between the infinite dimensional algebra Aq introduced in [6] and the q-Onsager algebra Concluding remarks follow in Section 13 14 15 Notation The q-commutator X, Y q = qXY tion parameter assumed not to be a root of unity is introduced, where q is the deforma- 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 41 42 43 10 11 12 14 18 In the first part of this Section, four different types of coideal subalgebras of Uq (sl2 ) with central extension are introduced through basic generators and relations The coaction and counit maps4 are given Note that two of these algebras have already appeared in the literature: the q-Onsager algebra [27,1] and the augmented q-Onsager algebra [17,11,5] The two others, the so-called triangular q-Onsager algebra and Uq (gl2 ) invariant q-Onsager algebra5 are new In a second part, depending on the choice of boundary conditions, by analogy with [18, eq (3.2)] it is shown that the Hamiltonian (1.1) commutes with all generators of one of these four algebras Define the extended Cartan matrix {aij } (aii = 2, aij = −2 for i = j ) The quantum affine algebra Uq (sl2 ) is generated by the elements {hj , ej , fj }, j ∈ {0, 1} which satisfy the defining relations [hi , hj ] = , [hi , ej ] = aij ej , [hi , fj ] = −aij fj , q hi − q −hi [ei , fj ] = δij q − q −1 20 21 22 23 24 25 26 27 28 29 31 32 together with the q-Serre relations [ei , [ei , [ei , ej ]q ]q −1 ] = , 19 30 33 and [fi , [fi , [fi , fj ]q ]q −1 ] = (2.1) The sum c = h0 + h1 is the central element of the algebra The Hopf algebra structure is ensured by the existence of a comultiplication : Uq (sl2 ) → Uq (sl2 ) ⊗ Uq (sl2 ), antipode S : Uq (sl2 ) → Uq (sl2 ) and a counit E : Uq (sl2 ) → C with 34 35 36 37 38 39 The corresponding Hamiltonian can be understood as the thermodynamic limit of the U (sl ) invariant spin chain q studied in [25] In general, given a Hopf algebra H with comultiplication and counit E, I is called a left H-comodule (coideal subalgebra of H) if there exists a coaction map δ : I → H ⊗ I such that (right coaction maps are defined similarly) ( × id) ◦ δ = (id × δ) ◦ δ , (E × id) ◦ δ ∼ = id 46 47 17 Four different types of q-Onsager symmetry algebras 44 45 16 39 40 15 18 19 13 − q −1 Y X 16 17 40 41 42 43 44 45 46 However, note that the generators of the U (gl ) invariant q-Onsager algebra have already appeared in [26,21] q 47 JID:NUPHB AID:13972 /FLA (ei ) = ei ⊗ + q hi ⊗ ei , 2 (fi ) = fi ⊗ q −hi + ⊗ fi , 3 (hi ) = hi ⊗ + ⊗ hi , S(ei ) = −q −hi ei , [m1+; v1.242; Prn:18/01/2017; 10:20] P.4 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• (2.2) S(fi ) = −fi q hi , S(hi ) = −hi S(1) = and 10 11 12 E(ei ) = E(fi ) = E(hi ) = , E(1) = Note that the opposite coproduct can be similarly defined with tation map σ (x ⊗ y) = y ⊗ x for all x, y ∈ Uq (sl2 ) is used 10 ≡σ ◦ where the permu- 13 14 2.1 The q-Onsager algebra 15 16 17 18 19 20 21 22 15 The q-Onsager algebra Oq (sl2 ) is an example of tridiagonal algebra [27] Including a central extension, it is generated by two elements W0 , W1 , a central element and unit The defining relations are: W0 , W0 , W0 , W1 q q −1 = ρ W0 , W1 , 25 26 27 28 29 30 (2.3) W1 , W1 , W1 , W0 q q −1 = ρ W1 , W0 , 23 24 W0 , = W1 , 33 34 35 36 37 40 41 42 43 44 45 46 47 18 19 20 21 24 25 −1 26 (2.4) ) k+ k− where k± are nonzero scalars By analogy with the situation for Hopf algebras, one endows the q-Onsager algebra with the coaction map δ : Oq (sl2 ) → Uq (sl2 ) ⊗ Oq (sl2 ) defined by: 27 28 29 30 31 δ(W0 ) = (k+ e1 + k− q −1 f1 q ) ⊗ + q h1 h1 ⊗ W0 , (2.5) 32 33 δ(W1 ) = (k− e0 + k+ q −1 f0 q h0 ) ⊗ + q h0 ⊗ W1 , 34 δ( ) = q ⊗ c 35 36 and counit E : Oq (sl2 ) → C: 37 38 39 17 23 = 0, where ρ is a scalar Without loss of generality, let us define ρ = (q + q 16 22 31 32 11 12 13 14 38 E(W0 ) = +, E(W1 ) = −, E( ) = E(1) = (2.6) This induces an homomorphism ψ = (id × E) ◦ δ : Oq (sl2 ) → Uq (sl2 ) from the q-Onsager algebra to a subalgebra of Uq (sl2 ): ψ(W0 ) = k+ e1 + k− q −1 f1 q h1 + ψ(W1 ) = k− e0 + k+ q ψ( ) = q c −1 f0 q h0 + 39 40 41 42 43 +q h1 −q h0 , , (2.7) 44 45 46 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.5 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• 2.2 The triangular q-Onsager algebra The triangular q-Onsager algebra Oqt (sl2 ) is generated by three elements T0 , T1 , P˜ , a central element and unit The defining relations are: ˜ −1 = ρt T0 , T1 , T0 , T0 , P q T1 , T0 q −1 = ρ˜t , 10 11 12 13 14 15 16 17 18 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 = P˜ , = ρ˜t = − and + − (q − q −1 ), (2.9) where k− , ± are nonzero scalars We endow this algebra with the coaction map δt : Oqt (sl2 ) → Uq (sl2 ) ⊗ Oqt (sl2 ) defined by: δt (T0 ) = k− q −1 f1 q h1 ⊗ + q h1 ⊗ T0 , (2.10) + (q − q −2 ) q −1 e1 q h0 Et (T1 ) = −1 22 23 Et (P˜ ) = p, ˜ −, f1 q h1 + +q h1 Et ( ) = Et (1) = (2.11) (2.12) , ψt (T1 ) = k− e0 + − q h0 , ψt (P˜ ) = (q − q −2 ) − q −1 e1 q h0 + 25 26 27 28 30 + f0 q h1 +h0 + k− [e1 , e0 ]q + q c [f1 , f0 ]q + pq ˜ c, 31 32 ψt ( ) = q 33 34 2.3 The augmented q-Onsager algebra 35 Oqd (sl2 ) The augmented q-Onsager algebra has been introduced in [5], as a generaliza6 tion of the augmented tridiagonal algebra introduced in [17] It is generated by four elements K0 , K1 , Z1 , Z˜ , a central element and unit The defining relations are: K0 K1 = K1 K0 = − + , 42 K0 Z1 = q −2 Z1 K0 , K0 Z˜ = q Z˜ K0 , 43 K1 Z1 = q Z1 K1 , K1 Z˜ = q −2 Z˜ K1 , (2.13) 44 47 24 29 c 41 46 15 21 40 45 14 20 This induces an homomorphism ψt = (id × Et ) ◦ δt from the triangular q-Onsager algebra to a subalgebra of Uq (sl2 ): ψt (T0 ) = k− q 13 19 Et : Oqt (sl2 ) → C: +, 12 18 c δt ( ) = q c ⊗ Et (T0 ) = 11 17 ⊗ T1 + q f0 q ⊗ T0 , c 10 16 δt (T1 ) = k− e0 ⊗ + q ⊗ T1 , δt (P˜ ) = k− [e1 , e0 ]q + q c [f1 , f0 ]q ⊗ + q c ⊗ P˜ h0 and counit ρt = k− (q + q −1 )2 20 22 = T1 , T0 , (2.8) Let us define: 19 21 ˜1 = ρt T0 , T1 , T1 , T1 , P q 36 37 38 39 40 41 42 43 44 In [17], the special case = is considered In [5], the central element is not explicitly introduced and the first relation is replaced by [K0 , K1 ] = Note that for the level one q-vertex operators representations constructed in [5], one has = q 45 46 47 JID:NUPHB AID:13972 /FLA Z1 , Z1 , Z1 , Z˜ q q −1 = ρd Z1 ( K1 K1 − K0 K0 )Z1 , = Z˜ , Z1 , = K0 , = K1 , 12 13 14 15 16 17 18 =0 with the identification: (q − q −3 )(q − q −2 )3 ρd = q − q −1 (2.14) We endow this algebra with the coaction map δd (K0 ) = q h1 ⊗ K0 , δd : Oqd (sl2 ) → Uq (sl2 ) ⊗ Oqd (sl2 ) δd (K1 ) = q h0 ⊗ K1 , −2 −1 h1 c 13 14 15 16 Ed : Oqd (sl2 ) → C: Ed (K0 ) = 17 Ed (K1 ) = +, Ed (Z1 ) = Ed (Z˜ ) = 0, −, Ed ( ) = − Ed (1) = +, (2.16) 20 22 23 24 25 26 27 28 29 This induces an homomorphism ψd from the augmented q-Onsager algebra to a subalgebra of Uq (sl2 ): ψd (K0 ) = +q h1 ψd (K1 ) = , ψd (Z1 ) = (q − q −2 ) ψd (Z˜ ) = (q − q −2 ) ψd ( ) = − +q c +q −q −q h0 (2.17) , −1 e0 q h1 + − f1 q −1 e1 q h0 + + f0 q h1 +h0 h1 +h0 32 33 34 35 36 37 38 39 40 41 19 20 21 22 23 25 , 26 27 28 29 2.4 The Uq (gl2 ) invariant q-Onsager algebra 30 The Uq (gl2 ) invariant q-Onsager algebra Oqi (sl2 ) is generated by six elements e, f, q h , X, Y, Y˜ , a central element and unit The defining relations are: q h − q −h [e, f] = , q − q −1 [q h , X] = 0, [q h , Y˜ ]q = 0, [e, q ]q = [q , f]q = 0, h (2.18) [e, Y]q = 0, 44 45 + (q − q −2 + (q − q −2 46 Y, = Y˜ , = X, 32 36 37 [f, Y˜ ]q −1 = 0, [q , Y]q −1 = 0, h + 1)Yf − (q + q )(q + q −1 = e, ) q 34 35 [f, X] = Y˜ , [X, e] = Y, 38 39 40 41 ˜ + q (q − q −2 )(q + q −1 )2 (q − q −2 − 1)Ye 43 31 33 h [e, Y˜ ] = (q + q −1 )Xq h , [Y, f] = (q + q −1 )Xq h , ˜ − q −1 Yeff + (q + q −1 )2 Xefq h )q h [X, [Y, Y˜ ]] = (q − q −2 )3 (q Yfee 42 47 18 24 , 30 31 11 12 δd ( ) = q c ⊗ and counit 10 (2.15) 19 21 defined by: δd (Z1 ) = q ⊗ Z1 + (q − q ) q e0 q ⊗ K0 + f1 q ⊗ K1 , δd (Z˜ ) = q c ⊗ Z˜ + (q − q −2 ) f0 q c ⊗ K0 + q −1 e1 q h0 ⊗ K1 , c 10 11 Z˜ , Z˜ , Z˜ , Z1 q q −1 = ρd Z˜ (K0 K0 − K1 K1 )Z˜ , [m1+; v1.242; Prn:18/01/2017; 10:20] P.6 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• −1 −2 ˜ = f, )Xq h q 42 43 2h 44 Ye − q Yf + q (q + q = qh , = −1 )Xq h , 45 46 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.7 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• We endow this algebra with the coaction map δi : Oqi (sl2 ) → Uq (sl2 ) ⊗ Oqi (sl2 ) defined by: δi (f) = f0 ⊗ q −h + ⊗ f, δi (e) = e0 ⊗ + q h0 ⊗ e, δi (X) = ([e1 , e0 ]q − [f1 , f0 ]q −1 q ) ⊗ + (q − q − q c+1 (q + q −1 )f1 q h0 ) ⊗ + q h0 +c ⊗ Y q −2 (2.20) ) q [e1 , e0 ]q ⊗ e + (q − 1)q h0 2h0 e1 ⊗ e + q c+h0 10 δi (Y˜ ) = q c 12 f1 , f , f 14 + q 1+h0 e1 ⊗ 15 16 17 18 19 20 21 22 23 24 25 26 (2.21) h and counit 37 38 39 40 16 18 19 Ei (q h ) = Ei ( ) = Ei (1) = (2.22) 20 This induces an homomorphism ψi = (id ⊗ Ei ) ◦ δi from the Uq (gl2 ) invariant q-Onsager algebra to a certain subalgebra of Uq (sl2 ): 21 ψi (e) = e0 , ψi (f) = f0 , ψi (q ) = q , h h0 (2.23) q −1 43 44 45 46 47 23 24 26 − q c+1 (q + q −1 )f1 q h0 , q f1 , f , f 22 25 ψi (X) = [e1 , e0 ]q − [f1 , f0 ]q −1 q c , 27 28 − q −1 (q + q −1 )e1 q h0 , 29 ψi ( ) = q c 30 31 32 2.5 Non-Abelian symmetry algebras of the Hamiltonians 33 In the thermodynamic limit of the XXZ spin chain Hamiltonian, it is well-known that the Hamiltonian commutes with the generators of the quantum affine algebra Uq (sl2 ) acting on an infinite tensor product representation [15,18,13] In this subsection, we basically use similar arguments in order to characterize the hidden non-Abelian symmetries of the open XXZ spin chain for four different types of boundary conditions As an example, let us consider the Hamiltonian (1.1) On the semi-infinite tensor product vector space (1.2), the generators (2.7) of the q-Onsager algebra act as7 (see also [5]): 41 42 11 15 33 36 10 14 q + (q − q −1 )q c f1 ⊗ f2 q −h , q − q −1 31 35 13 − q −h Ei (e) = Ei (f ) = Ei (X) = Ei (Y) = Ei (Y˜ ) = 0, ψi (Y˜ ) = q c 34 17 28 32 12 Ei : Oqi (sl2 ) → C: ψi (Y) = e1 , e0 , e0 30 − q −1 (q + q −1 )e1 q h0 ⊗ q −h + q c ⊗ Y˜ q −1 δi ( ) = q c ⊗ 27 29 q 2h − f1 ⊗ , q −1 − (q − q −2 ) q c [f0 , f1 ]q ⊗ f 13 11 (2.19) + q c ⊗ X + (q − q −2 )(q h0 +1 e1 ⊗ e − q c−1 f1 ⊗ fq h ), δi (Y) = ( e1 , e0 , e0 δi (q h ) = q h0 ⊗ q h , c 7 34 35 36 37 38 39 40 41 42 The (evaluation in the principal gradation) endomorphism π : U (sl ) → End(V ) (V ≡ C2 ) is used: q ζ ζ πζ [e1 ] = ζ σ+ , πζ [e0 ] = ζ σ− , πζ [f1 ] = ζ −1 σ− , πζ [f0 ] = ζ −1 σ+ , πζ [q h1 ] = q σ3 , πζ [q h0 ] = q −σ3 43 44 45 46 (2.24) 47 JID:NUPHB AID:13972 /FLA (∞) W0 ∞ j =1 + (∞) W1 13 14 + − ··· ⊗ q (∞) hi , W0 16 ⊗q −σ3 , 10 = · · · ⊗ q σ3 ⊗ q σ3 ⊗ i+1,i ⊗ II ⊗ · · · ⊗ II (2.26) i+1,i = (2k+ σ+ − 2k− σ− ) ⊗ ( − q )σ3 + ( q σ3 σ3 26 27 30 31 32 33 36 37 38 39 40 41 42 43 44 45 (∞) hi , W0 i=1 23 = − (q − q −1 ) · · · ⊗ q σ3 ⊗ q σ3 ⊗ (k+ σ+ − k− σ− ) 24 25 On the other hand, the contribution from the boundary term in (1.1) is non-vanishing It gives: (∞) hB , W0 26 27 28 = (q − q −1 ) · · · ⊗ q σ3 ⊗ q σ3 ⊗ (k+ σ+ − k− σ− ) 29 30 (∞) W1 , Combining the above expressions together and repeating the analysis for we conclude that the q-Onsager algebra is a symmetry algebra of the Hamiltonian with generic boundary conditions: 31 32 33 34 (∞) H XXZ , W0 (∞) H XXZ , W1 (2.27) 35 Although the calculations become quickly more involved, the same technique is then extended to the Hamiltonian (1.1) with triangular ( ± = 0, k+ = 0, k− = 0), diagonal ( ± = 0, k± = 0) and special boundary conditions ( + = 1, − = 0, k± = 0), respectively With respect to the boundary conditions chosen, it is found that the corresponding Hamiltonian is commuting with all generators of the triangular q-Onsager, augmented q-Onsager and Uq (gl2 ) invariant q-Onsager algebras acting on (1.2), respectively In the next Section, we present an alternative and much simpler derivation of the symmetry operators generating the four different types of q-Onsager algebras 37 =0 and 46 47 16 21 34 35 15 22 ∞ 28 29 14 20 Summing up the local densities, after some simplifications one ends up with: 24 25 13 19 − 1)σ3 ⊗ (2k+ σ+ − 2k− σ− ) 22 23 12 18 19 21 11 17 where 18 20 , sitej −σ3 where the coaction map (2.5) is iterated repeatedly and (2.7) is used The action of the operators (2.25) on (1.1) is easy to compute On one hand, introduce the local density hi = σ1i+1 σ1i + σ2i+1 σ2i + σ3i+1 σ3i By straightforward calculations, one first observes: 15 17 ⊗q σ3 j =1 10 12 ··· ⊗ q σ3 · · · ⊗ q −σ3 ⊗ q −σ3 ⊗ (k+ σ+ + k− σ− ) ⊗II ⊗ · · · ⊗ II = (2.25) sitej + ∞ 11 · · · ⊗ q σ3 ⊗ q σ3 ⊗ (k+ σ+ + k− σ− ) ⊗II ⊗ · · · ⊗ II = [m1+; v1.242; Prn:18/01/2017; 10:20] P.8 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• = 36 38 39 40 41 42 43 44 45 46 To obtain (2.25), one fixes the evaluation parameter ζ = 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.9 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• An alternative derivation of the symmetry operators 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 In the q-Onsager approach of the finite open XXZ spin chain [7] with left and right generic non-diagonal boundary conditions, it is known that all local or non-local conserved quantities are generated from elements in a certain quotient8 of the infinite dimensional algebra Aq [7] (see also [6]) In the literature, the generators of Aq are usually denoted {W−k , Wk+1 , Gk+1 , G˜k+1 |k ∈ Z+ }, and W0 , W1 are called the fundamental generators Importantly, W0 , W1 satisfy the defining relations of the q-Onsager algebra (2.3) [1,2] By analogy with Onsager’s [24], Dolan–Grady’s [14] and Davies’ [12] works on the Onsager algebra, generalizing the results of [6, Subsection 2.4.2] it was shown in [4, Example 2] through a brute force calculation that the first few ‘descendants’ generators W−1 , W2 , G1 , G˜1 , G2 , G˜2 can be written as polynomials in the two fundamental generators W0 , W1 only Including a central element γ , for instance one finds: G1 = W , W 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 − (q − q ) + −γ (3.1) , + − γ W0 , + (q + q + 2q)W1 W0 W1 W0 + ρ(q − q − (q + q −2 )(q − q −1 )2 − ( + + + − γ (qW1 W0 −1 2 q −q )γ − q + q −2 − 2 + −γ 46 47 10 11 12 13 14 15 −1 )(W02 22 23 + W12 ) 24 25 (q − q −1 )3 (γ − 1) − ρq ρ(q + q −2 ) (q + q −1 )3 (q + q −2 ) The expressions for the elements G˜1 , W2 , G˜2 are obtained from G1 , W−1 , G2 by exchanging W0 ↔ W1 in the above formulae Actually, under certain assumptions it is possible to show that all descendants generators of the algebra Aq admit a unique explicit polynomial formulae in terms of the two fundamental generators W0 , W1 [4] Roughly speaking, they take the form: Gk+1 = Fk+1 (W0 , W1 ), Wk+1 = F−k (W1 , W0 ), G˜k+1 = Fk+1 (W1 , W0 ) 18 21 − q −1 W0 W1 ) W−k = F−k (W0 , W1 ), 17 20 (3.2) for all k ∈ N, where {Fk (X, Y ), k ∈ Z} is a family of two-variable polynomials in X, Y Note that the polynomial formulae that are obtained in [4] can be independently derived using the connection between the algebra Aq and the reflection equation algebra [10] Details will be reported elsewhere We now show how each realization of the four different types of q-Onsager algebras in terms of Uq (sl2 ) Chevalley generators (2.7), (2.12), (2.17), (2.23) can be recovered in a straightfor- 44 45 19 + (q −3 − q )(W0 W12 W0 + W1 W02 W1 ) − (q −5 + q −3 + 2q −1 )W0 W1 W0 W1 16 (q − q −1 )2 (q + q −2 )W0 W1 W0 − W02 W1 − W1 W02 + W1 − W−1 = ρ ρ G2 = (q −3 + q −1 )W02 W12 − (q + q)W12 W02 ρ(q + q −2 ) 26 27 q −1 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 Because the generators of A act on a finite dimensional vector space V (N ) of dimension 2N , they satisfy additional q relations that are q-deformed analogs of Davies’ relations Compare the first reference of [7, eqs (17), (18)] to the first reference of [12, eq (2.6a), (2.6b)] for details 45 46 47 JID:NUPHB AID:13972 /FLA 10 [m1+; v1.242; Prn:18/01/2017; 10:20] P.10 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• 10 ward manner starting from the polynomial formulae (3.1) for the first few descendant generators Define: W0 = ψ(W0 ), W1 = ψ(W1 ), 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 (3.3) with (2.7) Observe that the polynomials (3.2) can be systematically expanded in terms of the Chevalley generators and parameters k± , ± The explicit expressions for the first few examples (3.1), sufficient for our purpose, are reported in Appendix A Then, we specialize case by case the boundary parameters into these expressions According to the specialization chosen, the corresponding ‘reduced’ descendant generators produce the symmetry operators of each of the q-Onsager algebras exhibited in the previous Section Explicitly, we obtain: • Triangular boundary conditions k− = 0, k+ = 0, W0 |k+ →0 = ψt (T0 ), • Diagonal generic boundary conditions k± = 0, W0 |k± →0 = ψd (K0 ), W1 |k± →0 = ψd (K1 ), G1 k− G˜1 k± →0, + →1, − →0 + →1, − →0 k+ k± →0, + →1, − →0 W2 |k± →0, + →1, − →0 32 W−1 |k± →0, k± →0, 36 G2 k− G˜2 37 k+ k± →0, 33 34 35 = 0, 38 39 40 41 42 43 46 47 10 − 12 13 k+ →0 14 = ψt (P˜ ) 15 16 = k± →0 = 0, 17 = ψd (Z1 ), + G˜1 k+ k± →0 = ψd (Z˜ ) W0 |k± →0, (∞) H XXZ , Fk (W0 18 19 20 = 21 + →1, − →0 = ψi q −h , 22 23 24 25 26 = ψi (q − q −2 ) f , = ψi = ψi q + q −1 = ψi q 1+h + q −1−h + (q − q −1 )2 fe q − q −1 Xq −h (q + q −1 )2 = ψi −q −2 27 28 , 29 30 31 , − q −1 ) 32 − q −1 )2 (q (q Yq −h + q −1 (q + q −1 ) (q + q −1 ) 33 eXq h , 34 35 (q − q −1 )2 (q − q −1 ) ˜+ Y fX −q (q + q −1 ) (q + q −1 ) 36 37 Finally, as an alternative check of the analysis of the previous Section, we now show that according to the choice of boundary conditions, the corresponding above set of operators acting on the vector space (1.2) commute with the associated Hamiltonian For generic boundary conditions k± , ± , from (2.27) recall that the two operators W0 , W1 acting on (1.2) commute with the Hamiltonian As a corollary of (2.27) and (3.2), for generic boundary conditions it implies that any descendant generator is commuting with (1.1): 44 45 + →1, − →0 = = ψi (q − q −2 ) q −1 eq −h , + →1, − →0 + →1, − →0 ± G1 k− • Special diagonal boundary conditions k± = 0, W1 |k± →0, ± G˜1 k+ W1 |k+ →0 = ψt (T1 ), 30 31 11 28 29 γ = ψ( ), 11 12 (∞) , W1 38 39 40 41 42 43 44 ) = 0, (∞) (∞) H XXZ , Fk (W1 , W0 ) =0 (3.4) for all k ∈ Z 45 46 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.11 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• 11 Specializing the boundary parameters accordingly, it implies that the four different types of q-Onsager algebras are symmetry algebras of the Hamiltonian Let us remark that the defining relations of the current algebra associated with the q-Onsager algebra are given in [10] For the triangular, augmented and Uq (gl2 ) invariant q-Onsager algebras, the defining relations of the corresponding current algebras can be derived, respectively, by taking appropriate limits of the relations in Aq , given by [10, Definition 3.1] and [10, Proposition 3.1] 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 Concluding remarks 11 12 10 10 11 It is instructive to consider the limit q → of the four different types of q-Onsager algebras described in Section In particular, in this limit the q-Onsager, the augmented q-Onsager and the Uq (gl2 ) invariant q-Onsager algebras specialize to three different invariant fixed-point subalgebras of U (sl2 ) For the q-Onsager algebra with q → 1, the corresponding automorphism is the Chevalley involution of U (sl2 ), given by θ (ei ) = fi , θ (hi ) = −hi For the augmented q-Onsager algebra with q → 1, one considers the composition of the Chevalley involution with the outer automorphism of U (sl2 ) It reads: θd (e0 ) = f1 , θd (e1 ) = f0 , θd (f0 ) = e1 , θd (f1 ) = e0 and θd (h0 ) = −h1 For the Uq (gl2 ) invariant q-Onsager algebra at q → 1, one considers the composition of the Chevalley involution with the Lusztig’s automorphism of U (sl2) [23] It reads: θi (e0 ) = e0 , θi (f0 ) = f0 , θi (h0 ) = h0 , θd (f1 ) = [[e1 , e0 ], e0 ]/2, θd (e1 ) = [[f1 , f0 ], f0 ]/2 and θi (h1 ) = −h1 − 2h0 From that point of view, three of the algebras introduced in Section can be understood as a q-deformation of these invariant fixed-point subalgebras of U (sl2 ) [21] For the q-Onsager and augmented q-Onsager algebras, see [11] for details Note that it is a simple exercise to apply the technique of [11] to the Uq (gl2 ) invariant q-Onsager algebra Besides these three algebras, let us point out that the triangular q-Onsager algebra gives an example of coideal subalgebra that does not correspond to an invariant fixed-point subalgebra of Uq (sl2 ) even at q → From the point of view of physics, whereas the existence of infinite Abelian symmetries in a lattice system – that are associated with infinitely many mutually commuting conservation laws – reduces the problem of degeneracies of the Hamiltonian’s spectrum from infinite to finite, the existence of non-Abelian symmetries imply that common eigenspaces can be understood as irreducible modules of the symmetry algebra For the thermodynamic limit of the open XXZ spin chain with Hamiltonian (1.1) and certain boundary conditions it follows that the generators of the corresponding q-Onsager algebra discussed in Section change eigenvectors of the Hamiltonian without changing the eigenvalues Then, the solution of the model (spectrum, eigenstates, multiple integral representations of correlation functions and form factors) can be derived using infinite dimensional (q-vertex operators) representations of the symmetry algebra Clearly, the q-vertex operators for each symmetry algebra follow from the representation theory of Uq (sl2 ) Note that the solution for special, diagonal and triangular boundary conditions is given in [19,5, 8,9] For the Hamiltonian with generic boundary conditions, although q-vertex operators for the q-Onsager algebra are known [5], the solution remains an open problem Finally, an interesting problem would be to extend the analysis here presented to integrable models with higher rank symmetries of q-Onsager’s type (see [3]) The higher rank infinite dimensional algebra extending Aq , yet unknown, could be a starting point for the identification of the symmetry algebras 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.12 (1-13) P Baseilhac, S Belliard / Nuclear Physics B ••• (••••) •••–••• 12 Acknowledgements 10 11 P.B thanks T Kojima for discussions S.B thanks V Regelskis for discussions and also LMPT for hospitality, where part of this work has been done S.B is supported by a public grant as part of the “investissement d’avenir” project, reference ANR-11-LABX-0056-LMH, LabEx LMH P.B is supported by CNRS 13 14 W−1 = 15 17 18 19 20 23 24 25 G2 = k− 32 38 39 + k− f q e e0 e1 + (qf0 f1 − q −1 (A.1) 12 13 q 1+h1 + q −1−h1 + (q − q −1 )2 f1 e1 q c −2 (A.2) 16 17 18 19 20 )e1 e0 e1 − e0 (e1 ) 21 − q c−1 (f1 )2 f0 − (q + q −2 )f1 f0 f1 + f0 (f1 )2 24 23 , 25 q − q −1 q − q −2 c −1 c h1 )q e q + q f1 (qe0 e1 − q −1 e1 e0 ) (q + q − q + q −1 q + q −1 26 (A.3) + − q −1 q2 44 45 46 47 27 28 29 − q −2 q q c−1 (qf0 f1 − q −1 f1 f0 )e0 q h1 (q + q −1 )q 2c+1 f1 + q + q −1 q + q −1 + O(k+ ) + O(k− ) Note that for simplicity, the terms of order k+ and k− in kG−2 are not explicitly written, as they G˜1 k+ , ˜ not contribute in the limit k± = Also, the descendant generators W2 and kG+2 can be derived in terms of the generators of the Chevalley-type presentation using the map x0 → x1 , x1 → x0 , with x ∈ {h, e, f }, k± → k∓ and ± → ∓ on the expressions (A.1), (A.2) and (A.3) respectively 30 31 32 33 34 35 36 37 38 39 40 References 42 43 14 15 40 41 10 11 c f1 f0 )q , − q −1 (e0 )2 e1 − (q + q −2 )e0 e1 e0 + e1 (e0 )2 33 37 e0 q h1 22 + 31 36 −1 +q h1 k− + q −1 (q + q −1 )e0 + q −1 (q − q −1 )f1 q h1 (qe1 e0 − q −1 e0 e1 ) (q + q −1 )2 30 35 f1 + −1 − q 2c (f1 )2 f0 − (q + q −2 )f1 f0 f1 + f0 (f1 )2 29 34 c − (e1 ) e0 + (q + q 22 28 −q 21 27 q + q −1 (q − q −1 ) + + qe1 e0 − q −1 e0 e1 + (qf0 f1 − q −1 f1 f0 )q c q h1 (q + q −1 )2 k+ + (q − q −1 )(qf0 f1 − q −1 f1 f0 )q c e1 + q −2 (q + q −1 )f0 q h0 (q + q −1 )2 16 26 − + k+ qe1 e0 − q 12 Appendix A Descendant generators in the Chevalley-type presentation of Uq (sl2 ) G1 = (q − q −2 ) k− 41 42 [1] P Baseilhac, Deformed Dolan–Grady relations in quantum integrable models, Nucl Phys B 709 (2005) 491–521, arXiv:hep-th/0404149 [2] P Baseilhac, An integrable structure related with tridiagonal algebras, Nucl Phys B 705 (2005) 605–619, arXiv: math-ph/0408025 [3] P Baseilhac, S Belliard, Generalized q-Onsager algebras and boundary affine Toda field theories, Lett Math Phys 93 (2010) 213–228, arXiv:0906.1215 43 44 45 46 47 JID:NUPHB AID:13972 /FLA [m1+; v1.242; Prn:18/01/2017; 10:20] P.13 (1-13) P Baseilhac, S Belliard 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