Generalized q-Onsager algebras and boundary affine Toda field theories

GENERALIZED q − ONSAGER ALGEBRAS AND BOUND AR Y AFFINE TOD A FIELD THEORIES P . BASEILHAC AND S. BELLIARD Abstract. Generalizations of the q − Onsager algeb ra are int ro duced and studied. In one of the si m plest case and q = 1, the algebra reduces to the one prop osed by Uglov-Iv a nov . In the general case and q 6 = 1, an explicit algebra homomorphism associated with coideal subalgebras of quantum affine Lie algebras (simply and non-simply laced) i s exhibited. Boundary (soliton non-preserving) i ntegrable quantu m T oda field theories are then considered in light of these resul ts. F or the first time, all defining relations f or the underlying non-Ab elian symmetry algebra are explicitely obtained. As a consequence , based on purely algebraic argumen ts all integrable (fixed or dynamical) b oundary conditions are classified. MSC: 81R50; 81R10; 81U15; 81T40. Keywords : q − O nsager algebra; Quantum group symmetry; Boundary affine T o da field theory 1. Introduction In recent years, a new algebra ic structure calle d the q − Onsag er alg ebra (or equiv alen tly the tridiagona l algebra) ha s emerged in differen t pro blems of mathematical physics. On one side, it app ears in the mathematical litter a ture of P − and Q − polyno mial asso ciation schemes and their relations hip with the Askey scheme of or thogonal po lynomials [Zhed, ITT er, GruHa, T er 1, T er2], related Jacobi matrices a nd, more generally , certain fa milies of sy mmetric functions o f one v ariable a nd related blo ck tridiago nal matrices (see e.g. [T er3, Bas3]). On the other side, this alg ebra app ears in several q ua n tum integrable s y stems. Playing a crucial role at q = 1 in the exact s olution of the planar Ising [Ons] and super in tegrable Potts mo del [v oGR], it also finds applications in solv ing the XXZ op en spin chain with non-diag onal b oundary para meters and ge neric deformation parameter q . Indeed, the transfer matrix o f this mo del has b een shown to admit a n expa nsion in ter ms of the elements of the q − Onsager algebr a [B asK0, BasK 1 ] acting on so me finite dimensio nal representation. As a consequence, the solution of the mo del i.e. the co mplete s pectrum and eigenstates can b e derived using solely its representation theory , bypassing the Bethe ansatz appr oach which do es not apply in the g eneric regime of parameters [BasK2]. Appart fro m lattice mo dels, in quan tum field theor y the q − Onsag e r a lg ebra is kno wn to be the hidden non-Ab elian symmetry of the b oundary sine-Gordon mo del [Bas 1 , Bas2]. By definition, the q − Onsager algebra is an a s so c iativ e alg ebra with unit y generated by t w o elements (called the standard gener ators), say A 0 , A 1 . Int ro ducing the q − commutator 1  X , Y  q = X Y − q Y X , the fundamental (so metimes called q − Do lan-Grady) relatio ns take the form [ A 0 , [ A 0 , [ A 0 , A 1 ] q 2 ] q − 2 ] = ρ 0 [ A 0 , A 1 ] , [ A 1 , [ A 1 , [ A 1 , A 0 ] q 2 ] q − 2 ] = ρ 1 [ A 1 , A 0 ] (1.1) where q is a defor mation pa rameter (assumed to be not a ro ot of unity) and ρ 0 , ρ 1 are fixed scalars . Note that for ρ 0 = ρ 1 = 0 this algebra reduces to the q − Serre relations o f U q ( c sl 2 ), and for q = 1, ρ 0 = ρ 1 = 16 it lea ds to the Onsager a lg ebra [Ons, Per] defined b y the Dolan-Grady r elations [DoG]. 1 F or further conv enienc e, definitions for the parameter q and the q − commut ator chosen here differ compared to [Bas3, BasK0, BasK1, BasK2]. 1 2 P . BASEILHAC AND S. BELLIARD Similarly to the well-established relationship betw een the Onsag er a lgebra and the affine Lie algebra c sl 2 [Dav, DaRo], the q − O nsager algebra (1.1) is actually closely related with the U q ( c sl 2 ) a lg ebra, a fact that ma y be also expected from the structure of the l.h.s. of (1.1) c o mpared with the q − Serre relations of U q ( c sl 2 ). Indeed, ex a mples o f algebra homomorphisms for the standard generato rs A 0 , A 1 hav e b een prop osed for ρ 0 6 = 0 , ρ 1 6 = 0, and related finite dimensional repr esent ations studied in details. W e r efer the reader to [IT er1, Bas2, AlCu, IT er2] for details. In particular , the following realization immediately follows from [Bas 2 ]: A 0 = c 0 e 0 q h 0 / 2 + c 0 f 0 q h 0 / 2 + ǫ 0 q h 0 , A 1 = c 1 e 1 q h 1 / 2 + c 1 f 1 q h 1 / 2 + ǫ 1 q h 1 , (1.2) where 2 { h i , e i , f i } denote the generators of U q ( c sl 2 ) and one iden tifies ρ i = c i c i ( q + q − 1 ) 2 for i = 0 , 1. Thanks to the Hopf algebra structure of U q ( c sl 2 ), finite dimensiona l representations hav e b een studied in details (see for instance [Bas3, IT er2]). In addition, a new type o f current algebra has b een recently derived [Ba sS1] which r igorously establishes the iso mo rphism betw een the reflection eq uation algebra asso ciated with U q ( c sl 2 ) R − matrices and the q − Onsag er algebra (1.1). In the cont ext of quantum integrable sy s tems, the elements A 0 , A 1 take the form of non lo cal op erator s on the lattice o r contin uum. According to the mo del and ob jectiv e co nsidered, they are used either to even tually der iv e seco nd order difference equations fix ing the spe ctrum of the mo del [B asK2], or the complete set of sca ttering amplitudes of the fundamen tal particles [MN98, DeM, BasK3]. In v iew o f all these results, finding an a nalogue o f the deformed relatio ns (1.1) that may be related to high er r ank affine Lie algebr a s in a similar manner , as well as considering p otential implica tio ns for quantum in tegra ble s ystems with extended sy mmetries seems to b e a rather in tere s ting problem. In the undeformed case q = 1, a step towards this direction has b een made by Uglov and Iv anov who intro duced the so-called sl n − Onsag er’s algebra for n ≥ 2 . How ever, to our knowledge since these res ults no further progre s s in this directio n were ever published. In the present letter, we remedy this situatio n. Namely , to ea c h affine Lie algebr a (of classical or exceptional t y p e) b g we asso ciate a q − Onsage r a lgebra denoted O q ( b g ). Then, by analogy with the c sl 2 case, we prop ose an algebr a homomorphism from O q ( b g ) to the coideal subalg ebra 3 of U q ( b g ) gener alizing (1.2). Applications to b oundary quantum a ffine T oda field theor ies in tro duced in [F rK, BCDRS] - with soliton non-preserving b oundary conditions - are then co nsidered. Despite of the fa ct that defining relations of the underlying hidden sy mmetry in these models were not kno wn up to now (except for the sine-Gordon mo del [Bas1, Bas2]), the explicit kno wledge of non-lo cal conserved ch arg es hav e provided a powerful to ol to constr uc t b oundar y reflection matrices at least for b g ≡ a (1) n , d (1) n cases [MN98, DeM, DeG]. Here and for the fir st time, we show that each b oundary a ffine T o da field theory of the family defined in [F rK, BCDRS] a sso ciated with b g enjoys a hidden non-Ab elian symmetry of type O q ( b g ). As a conse q uence, all known sca la r int egra ble b oundary conditions [BCDRS] simply follow from the a lg ebraic structure, with no reference to its representation theory 4 . More generally , all p ossible integrable dynamica l b oundary conditions (additional degrees of freedom a re lo ca ted at the b oundar y) admissible in these mo dels are also classified accor ding to this new fra mework, gener alizing the results of the b oundary sine-Gordon mo del with dynamical b oundary conditions [BasDel, B asK3]. 2 Defining relations of U q ( c sl 2 ) are gi ven in the next section. 3 F or definitions, see e. g. [MRS, Le] 4 Con trary to previous works, which are represent ation’s dep enden t. GENERALIZED q − ONSA GER ALGEBRAS AND BOUNDAR Y TODA FIELD THEORIES 3 2. Generaliza tions of the q − Onsa ger algebra As mentionned in the intro duction, generalized q − Onsa ger algebras can b e introduce d by ana lo gy with (1.1 ). Having in mind the s tructure of q − Serre r e lations for higher r a nk affine Lie alge bras and their p oten tial r elations with coideal s ubalgebras of quantum affine algebras , a g eneral formulation can be prop osed. Definition 2.1 . L et { a ij } b e the extende d Cartan matrix of the affine Lie algebr a b g with Dynkin diagr am r ep orte d in App endix A . Fix c oprime inte gers d i such t hat d i a ij is symmetric. The gener alize d q − Onsager algebr a O q ( b g ) is an asso ciative algebr a with unit 1 , elements A i and sc alars ρ k ij , γ kl ij ∈ C with i, j ∈ { 0 , 1 , ..., n } , k ∈ { 0 , 1 , ..., [ − a ij 2 ] − 1 } 5 and l ∈ { 0 , 1 , ..., − a ij − 1 − 2 k } ( k and l ar e p ositive inte ger). The defining r elations ar e : 1 − a ij X r =0 ( − 1) r  1 − a ij r  q i A 1 − a ij − r i A j A r i = [ − a ij 2 ] − 1 X k =0 ρ k ij − 2 k − a ij − 1 X l =0 ( − 1) l γ kl ij A − 2 k − a ij − 1 − l i A j A l i , (2.1) wher e the c onstants γ kl ij ar e such that: F or a ij = a j i = − 1 : γ 00 ij = γ 00 j i = 1 ; F or a ij = − 1 and a j i = − 2 : γ 00 ij = γ 00 j i = γ 01 j i = 1 ; F or a ij = − 1 and a j i = − 3 : γ 00 ij = 1 , γ 00 j i = γ 02 j i = γ 10 j i = 1 , γ 01 j i = ( q + q − 1 )( q 2 + q − 2 )( q 2 + 3 + q − 2 ) ( q 4 + 2 q 2 + 4 + 2 q − 2 + q − 4 ) ; F or a ij = − 1 and a j i = − 4 : γ 00 ij = 1 , γ 00 j i = γ 03 j i = γ 1 l j i = 1 , γ 01 j i = γ 02 j i = [3] q [5] q q 4 + q − 4 + 3 . Remark 1. F or b g ≡ a (1) n , q = 1 and ρ 0 ij = 1 , the r elations r e duc e to the ones of Uglov-Ivanov’ s sl n − Onsager’s algebr a [UgIv] . F or b g ≡ a (1) n and q 6 = 1 , t he r elations alr e ady app e ar e d in [Bas1] without detaile d explanations. F or simply lac e d c ases, note the close r elationship with the defining r elations of c oide al su b algebr as or the non-standar d deformation of finite dimensional Lie algebr as [Le, Ga vr, K lim] . F or q 6 = 1, a n e xplicit relationship with coidea l subalg ebras of U q ( b g ) can b e easily exhibited. T o this end, let us first recall some definitions that will b e use ful below. Define fo r q ∈ C ∗  a b  q = [ a ] q ! [ b ] q ! [ a − b ] q ! , [ a ] q ! = [ a ] q [ a − 1] q . . . [1] q , [ a ] q = q a − q − a q − q − 1 , [0] q = 1 . Definition 2.2. [Jim] L et { a ij } b e the ex tende d Cartan matrix of the affine Lie algebr a b g with Dynkin diagr am given in A pp endix A. Fix c oprime int e gers d i such t hat d i a ij is symmetric. U q ( b g ) is an asso ciative algebr a over C with unit 1 gener ate d by the elements { e i , f i , q ± h i 2 i } , i ∈ 0 . . . n subje ct to the r elations: 5 [ a ] means the nearest higher integer of a wi th [1/2]=1. 4 P . BASEILHAC AND S. BELLIARD q ± h i 2 i q ∓ h i 2 i = 1 , q h i 2 i q h j 2 j = q h j 2 j q h i 2 i , q h i 2 i e j q − h i 2 i = q a ij 2 i e j , q h i 2 i f j q − h i 2 i = q − a ij 2 i f j , [ e i , f j ] = δ ij q h i i − q − h i i q i − q − 1 i , e i e j = e j e i , f i f j = f j f i , for | i − j | > 1 , 1 − a ij X r =0 ( − 1) r  1 − a ij r  q i e 1 − a ij − r i e j e r i = 0 , 1 − a ij X r =0 ( − 1) r  1 − a ij r  q i f 1 − a ij − r i f j f r i = 0 . The Hopf algebr a structur e is ensu r e d by the existenc e of a c omultiplic ation ∆ : U q ( b g ) 7→ U q ( b g ) ⊗ U q ( b g ) , antip o de S : U q ( b g ) 7→ U q ( b g ) and a c ounit E : U q ( b g ) 7→ C with ∆( e i ) = e i ⊗ q − h i / 2 i + q h i / 2 i ⊗ e i , ∆( f i ) = f i ⊗ q − h i / 2 i + q h i / 2 i ⊗ f i , ∆( h i ) = h i ⊗ I I + I I ⊗ h i , (2.2) S ( e i ) = − e i q − h i i , S ( f i ) = − q h i i f i , S ( h i ) = − h i , S ( I I ) = 1 and E ( e i ) = E ( f i ) = E ( h i ) = 0 , E ( I I ) = 1 . Based on the realization (1.2) of the algebr a (1.1) for the simplest ca se c sl 2 ≡ a (1) 1 , and the results in [DeM, DeG] it looks rather natural to consider the following r ealizations for the genera lized q − O nsager algebras . Prop osition 2.1. L et { c i , c i } ∈ C and { w i } ∈ C ∗ . Ther e is an algebr a homomorph ism O q ( b g ) → U q ( b g ) such that A i = c i e i q h i 2 i + c i f i q h i 2 i + w i q h i i (2.3) iff the p ar ameters w i ar e subje ct to the fol lowing c onstr aints: F or b g = a (1) n ( n > 1 ) , d (1) n , e (1) 6 , e (1) 7 , e (1) 8 : n w i  w 2 j + c j c j q + q − 1 − 2  = 0 w j  w 2 i + c i c i q + q − 1 − 2  = 0 wher e i, j ar e simply linke d . F or b g = b (1) n , c (1) n , a (2) 2 n , a (2) 2 n − 1 , d (2) n +1 , , e (2) 6 , f (1) 4 : w j  w 2 i + c i c i q i + q − 1 i − 2  = 0 if i, j ar e doubly linke d with i the longest r o ot ; n w i  w 2 j + c j c j q j + q − 1 j − 2  = 0 w j  w 2 i + c i c i q i + q − 1 i − 2  = 0 if i , j ar e simply linke d . GENERALIZED q − ONSA GER ALGEBRAS AND BOUNDAR Y TODA FIELD THEORIES 5 F or b g = g (1) 2 , d (3) 4 : n w j  w 2 i + c i c i ( q i + q − 1 i − 2)  = 0 w i  w 2 j + c j c j ( q j + q − 1 j − 2)  w 2 j + c j c j ( q j + q − 1 j − 1) 2 ( q j + q − 1 j − 2)  = 0 if i , j ar e triply linke d with i the longest r o ot . n w i  w 2 j + c j c j q j + q − 1 j − 2  = 0 w j  w 2 i + c i c i q i + q − 1 i − 2  = 0 if i, j ar e simply linke d . F or b g = a (2) 2 : w j  w 2 i + c i c i ( q i + q − 1 i − 2)  = 0 with i the longest r o ot . Pr o of. Plug ging (2.3) into the rela tio ns of Definition 2.1, stra ight forward calc ulations leav e few unw a n ted terms that cancel provided the ab ov e constr ain ts on par ameters w i are satisfied. The structure constants ρ k ij - with resp ect to the indices i, j - are iden tified as follows: F or b g = a (1) n ( n > 1) , d (1) n , e (1) 6 , e (1) 7 , e (1) 8 : ρ 0 ij = c i c i and ρ 0 j i = c j c j . F or b g = b (1) n , c (1) n , a (2) 2 n , a (2) 2 n − 1 , d (2) n +1 , f (1) 4 : ρ 0 ij = c i c i and ρ 0 j i = c j c j ( q + q − 1 ) 2 if i, j are doubly link ed with i the longes t r oo t ; ρ 0 ij = c i c i and ρ 0 j i = c j c j if i, j are simply link ed . F or b g = g (1) 2 , d (3) 4 : ρ 0 ij = c i c i , ρ 0 j i = c j c j ( q 4 + 2 q 2 + 4 + 2 q − 2 + q − 4 ) and ρ 1 j i = − c 2 j c 2 j ( q 4 + 1 + q − 4 ) 2 if i, j are triply link ed with i the longes t r o ot ; ρ 0 ij = c i c i and ρ 0 j i = c j c j if i, j are s imply linked . F or b g = a (2) 2 : ρ 0 ij = c i c i , ρ 0 j i = c j c j ( q + q − 1 ) 2 ( q 4 + 3 + q − 4 ) and ρ 1 j i = − c 2 j c 2 j ( q + q − 1 ) 4 ( q 2 + q − 2 ) 4 with i the longes t ro ot .  Remark 2. Al l the structur e c onstants ar e inva riant by the change q → q − 1 , whi ch yield s to the obvio us r e alization A i = c i e i q − h i 2 i + c i f i q − h i 2 i + w i q − h i i . Quantum affine algebras U q ( b g ) a re known to b e Hopf algebra s, thanks to the existence of a copro duct, counit and an tip ode actions. F or generaliz ed q − Onsa g er algebra s O q ( b g ), a coaction map [Cha] can b e int ro duced: Prop osition 2.2. L et c i , c i ∈ C . The gener alize d q-Onsager algebr a O q ( b g ) is a left U q ( b g ) − c omo dule algebr a with c o action map δ : O q ( b g ) → U q ( b g ) ⊗ O q ( b g ) such that δ ( A i ) = ( c i e i q h i 2 i + c i f i q h i 2 i ) ⊗ I I + q h i i ⊗ A i . (2.4) Pr o of. The verification of the como dule algebra a xioms (see [Cha]) is immediate using (2.2). W e hav e also to show that δ ( A i ) statisfy (2.1 ). Assume A i satisfy (2 .1). Plugging (2.4) in (2.1), expanding and using the comm utation r elations of U q ( b g ) given in Definition 2.2, the claim follo ws.  6 P . BASEILHAC AND S. BELLIARD Remark 3. If one emb e ds O q ( b g ) into U q ( b g ) ac c or ding to Pr op. 2.1, the c o action δ is identifie d with the c omultiplic ation ∆ of U q ( b g ) . 3. Boundar y affine Toda field theories revisited Among in tegrable quantum field theories, the sine-Gor don mo del is known to enjoy a hidden non- Abelia n U q ( c sl 2 ) symmetry , a fact that r e lies on the existence of non-lo c al cons e r v ed charges generating the algebra [BeLe]. Restricted to the half-line and p erturb ed at the boundar y by cer ta in lo cal vertex o p- erators , the b oundary sine-Gordon mo del remains integrable [GZ]. Co rresp onding scattering amplitudes of the fundamental s olitons and breathers reflecting on the b oundary hav e b een derived e ither solving directly the so-ca lle d b oundary Y ang- Baxter eq uation (i.e. the reflection equation) [GZ, Gh], or using the existence o f non-lo cal conserved charges [MN98, DeM] that generate a remnant of the bulk U q ( c sl 2 ) quantum g roup symmetry . Ho wev er , the explicit defining relations of this remnant hidden non-Abe lia n symmetry algebra were only iden tified later on: for both integrable fixed or dynamical boundar y co ndi- tions, the symmetry algebra is the q − Onsager alg ebra 6 (1.1) [Bas1, Bas2]. In particular, in agreement with pr evious results fixed in tegrable boundar y conditions ar e not restr icted by the algebr aic structure whereas dynamical ones [Bas Del, BasK 3] a re ass ocia ted with b oundary op erator s acting o n finite or infinite dimensio na l representations of the q − Onsa ger algebra. Affine T o da field theorie s a r e natural gener a lizations of the sine-Gor don field theory , each being as- so ciated with an affine L ie algebra b g . Similarly to the sine-Gordon case, in the bulk they enjo y a U q ( b g ) quantum gr o up symmetry which determines completely all scattering amplitudes. Restricted on the half-line, tw o types of b oundary conditions may b e added that preserve integrability: either soliton non-preser ving - the most studied case 7 since [F rK, BCDRS] - or soliton preser ving [Sk, Del] b oundary conditions. In the follo wing, we fo cus o n the first family of integrable models which Euclidea n actio n 8 reads [F r K, BCDRS]: S = 1 4 π Z x< 0 d 2 z  ∂ φ ∂ φ + λ 2 π n X j =0 n j exp  − i ˆ β 1 | α j | 2 α j · φ   + λ b 2 π Z dt n X j =0 ǫ j exp  − i ˆ β 2 α j · φ (0 , t )  , (3.1) where φ ( x , t ) is an n − comp onent b oso nic field in tw o dimensions, { α j } and n j are the simple ro ots and Kac lab els, re spectively , o f b g , λ, λ b are rela ted with the mass sc ale, b β is the coupling constant and { ǫ j } ar e the bo undary par ameters or op erators. This action remains how ever integrable for certain scalar b oundary conditions ǫ j that have b een identified either at the classica l [B C DRS ] or quantum [PRZ] level based o n the existence o f lo c al higher spin co nserved charges 9 . F o r the simply laced ca ses b g = a (1) n , d (1) n , non-lo c al conserved charges tha t genera te a (coideal) subalgebra of U q ( b g ) hav e a lso b een de r iv ed [DeM, DeG]. They 6 F or the XXZ op en s pi n chain with generic i n tegrable b oundary conditions, the symmetry is asso ciated w i th an (Ab elian) q − Onsager’s subalgebra. But in the thermodynamic limit, it i s possible to show that the Hamiltonian b ecomes in v ariant under the action of the elements of the q − Onsager algebra [BBS] . 7 Among the kno wn non-per turbativ e results in b oundary affine T oda field theories, scattering amplitudes (for a (1) n , n > 1) ha ve b een considered in details in [Gan, DelGan] , and mass-parameter as well as v acuum expectation v alues of lo cal fields ha ve b een prop osed in [F aOn]. See also r elated results and non-perturbative c hec ks in [AhKR]. 8 According to a recent pap er [AvDoik] (see al so [Doik1]), a Hamiltonian has b een prop osed f or soliton preserving boundary conditions. 9 At classical lev el, an extended Lax pair formal i sm has also b een prop osed [BC D RS]. Given few assumptions, it giv es further support for the boundary conditions previously deriv ed. GENERALIZED q − ONSA GER ALGEBRAS AND BOUNDAR Y TODA FIELD THEORIES 7 read: ˆ Q j = Q j + Q j + b ǫ j q T j , j = 0 , 1 , , ..., n with b ǫ j = λ b 2 π c ˆ β 2 1 − ˆ β 2 ǫ j (3.2) where c = q λ ( ˆ β 2 / (2 − ˆ β 2 )) 2 ( q 2 − 1 ) / 2 iπ , the c harg e s Q j , Q j are realized in terms of v ertex op erators of holo morphic/antiholomorphic fields and T j has a form analog to the bulk top olog ical charge but restricted to the half-line. F or more details, explicit expressions ca n b e found in [DeM]. Generaliz a tions of the express io ns (3.2) to the non-simply laced ca ses a re straig h tforward. Although in [DeM] o nly scalar bo undary conditions were considered, calculations leading to (3.2) also hold assuming instead b oundary op erators ǫ j provided [ x, b ǫ j ] = 0 ∀ x ∈ { Q j , Q j , T j } . (3.3) Despite o f the results in [DeM, DeG] that pr ovide a p ow erful too l to derive efficiently all sca ttering amplitudes of the solitons r eflecting on the b oundary 10 , the e xplicit defining relatio ns of the U q ( b g )’s coideal s ubalgebra gener alizing (1.1) to higher ra nk b g ar e still unknown up to now. Beyond the interest of having a prop er mathematical frame, this problem is relev ant in the study o f (3 .1) as a dmissible fixed or dynamical bo undary conditions and b o undary states should b e classified accor ding to the representation theory o f the a lg ebra genera ted b y (3.2). T o identify the underlying no n-Abelian hidden symmetry of (3.1 ) and classify co rresp onding b oundary conditions, in b oth situations (fixed or dynamical bounda ry co nditions) it is then importa n t to reca ll that the existence o f no n-lo cal co nserved charges of the for m (3.2) with q → q j for non-s imply laced cases essentially r elies on the s tructure of the boundar y terms app earing in (3.1). Given s uc h bo undary terms and having der iv ed the non-lo cal conserved charges [MN98, DeM, DeG], the integrability condition for the model (3 .1) req uir es that all non-lo cal charges c lo se among a finite n umber of alg ebraic rela tions, yet to be iden tified. The a nsw er to this problem - finding these a lgebraic r elations - actually fo llo ws from the results of the previo us Section. Indeed, pre sen ted in terms o f U q ( b g ) generators and up to an ov e rall scalar factor the non- lo ca l co nserved charges (3.2) with q → q j for non-simply lace d ca ses turn out to b e exactly o f the form (2.3). Then, t wo situations can b e cons idered: 3.1. Fixed b oundary conditions . Assume b ǫ j (or equiv alently ǫ j ) are sca lars. According to Prop osition 2.1, g iv en any simply or non-simply laced affine Lie a lgebra the co r resp onding non-lo cal conserved charges (3.2) with q → q j close over the relations (2.1 ) provided the b oundary conditions are c o nstrained by the relations below (2.3) setting c j = c j ≡ 1, b ǫ j ≡ w j . Appart from the simple solutions w j ≡ 0 ∀ j , solving all cons tr ain ts case by case yields to the following families of admissible in tegr able bo unda ry conditions: F or b g = a (1) n ( n > 1 ) , d (1) n , e (1) 6 , e (1) 7 , e (1) 8 : b ǫ j = ± i q 1 / 2 − q − 1 / 2 ∀ j ; F or b g = b (1) n : b ǫ j = ± i q − q − 1 for j ∈ { 0 , 1 , ..., n − 1 } , b ǫ n arbitrar y ; F or b g = a (2) 2 n − 1 : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j ∈ { 0 , 1 , ..., n } or b ǫ j = 0 for j ∈ { 0 , 1 , ..., n − 1 } , b ǫ n arbitrar y ; F or b g = c (1) n : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j ∈ { 0 , ..., n } , or b ǫ j = 0 for j ∈ { 1 , ..., n − 1 } , b ǫ 0 , b ǫ n arbitrar y ; 10 Deriving all scatt ering amplitudes solely using the reflection equation - as done in [Gan] for th e ca se a (1) n - is more difficult. 8 P . BASEILHAC AND S. BELLIARD F or b g = d (2) n +1 : b ǫ j = ± i q j − q − 1 j for j ∈ { 1 , ..., n − 1 } , b ǫ 0 , b ǫ n arbitrar y ; F or b g = a (2) 2 n ( n > 2) : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j ∈ { 1 , ..., n } b ǫ 0 arbitrar y or b ǫ j = 0 for j ∈ { 0 , ..., n − 1 } , b ǫ n arbitrar y ; F or b g = a (2) 2 : n either b ǫ 0 = ± i q 2 − q − 2 , b ǫ 1 arbitrar y or b ǫ 1 = 0 , b ǫ 0 arbitrar y ; F or b g = a (2) 4 : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j = 1 , 2 , b ǫ 0 arbitrar y or b ǫ 2 = ± i q 2 − q − 2 , b ǫ 0 = 0 , b ǫ 1 arbitrar y or b ǫ 0 = b ǫ 1 = 0 , b ǫ 2 arbitrar y ; F or b g = g (1) 2 : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j or b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j = 0 , 1 , b ǫ 2 = ± i ( q + q − 1 − 1) q 1 / 2 − q − 1 / 2 ; F or b g = d (3) 4 : b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j ∈ { 0 , 1 , 2 } ; F or b g = f (1) 4 : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j ∈ { 0 , ..., 4 } or b ǫ j = ± i q j − q − 1 j for j ∈ { 0 , 1 , 2 } , b ǫ j = 0 for j ∈ { 3 , 4 } ; F or b g = e (2) 6 : n either b ǫ j = ± i q 1 / 2 j − q − 1 / 2 j for j ∈ { 0 , ..., 4 } or b ǫ j = 0 for j ∈ { 0 , 1 , 2 } , b ǫ j = ± i q j − q − 1 j for j ∈ { 3 , 4 } . Note that for the ca ses a (1) n , d (1) n , a bove results are in p erfect a greement with [DeM, DeG]. In the classica l limit q → 1 , except for the exce ptio nal cases g (1) 2 , d (3) 4 all ab ov e integrable b oundary co nditions agr ee with the r esults in [BCDRS]. 3.2. Dynamical b oundary conditions. By a nalogy with [BasK 3], instead o f scalar b oundary co ndi- tions an interesting pr oblem is to co ns ider additional o per a tors b ǫ j lo cated at the b oundar y , and interacting with the bulk fields accor ding to (3.1). As mentionned ab ov e, following the arguments o f [DeM] non-lo ca l conserved c harge s of the form (3.2) with q → q j can b e constructed. These charges can be written: ˆ Q j = ( Q j + Q j ) ⊗ I I + q T j j ⊗ b ǫ j , j = 0 , 1 , , ..., n (3.4) where the first and seco nd representation spa ces are asso ciated with the par ticle/b o undary s pa ce of s tates, resp ectively . In tegra bilit y requires that the charges (3.4) form an alg ebra, ensuring the existence of a factorized scattering theo ry a nd, in particular, of a so lition re flection matr ix commuting with (3.4). W e are then loo king for a set of algebraic relations satisfied b y the elements (3.4). T o this end, it is crucial to notice the following: acc ording to the defining relations of U q ( b g ) a nd the term ( Q j + Q j ) ⊗ I I in (3.4) such non linear c om binations o f (3 .4) for different j ca n o nly simplify if q − Se r re relations are used. More precisely , a stra igh tforward calculation shows that c om binations o f ( Q j + Q j ) ⊗ I I for different j only close on the alge br aic rela tions (2.1) - a consequence of Pr opo sition 2 .1 fo r w i ≡ 0 ∀ i . So, if the conser v ed charges form an algebra , due to the term ( Q j + Q j ) ⊗ I I its defining relatio ns are nec essarely given by (2.1). Let us then s ee under which conditio ns on the b o undary op erators b ǫ j the whole combination (3.4) could satisfy (2.1) setting A j ≡ ˆ Q j . P lug ging (3.4) in (2.1) and expa nding one finds that (3.4) satisfy GENERALIZED q − ONSA GER ALGEBRAS AND BOUNDAR Y TODA FIELD THEORIES 9 the algebr aic rela tions (2 .1) if and only if the ter ms b ǫ j also satisfy (2.1). N ote tha t these calculations are analogous to the ones of Prop osition 2.2, which explains the for m of the coaction map as de fined in (2.4). Under these conditions, it follows that ˆ Q j generate the q − O nsager alg e bra. F or g = sl 2 , a simple realization has b een prop osed in [B asK3]. F or higher ra nk cases, a n interesting pr oblem would b e to construct r ealizations in terms of q − defo rmed oscillato rs, gener alizing the results o f the massless case (see eq. (1.1 7) in [BHK]). In an y case, given the family o f bo undary integrable affine T o da field theories (3.1) all admissible dyna mical b oundary c o nditions b ǫ j are r equired to satisfy (2.1). 4. Discussion In this letter, a new family of qua n tum algebra s that w e call the gener alize d q − Onsager algebr as O q ( b g ) asso ciated with the affine Lie a lg ebras b g ha s b een introduced and studied. Some pro perties a nd the explicit relationship with coideal suba lgebras of U q ( b g ) ha ve been clarified, and simple consequences for quantum integrable systems - namely b oundary affine T o da field theories with soliton non- preserving bo undary conditions - hav e b een explore d. Clearly , extending all kno wn results of the c sl 2 − case (1.1) to the who le family O q ( b g ) is rather interesting fro m different po in ts of view. F rom the mathematical side, it is now well understoo d thanks to T erwilliger et al. ’s w orks (see some references b elow) that (1.1) pr ovides an alg ebraic framework to classify all othog o nal p olynomials of the Askey scheme. W either the genera lized q − Onsa ger a lgebras O q ( b g ) provide an a lgebraic framew ork for m ultiv ariable o rthogonal poly nomials - known or new - is an interesting problem. Another interesting problem is to construct new curr en t algebr as as s ocia ted with O q ( b g ) by analo gy with [BasS1] and e s tablish the iso morphism b e w tee n O q ( b g ) a nd the fa mily of reflec tio n equation a lgebra asso ciated with q − twisted Y angians [MRS] for R − matrices ass ocia ted with higher r ank quantum affine Lie algebras. F rom the ph ysics side - b eyond the explicit construc tio n of b oundary reflection matrices for bo undary affine T o da field theories (see [DeM , DeG] for the simply la ced cases) - g eneralized q − Onsager algebras should provide a p ow e rful to ol in order to study quant um integrable systems with extended symmetries. In this dir e ction, irr educible representations of O q ( b g ) will find applications to the s pectrum of b ound- ary states in b oundary in teg rable qua n tum field theories. Also, studying the explicit construction o f a hierarch y of comm uting q uan tities that generaliz es the Dolan-Grady hierarchy [DoG] or its q − deformed analogue [Bas1] will find a pplications in studying the sp ectrum and eigenstates in related spin chains. The r esults in [Doik2] might be a go o d star ting p oin t. Some of these problems will b e co nsidered elsewhere. Ac knowledgemen ts: S.B thanks LMPT for hospitality where part of this w ork has been done, INFN iniziativ a spe cifica FI11 for financia l supp ort and Ita lian Ministry of E ducation, University and Resear c h grant P RIN-2007JHLP EZ. App endix A. Dynkin diagrams for affine Lie alge bras The upper (resp. low e r ) indices denote the num b er (resp. v a lue of ( d i , n i )) a sso ciated with each no de. The explicit v alue s of the co efficient o f the extended Cartan matrix a ij for eac h affine Lie algebra [K ac] can b e found using a ii = 2 and the r ules: ✐ ✐ i j a ij = a j i = 0, ✐ ✐ i j a ij = a j i = − 2, ✐ ✐ i j a ij = a j i = − 1, ✐ ✐ ❅  i j a ij = − 1 a j i = − 2 , ✐ ✐ ❅  i j a ij = − 1 a j i = − 3, ✐ ✐  ❅ i j a ij = − 1 a j i = − 4. - Simply laced Dynkin diagrams (all d i = 1 and the lo wer indices corresp ond to ( n i )): 10 P . BASEILHAC AND S. BELLIARD a (1) 1 ✐ ✐ 0 1 (1) (1) a (1) n ( n ≥ 2 ) ✐ ❆ ❆ ❆ ❆ ✁ ✁ ✁ ✁ ✐ ✐ ✐ 0 i n − 1 n (1) (1) (1) (1) d (1) n ( n ≥ 4) ✐ ✐ ✟ ✟ ❍ ❍ ✐ ✐ ✐ ✐ ✐ ✟ ✟ ❍ ❍ 2 i n − 2 (2) (2) (2) n − 1 n (1) (1) 0 1 (1) (1) e (1) 6 ✐ ✐ ✐ ✐ ✐ ✐ ✐ 1 2 3 4 5 6 0 (1) (1) (2) (3) (2) (1) (2) e (1) 7 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ (1) (2) (3) (4) (3) (2) (1) (2) 0 1 2 3 4 5 6 7 e (1) 8 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ (2) (4) (6) (5) (4) (3) (2) (3) (1) 1 2 3 4 5 6 7 0 8 - Non-simply laced Dynkin diagrams (the lo wer n um b ers cor resp ond to ( d i , n i )): b (1) n ( n ≥ 3 ) ✐ ✐ ✟ ✟ ❍ ❍ ✐ ✐ ✐ ✐ ❅  0 1 (2,1) (2,1) 2 3 n -1 n (2,2) (2,2) (2,2) (1,2) a (2) 2 n − 1 ( n ≥ 3 ) ✐ ✐ ✟ ✟ ❍ ❍ ✐ ✐ ✐ ✐ ✐  ❅ 0 1 (1,1) (1,1) 2 3 n - 2 n − 1 n (1,2) (1,2) (1,2) (1,2) (2,1) c (1) n ( n ≥ 2 ) ✐ ❅  ✐ ✐ ✐ ✐ ✐  ❅ 0 (2,1) 1 (1,2) 2 (1,2) n − 2 (1,2) n − 1 (1,2) n (2,1) d (2) n +1 ( n ≥ 2 ) ✐ ❅  ✐ ✐ ✐ ✐ ✐  ❅ 0 (1,1) 1 (2,1) 2 (2,1) n − 2 (2,1) n − 1 (2,1) n (1,1) a (2) 2 n ( n ≥ 2 ) ✐ ❅  ✐ ✐ ✐ ✐ ✐  ❅ 0 (1,2) 1 (2,2) 2 (2,2) n − 2 (2,2) n − 1 (2,2) n (4,1) e (2) 6 ✐ ✐ ✐ ✐ ✐ ❅  0 1 2 3 4 (1,1) (1,2) (1,3) (2,2) (2,1) f (1) 4 ✐ ✐ ✐ ✐ ✐ ❅  0 1 2 3 4 (2,1) (2,2) (2,3) (1,4) (1,2) g (1) 2 ✐ ✐ ✐  ❅ 0 1 2 (3,1) (3,2) (1,3) d (3) 4 ✐ ✐ ✐  ❅ (1,1) (1,2) (3,1) 0 1 2 a (2) 2 ✐ ✐  ❅ 0 1 (4,1) (1,2) GENERALIZED q − ONSA GER ALGEBRAS AND BOUNDAR Y TODA FIELD THEORIES 11 References [AhKR] C. 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GENERALIZED q − ONSA GER ALGEBRAS AND BOUNDAR Y TODA FIELD THEORIES 13 Labora toire de Ma th ´ ema tiques et Physique Th ´ eorique CNRS/UMR 6083 , F ´ ed ´ era tion Denis Poisson, Un iver- sit ´ e de Tours, P a rc de G rammont, 3720 0 Tours, FRANCE E-mail addr e ss : baseilha@l mpt.univ-to urs.fr Istituto Nazionale di Fisica Nu cleare, Sezione di Bologn a, Via Irnerio 4 6, 401 2 6 Bologna, It al y E-mail addr e ss : belliard@b o.infn.it