Quantum deformations of associative algebras and integrable systems

Quan tum deformations of asso ciativ e algebras and in tegrable systems B.G.Konop elc henk o Dipartimen to di Fisica, Univ ersita del Salen to and INFN, Sezione di Lecce, 73100 Lecc e, Italy No ve mber 5, 2018 Abstract Quantum deformations of the structure constan t s for a class of as- sociative n oncommutativ e algebras are studied. It is shown that these deformations a re go verned by the quan t um cen tral systems whic h has a geometrical meaning of v anishing Riemann curv ature tensor for Christof- fel sy mb ols identified with the structure constants. A sub class of isoas- sociative q uantum deformations is describ ed b y the oriented associativity equation and, i n particular, b y the WD VV equation. It is d emonstrated that a wider class of w eakly ( non)associative quantum d eformations is connected with the integrable soliton equations too. I n particular, suc h deformations for the th ree-dimensional and infi n ite-dimensional algebras are describ ed by the Boussinesq equation and KP hierarc hy , respectively . MSC:16xx,35 Q53,3 7 K10,5 3Axx Key words: Algebra ,Quantum deformation, integrable systems, WDVV equa- tion 1 In tro d uction Mo dern theory o f deformations fo r asso cia tive algebra s whic h w a s formulated in the cla ssical w or ks by Ger stenhab er [1 ,2] got a fresh imp etus with the discovery of the Witten-Dijkgra af-V erlinde-V erlinde (WDVV) equa tio n [3 ,4]. Beautiful formalization of the theory of WDVV equation in terms of the F rob enius mani- folds given by Dubrovin [5,6] a nd its subs e quent e xtension to F-manifolds [7,8] hav e provided us with the re ma rk able realization (see e.g. [5-11] ) of o ne of Ger- stenhab er’s appro ches to the deformatio n of asso c ia tive algebras which consists in the treatment of ” the set of structure constants a s para meter space for the deformation theory” ( [1], Chapter I I, section 1 ). A c ha racteristic feature of the theory of F ro benius and F-manifolds is that the action of the algebra is defined on th e tangent shea f of these manifolds [5-11]. 1 A differen t metho d to describe deformations o f the str ucture constan ts for asso ciative commut a tive algebra in a giv e n basis has been prop osed r ecently in [12,13,1 4]. This a pproach consists 1) in conv erting the table of m ultiplication for an asso c ia tive algebra in the basis P 0 , P 1 , ..., P N − 1 , i.e. P j P k = C l j k ( x ) P l , j, k = 0 , 1 , ..., N − 1 (1.1) int o the zero s e t Γ of the functions f j k = − p j p k + C l j k ( x ) p l , j, k = 0 , 1 , ..., N − 1 (1.2) with p 0 , p 1 , ..., p N − 1 and deforma tion parameter s x 0 , x 1 , ..., x N − 1 being the Dar- bo ux canonical co o rdinates in the symplectic spa ce R 2 N and 2) in the re q uire- men t that the ideal J genera ted b y the funct io ns f j k is the Poisson ideal, i.e. { J, J } ⊂ J (1.3) with re sp ect to the sta ndard Poisson bra ck et { , } in R 2 N . Her e and below the summation ov er rep eated index is as sumed and this index a lwa ys run from 0 to N-1. Deformations of the structure constants C l j k defined by these co nditions are gov er ned by the cen tr al system (CS) of equa tions co ns isting from the asso cia- tivit y condition C l j k ( x ) C n lm ( x ) − C l mk ( x ) C n lj ( x ) = 0 (1.4) and t he coiso tropy condition [ C, C ] m j klr + C m sj ∂ C s lr ∂ x k + C m sk ∂ C s lr ∂ x j − C m sr ∂ C s j k ∂ x l − C m sl ∂ C s j k ∂ x r + + C s lr ∂ C m j k ∂ x s − C s j k ∂ C m lr ∂ x s = 0 . (1.5) Such deformations o f the structure co nstants ha ve been called the co is otropic deformations in [1 2-14 ]. CS (1.4 ), (1.5) is inv aria nt under the general transfor- mations x j → e x j of the deformation parameters with C l j k being the (1,2 ) t y pe tensor [1 4]. It has a n umber of other in tere sting prop er ties. F or the finite-dimensional algebr as this CS co nt a ins as the particular cases the or iented asso ciativity eq ua tion, the WD VV equa tion and certain h ydro dynamical t yp e equations like the stationary disp ersionles s Kadomtsev-Petviash vili (KP) equa- tion ( or Khokhlov-Zab olotsk aya equatio n) [14]. F or the infinite-dimensional po lynomial algebras in the F aa’ di Bruno basis the coisotropic deformations are describ ed b y the universal Whitham hie r arch y of zero gen us , in particular, b y the dispersio nless KP hierarch y [1 2]. It was demonstr a ted in [14] that the theory of coisotropic defor mations and the theo ry o f F-manifolds are e ssentially equiv alent as far as the characteriza tion (1.4),(1.5) o f the structure cons tants is concer ned. One of the adv antages of 2 the for mer is that it is formulated basically in a simple framework of classical mechanics with the standa rd ingredients like the phase space with the canonical co ordinates p j , x j and the co nstraints f j k = 0 whic h a re nothing else than the Dirac’s first class constraints. This featur e of the a pproach prop osed in [12 - 14] strong ly sugg e sts a wa y to built a natural and simple q uantum version of coiso tropic deforma tions in para llel with the passag e from the classic al to quantum mechanics. In this pa per we pr esent the basic elements o f the theor y o f quan tum de- formations for a c lass of asso c ia tive nonco mm utative algebras. A ma in idea of the approach is to as so ciate the elemen ts of the Heisenberg algebra with the elements P j of the basis for the algebra and deforma tion parameters x j . Rea l- ising the table of m ultiplication , following the Dirac’s pres cription, a s the set of equations selecting ”ph ysica l” subspa ce in the infinite-dimensional linear space and requireing that this subspace is not empt y , one gets a system of equatio ns , the quan tum cen tral system (Q CS), ~ ∂ C n j k ∂ x l − ~ ∂ C n kl ∂ x j + C m j k C n lm − C m kl C n j m = 0 (1.6) which g ov erns quantum defor mations of the structure constants. Here ~ is the Plank’s constan t. It is sho wn that a subclas s of isoasso ciative quantum deformations , for which the cla ssical c o ndition (1.4) is v alid for all v alues o f qua n tum deformation pa- rameters, is describ ed by the or ie n ted a sso ciativity equation a nd , as the re- duction, b y the WDVV equation. A wider clas s of w eakly (non)asso ciative quantum deformatio ns is considered too . It is c har acterised b y nonv anis hing quantum anomaly ( defect o f a sso ciativity). These deformations a re a lso as- so ciated with integrable sy stems. It is shown that for the three-dimensiona l algebra a c lass o f such deformations is describ ed b y the Boussinesq equation. F or infinite-dimensiona l p olynomial algebr as in the F aa ’ di Br uno basis the weakly (non)asso ciative quantum deformations of the s tr ucture cons tants are given b y the K P hier arch y or , more g enerally , by the multi-compone nt K P hierarch y . The pap er is or ganised as follows. The definition of qua nt um defor mations and the deriv a tion o f the QCS (1.6) are giv en in the section 2. Isoasso ciativ e quantum deforma tions and c orresp o nding oriented asso cia tivity equa tio n are discussed in the sectio n 3 . In the next section 4 the weakly (non)asso cia tivite deformations and an example of such deforma tion des crib ed b y the Bouss inesq equation are cons idered. Quantum defor ma tions of the infinite-dimensio nal algebra and asso ciated K P hierarch y are studied in the section 5 .2. 2 Quan tum deformati ons of asso c iativ e algebras. In the construction of quantum version o f the coistropic deformations we will follow basically the same lines as in the sta ndard passag e from the classica l 3 mechanics to the quantum mec hanics : subsitute a phase space by the infinite- dimensional linear (Hilber t) spa ce, introduce op er ators instead of the cano ni- cally conjugated momenta a nd co ordinates etc. So, let A be a N-dimensio nal asso ciative algebra with ( or without) unity ele - men t P 0 . W e will consider a class of a lg ebras which p oss es a basis comp osed b y pairwise comm uting elements. Denoting elements of a basis as P 0 , P 1 , ..., P N − 1 one writes the table of multiplication P j P k = C l j k ( x ) P l , j, k = 0 , 1 , ..., N − 1 (2.1) where x 0 , x 1 , ..., x N − 1 stand fo r the defor mation parameter s of the structure constants. The comm utativity of the elements of the basis implies that C l j k = C l kj . In order to define quan tum defor mations w e first asso cia te a set o f linear op erator s b p j and b x j ( j = 0 , 1 , ..., N − 1) with the elemen ts P j of the basis and the deformation parameters x j and require that these op erator s ar e elemen ts of the Heisen b erg algebra [ b p j , b p k ] = 0 ,  b x j , b x k  = 0 ,  b p j , b x k  = ~ δ k j , j, k = 0 , 1 ..., N − 1 (2.2) where ~ is the Planc k’s constant and δ k j is the Kroneck er sym b ol. The second step is to giv e a realization of the table of multiplication (2.1 ) in terms of these op erator s . F or this purp ose w e in tr o duce the set of o pe r ators c f j k defined b y b f j k = − b p j b p k + C l j k ( b x ) b p l , j, k = 0 , 1 , ..., N − 1 . (2.3) T o s implify notations we will omit in wha t follows the label b in the sym b ols of o p e r ators. It is easy to see that the representation of the table o f multiplication (2.1) by the o p er ator equations f j k = 0 is to o restrictive. Indeed, it imp lie s the relatio n [ f j k , p n ] = 0 which due to the identit y  p n , C l j k  = ~ ∂ C l j k ∂ x n (2.4) gives ∂ C l jk ∂ x n = 0. A right way to quantize the first-clas s constraints fr om the classical mechanics has b een suggested long tim e ago by Dirac [1 5]. It consists in the treatment of the first - class constraints as the conditions selecting a subspace H Γ of physical states in the Hilbert space H by the equations f j k | Ψ i = 0 , j, k = 0 , 1 , ..., N − 1 (2.5) where v ector s | Ψ i ⊂ H. The prescription (2.5 ) is the key p oint of the follo wing Definition. The str ucture constants C l j k ( x ) are sa id to define quantum deformations of a n asso ciativ e alge bra if the opera tors f j k defined by (2.3 ) hav e a nontrivial common k ernel. 4 If the conditions (2.5 ) are satisfied then any vector | Ψ i b elonging to H Γ is in v ar iant under the gr o up of transfo r mations gener ated by op er ators G = exp( α j k f j k ) where α j k are parameters , i.e. G | Ψ i = | Ψ i . In suc h a form the ab ov e definitio n is the quan tum version of the classical condition (1.3). The requirement (2 .5) for the existence of the common eigenv ecto rs with zero eigenv alues for all op eators f j k impo ses severe co nstraints on the functions C l j k ( x ) . W e be gin with the well-kno wn consequence o f (2.5) that is [ f j k , f ln ] | Ψ i = 0 , j, k , l , n = 0 , 1 , ..., N − 1 . (2.6) This condition is the quantu m version of the cois otropy condition { f j k , f ln } | Γ = 0 in the classical case. Using (2.2) and (2.4 ), o ne gets from (2.6 ) the relation ~ 2 ∂ 2 C m j k ∂ x l ∂ x n − ~ 2 ∂ 2 C m ln ∂ x j ∂ x k − ~ [ C, C ] m j k ln ! p m | Ψ i = 0 (2.7) where the bra cket [ C, C ] m j k ln is defined in (1.5). So, equations (2.6) are satisfied if ~ ∂ 2 C m j k ∂ x l ∂ x n − ~ ∂ 2 C m ln ∂ x j ∂ x k − [ C, C ] m j k ln = 0 , j, k , l , n , m = 0 , 1 , ..., N − 1 . (2.8) This c onstraint is the qua nt um version of the cos isotropy condition (1.5). T o derive the quantum version of the as s o ciativity condition (1.4) we use the ident ity ( p j p k ) p l − p j ( p k p l ) = p j f kl − p l f j k + C m kl f j m − C m j k f lm + +  ~ ∂ C n jk ∂ x l − ~ ∂ C n kl ∂ x j + C m j k C n lm − C m kl C n j m  p n . (2.9) It implies that (( p j p k ) p l − p j ( p k p l )) | Ψ i =  ~ ∂ C n j k ∂ x l − ~ ∂ C n kl ∂ x j + C m j k C n lm − C m kl C n j m  p n | Ψ i (2.10) for | Ψ i ⊂ H Γ . Hence, if the structure co ns tants ob ey the equations ~ ∂ C n j k ∂ x l − ~ ∂ C n kl ∂ x j + C m j k C n lm − C m kl C n j m = 0 , j, k , l , n = 0 , 1 , ..., N − 1 (2 .11) then (( p j p k ) p l − p j ( p k p l )) | Ψ i = 0 . (2.12) W e will r e fer to the co nditions (2.1 2 ) and (2.11) as the quantum or weak asso ciativity conditions. W e no te that in vir tue of (2.9 ) the reques t for the ”strong” asso ciativity ( p j p k ) p l − p j ( p k p l ) = 0 implies f j k = 0 and so it is to o rigid. T hus , s imilar to the table of m ultiplication (2.5 ) w e r equire the fulfilment of the asso ciativity condition only on the ” physical” subspace H Γ instead of the whole linear space H . 5 Equations (2.11) a nd (2.8) repre s ent the quantum counterpart o f the classic al CS (1 .4),(1.5). Thes e system of equations has in fact a muc h simpler form since only part of them are indep e nden t. Indeed, o ne ha s th e following identit y ~ T m j k, ln = ~ ∂ R m j lk ∂ x n − ~ ∂ R m nlk ∂ x j − C m j s R s lkn − C m ns R s kj l − C s ln R m ksj − C s j k R m ln s − C s lk R m sj n where T m j k, ln denotes the l.h.s. of equatio n (2.8) and R n klj stands for the l.h.s. o f equation (2.11). Thus, we hav e Prop ositi o n 2 . 1 The structure c o nstants C l j k ( x ) define a q uantum defor- mation of an asso ciative algebra if they obey the equations R n klj + ~ ∂ C n j k ∂ x l − ~ ∂ C n kl ∂ x j + C m j k C n lm − C m kl C n j m = 0 .  (2.13) W e will refer to the system (2.1 3) as the quant um cen tr al system (Q CS). W e emphasize that quantum deformations are defined in the categor y of as s o ciative noncommutativ e a lgebras whic h posses s c ommut a tive basis. Prop ositi o n 2.2 If the structur e c o nstants define a q uantum deformatio n then [ f j k , f lm ] = − ~ K st j k,lm f st , j, k , l , m = 0 , 1 , ..., N − 1 (2.14) where K st j k,lm = 1 2  δ t m ∂ C s j k ∂ x l + δ t l ∂ C s j k ∂ x m − δ t k ∂ C s lm ∂ x j − δ t j ∂ C s lm ∂ x k + + δ s m ∂ C t j k ∂ x l + δ s l ∂ C t j k ∂ x m − δ s k ∂ C t lm ∂ x j − δ s j ∂ C t lm ∂ x k ! (2.15) The proo f is b y direct calculation. W e note that the exp ession (2.15) exactly coincides with that which app ea r in the co isotropic case for the Poisson br ack ets b etw een the functions f j k . So, one has the same closed algebra for the basic o b jects f j k and b f j k for the coiso tropic and quan tum deformations up to the standard corresp ondence [ , ] ← → − ~ { , } [15] betw een comm utator s and Poisson brack ets. The central systems (1.4), (1.5) and (2.13 ) which define co isotropic and quantum defo r mations have rather different fo rms. In spite o f this they hav e some general prop erties in common. The inv ariance under the gener al transfor- mations of deformation para meters is o ne of them. Similar to the coisotropic case [12-14] the qua n tum deformation parameter s x j and corres po nding p k are strongly interrelated: they should ob ey the co nditio ns (2.2). So a ny change x j → e x j require an adequate c hange p k → e p k in order the relations (2.2) to be preserved. Th us, for the general transformation of the defo rmation parametrs x j in our scheme o ne has x j → e x j , p k → e p k = ∂ e x n ∂ x k p n , j, k = 0 , 1 , ..., N − 1 . (2.16) 6 Note tha t the transformations (2.16) pres erve the comm utativity of the bas is. The requirement of the inv ariance for equations (2.5) readily implies that C l j k ( x ) → e C l j k ( e x ) = ∂ e x l ∂ x t ∂ x s ∂ e x j ∂ x m ∂ e x k C t sm ( x ) + ~ ∂ e x l ∂ x m ∂ 2 x m ∂ e x j ∂ e x k (2.17) under transforma tions (2.16). Then, it is a straightforw ar d chec k that equation (2.13) is also in v ar iant. Hence, one ha s Prop ositi o n 2.3 The QCS (2.13) is inv ariant under the general transfor- mations of the deformation parameters. F urthermore, the r elation (2.17) evidently co incides with the transforma tio n law o f the Christoffel s y m b o ls and in the formula (2.1 3 ) the tensor R n klj is nothing but the Riemann curv ature tensor expr e s sed in terms of the Christoffel symbols (see e.g. [16 ,17]). Thu s, w e ha ve Geometrical in terpretation. The QCS system (2.13) which gov erns the quantum defor mations in geometrical terms means the v anishing of the Riemann curv ature tenso r R n klj for the torsionless Christo ffel sym b ols Γ l j k ident ified with the structure cons tants (Γ l j k = ~ C l j k ). In the standard terms of the matrix- v alued one-fo rm Γ with the matrix elements( see e.g. [17]) Γ l k = ( C j ) l k dx j = C l j k dx j (2.18) equation (2.13) lo o ks like ~ d Γ + Γ ∧ Γ = 0 (2.19) where d a nd ∧ denote the exterior differential and exter ior product, resp ectively . The fla tness condition [ ∇ j , ∇ l ] = 0 (2.20) for the torsionless connectio n ∇ j = ~ ∂ ∂ x j + C j is the another standar d form of equation (2.13). In the context of F robenius manifo lds the r e lation b etw een the structure consta nt s and Christoffel sym b o ls has b een discussed within a different approach in [6]. The iden tifica tion of the structure constants with the Chris toffel s ymbols leads to certain constra in ts w ithin such g eometrical interpretation. F or in- stance, for an algebra with the unit y element P 0 , for which C l 0 k = δ l k , equation (2.13) immediately implies ∂ C n j k ∂ x 0 = 0 , j, k , n = 0 , 1 , ..., N − 1 . (2.21) F urthermore, if one r equires that P 0 is inv ariant with respect to the tr ans- formations (2.16) then ∂ e x j ∂ x 0 = δ j 0 and ∂ x j ∂ e x 0 = δ j 0 , j = 0 , 1 , ..., N − 1 . F or alg ebras with different prop erties ( semisimple, nilpotent etc) the orbits generated by transformations (2.16) hav e quite differ e n t parametr izations. F or instance, for a semisimple algebra there is a bas is at which C l j k = δ j k δ l j (see 7 e.g.[5,6]). Let us denote the deformation parameter s asso ciated with this basis as u 0 , u 1 , ..., u N − 1 . Then the corr esp onding orbit has the following par ametriza- tion C l j k ( x ) = ∂ x l ∂ u m ∂ u m ∂ x j ∂ u m ∂ x k + ~ ∂ x l ∂ u m ∂ 2 u m ∂ x j ∂ x k (2.22) where x m ( u ) , m = 0 , 1 , ..., N − 1 are arbitrary functions. F or a nilp otent alg ebra for whic h all elemens hav e degree of nilpotency e qual to t wo ther e e xists a basis at w hich all C l j k = 0 . The general elemen t of the c o rresp onding orbit is given by the form ula C l j k ( x ) = ~ ∂ x l ∂ u m ∂ 2 u m ∂ x j ∂ x k (2.23) where again x m ( u ) a re arbitrary functions. In the construction presented ab ove we did not use concrete realization of op erator s p j and x k . Any such realization pro vide s us with a co ncrete real- ization of the asso ciative algebra under consideratio n and the for mulae derived . The most co mmon repr e sentation o f the Heisen b erg algebra (2.2) is given b y the so-called Schroding er representation at whic h o p er ators b x j are the op era - tors of multiplication by x j , p j are op erator s ~ ∂ ∂ x j and wa ve-functions Ψ( x ) are elements of the spa ce H . In this rea lization the asso ciative a lgebra A is the well-known a lgebra of diff e r ential polyno mials a nd equations (2.5) have the form − ~ ∂ 2 Ψ ∂ x j ∂ x k + C l j k ( x ) ∂ Ψ ∂ x l = 0 , j, k = 0 , 1 , ..., N − 1 . (2.24) It is a simple check that the usual compatibility co nditio n for the system (2.24) (equality of the mix e d third-or der deriv atives) is nothing else than the conditions (2.12) and it is equiv alent to equa tions (2.11). F or an algebra with the unit y elemen t one ha s ~ ∂ Ψ ∂ x 0 = Ψ and , in virtue of (2.21) one ha s Ψ( x ) = e x 0 ~ e Ψ( x 1 , ..., x N − 1 ) . (2.25) The system ( 2.24) is well-kno wn in geometry . In the theor y of the F robenius manifolds it is called the Gauss- Ma nin equation ( see e.g. [6,9] ). Such a system arises also in the theory of Gromov-Witten inv a riants [18,19]. The s tandard qua siclassica l approximation Ψ = exp( S ( x ) ~ ) , ~ → 0 (see e.g. [15] ) p erformed for equations (2.24) give rise to the Hamilton-Jacobi equations − ∂ S ∂ x j ∂ S ∂ x k + C l j k ∂ S ∂ x l = 0 . (2.26) These equations coincide with those for the gener ating function S for La - grangia n submanifols which arise in the theory of coisotropic deformatio ns [1 4]. In this classica l limit ~ → 0 the s y stem (2.8), (2.11 ) is reduced to the classical CS (1.4 ), (1.5) and the whole construction pres ented ab ov e is reduced to that of co isotropic deformations. Other realizations of the Heisen b er g algebra (2.2) are of in terest to o. Here we will mention only one of them given in terms of the standa rd crea tion and 8 annihilation op erator s a + j and a j and the F o ck space. The standard ba s is in he F o ck space is given by the v ectors | n 0 , n 1 , ..., n N − 1 i = ( n 0 ! n 1 ! ...n N − 1 !) − 1 2 ⊓ N − 1 k =0  a + k  n k | 0 i , n k = 0 , 1 , 2 ... where a j | 0 i = 0 , j = 0 , 1 , ..., N − 1 and h 0 | 0 i = 1 . Then | Ψ i = ∞ X n k =0 A n 0 ,n 1 ,...,n N − 1 | n 0 , n 1 , ..., n N − 1 i and t he constra int (2.5) t akes the form  − a j a k + C l j k ( a + ) a l  | Ψ i = 0 , j, k = 0 , 1 , ..., N − 1 . (2.27) This system of equations is equiv alent to the infinite system of discrete equa - tions for the c o efficien ts A n 0 ,...,n N − 1 while C l j k ( a + ) ob e y QCS (2.13). Equatio ns (2.27) define sort of coherent states which co uld b e relev ant to the theor y o f quantum deformations and its quasiclass ical limit. Finally , we note that several different ” quantization” sc hemes for the struc- tures asso ciated with the F rob enius manifolds , F-manifolds and co isotropic submanifolds ha ve been prop os ed in [18-22]. A co mparative analysis o f these approaches and our scheme will be done elsewhere. 3 Isoasso ciativ e quan tum deformations and ori- en ted asso ciativit y equation General quantum deformations describ e d in the pr evious section contain as a sub c lass of deforma tions for which the c la ssical as so ciativity condition (1 .4) is satisfied for all v alues o f quantum deformation parameter s. W e will refer to such deformations as isoasso ciative quantu m deformations by analo gy with the isomono dro m y a nd isospec tr al defor ma tions. The for mula (2.13) implies Prop ositi o n 3.1 Structure constants C l j k ( x ) define isoass o ciative quantum deformations of ass o ciative algebra if they obey the equations C m j k C n lm − C m kl C n j m = 0 , (3.1) ∂ C n j k ∂ x l − ∂ C n kl ∂ x j = 0 . (3.2) In terms of the o ne - form Γ (2.18) the sy stem (3.1), (3.2) loo ks lik e Γ ∧ Γ = 0 , d Γ = 0 . (3.3) Another wa y to ar r ive to the system (3.1 ), (3.2) co ns ists in the treatment of ~ in all the ab ov e for mulae beginning with (2.2 ) not as the fixed co nstant 9 but as a v ariable parameter. In such interpretation the QCS (2 .13) from the very beginning splits into t wo equations (3.1 ) and (3.2), the connection ∇ j from (2.20) b ecomes a p encil of flat torsionless connection discusse d in [5,6,8,9] and equations (2.24 ) coincide with the Dubro v in’s linear system f o r flat co o rdinates [5,6]. Thu s, quantum deformations for which p j and x j are elements of the penc il of Heisen b erg algebras (2.2) ar e o f par ticular interest. A w ay to deal with the sy stem (3 .1 ), (3.2) is to solve first equa tions (3.2 ). They imp ly that C l j k = ∂ 2 Φ l ∂ x j ∂ x k (3.4) where Φ l , l = 0 , 1 , ..., N − 1 are functions. E q uation (3.1) then become ∂ 2 Φ m ∂ x j ∂ x k ∂ 2 Φ n ∂ x m ∂ x l = ∂ 2 Φ m ∂ x l ∂ x k ∂ 2 Φ n ∂ x m ∂ x j . (3.5) The system (3.5) has app eared first in [5] (Prop os ition 2 .3) as the equation for the displacement v ecto r. It has b een reder ived in the different con text in [23] and has been called the or ient ed a sso ciativity equation there. In the form (3.3) it has a ppea red also in [24,25]. In our appro ach it describ es the iso asso- ciative q uantum deformation of the str ucture cons tants for a clas s of asso ciative noncommutativ e a lgebras. F or this class of defo r mations a ll o p e rators f j k hav e a simple gener ating ” function”, namely ~ 2 f j k = [ p j, [ p k , W ]] where W = − 1 2 ( x m p m ) 2 + Φ m p m . In the theory o f coisotropic defo r mations [14] the defo rmations given b y equations (3.4) , (3.5 ) constitute a sub class of all deforma tions. So, oriented asso ciativity equation describ es simultaneously bo th co is otropic and isoasso cia- tive quan tum deformations. In other w or ds, one of the characteristic featur e s of the class of deformations g iven by the form ulae (3.1), ( 3.2) is that they remain unch anged in the pro cess of ”quantization”. This means also that o ne c a n use bo th classic a l formulae [14] and the quantum one ( previous sec tio n) to des crib e the proper ties o f these deformations . F or instance, it was shown in [14] that in the natural parametrization of the str ucture constan ts C l j k by the eigenv alues of the matrices C j and in terms of canonical co ordinates u j the s ystem (3.2) bec omes the system of conditions for the commutativit y o f N h ydr o dynamical t y pe systems. At the same time, the functions Φ n hav e a meaning of conserved densities for these hydrodynamica l type s y stems. All these results are v alid for the isoasso ciative quan tum deformations too. The oriented asso ciativity equation (3.5) admits a well-known r eduction to a single sup erp otential F given by Φ n = η nl ∂ F ∂ x l 10 where η nl is a constant metric. In this case equations (3.5) b ecomes the famous WD VV equation [3,4] ∂ 3 F ∂ x j ∂ x k ∂ x s η st ∂ 3 F ∂ x t ∂ x m ∂ x l = ∂ 3 F ∂ x l ∂ x k ∂ x s η st ∂ 3 F ∂ x t ∂ x m ∂ x j . (3.6) Thu s, the WD VV equation also describes the isoasso ciative q ua nt um defor- mations. One more example of isoass o ciative quan tum defor mations is provided by the Riemann space with the flat Hessian met r ic g j k = ∂ 2 Θ ∂ x j ∂ x k considered in [26] ( see also [27], Prop osition 5.10). In this case [26] C l j k = ~ Γ l j k = ~ g lm ∂ 3 Θ ∂ x j ∂ x k ∂ x m , (3.7) equation (3.2) is satisfied ident ic a lly and the asso ciativity co ndition tak es the form ∂ 3 Θ ∂ x j ∂ x k ∂ x s g st ∂ 3 Θ ∂ x t ∂ x m ∂ x l = ∂ 3 Θ ∂ x l ∂ x k ∂ x s g st ∂ 3 Θ ∂ x t ∂ x m ∂ x j . (3.8) This equation repr esents a rather nontrivial single-field r eduction of equation (3.5). 4 W eakly (non)asso ciativ e quan tum deformati ons All co is otropic deformations are isoasso cia tive by construction [14]. F or a sub- class of them describ ed by the oriented asso ciativity equation the C S is reduced to the s ystem (3.1 ), (3.2). But, there is a nother sub class of co isotorpic de- formations for which the exactness conditions (3.2) are not satisfied. F or the finite-dimensional algebras suc h coisotro pic deforma tio ns are descr ibe d b y the stationary dispe r sionless KP equation and other h y dro dynamical t yp e systems [14] . In the infinite-dimensiona l c a se this t y p e of defor ma tions is descr ib ed by the univ e r sal Whitham hier arch y of zero g enus and, in particular, by the disp e rsionless KP hiera rch y [12 ]. What is the quantum version of coisotr o pic deformations of s uch type? One naturally exp ects that they will not b e the isoasso ciative one. On the other hand, quantum deformations, for which equations (3.2) ar e not satisfied, a re gov er ned b y equation (2.13) with a nice geo metrical meaning even if they a re not is oasso cia tive. All these s uggest that the general qua nt um defo r mations defined by QCS (2.13) without the additional exac tness constra int (3.2) should be of interest too. W e will refer to suc h deformatio ns as w ea k ly a sso ciative or weakly no nasso ciative quantum deformations. The fir st term is due to the fact that ac c ording to (2 .1 0) and (2.12) for such deformations one has as so ciativity for all v a lues of quantu m deformation parameters not on the oper a tor level, i.e. not o n the whole space H , but only 11 on the smaller ”physical” subspace H Γ . The second name reflects the fa ct that the defect of asso ciativity α n klj + C m j k C n lm − C m kl C n j m (4.1) for suc h deformations is given by α n klj = − ~  ∂ C n j k ∂ x l − ∂ C n kl ∂ x j  . (4.2) So, for small ~ o r slowly v arying structure constant s the defect of asso cia - tivit y is small. F or the matrix-v alued t wo-form α q with the matrix elemen ts ( α q ) n k + 1 2 α n klj dx l ∧ dx j one has α q = − ~ d Γ (4.3) where Γ is defined in (2.18). One may refer to α q also as a quan tum anomaly of asso ciativit y . Note that for an algebra with unit y elemen t all elements α n klj with k or l or j =0 v anish. Geometrical in terpreta tio n o f the Q CS (2.13 ) provides us with n umero us examples of weakly (non)asso ciative quantum deformations. An y tors ionless flat connectio n gives us such deformation for certain a sso ciative algebra. In the generic case , for instance, thes e deformations a r e given b y the formulae (2.22) and ( 2 .23) for the semisimple and nilp otent a lgebras, respectively . If there exists a metric g j k compatible with the Christoffel sym b ols Γ l j k = ~ C l j k then the generic deformation of the str ucture constants is describ ed b y the form ula C l j k = 1 2 ~ g nl  ∂ g nk ∂ x j + ∂ g j n ∂ x k − ∂ g j k ∂ x n  (4.4) where g j k is an arbitrary flat metric. Particular c hoice of the metric gives us a sp ecific deforma tio n. F or instance, for the diago na l flat metric g j k = δ j k H 2 j the weakly (non) a s so ciative deformations a re defined b y the solutions of the w ell- known Lame system whic h descr ib es the orthogona l systems of coo rdinates in the N-dimensional Euclidean spac e (see e.g. [16] ). F or certain metrics , as, for example, for the Hes s ian metric considered at the end o f the previous section, the quan tum anomaly v anis hes . Other examples are pr ovided by interpretation o f the system (2.24 ) a s the system of equa tio ns for p osition vector in the affine differential geometry ( see e.g. [28]). W e note also the pap er s [29,30 ] in which the equatio ns describing the g e o metry of submanifo lds for a fla t space have b een reduced to the WD VV t y pe equations . Different t yp e of (non)asso ciative defor mations is given b y the quan tum version of the coiso tropic deformatio ns of the finite-dimensio nal algebras s tudied in [14]. As an illustrativ e example w e will consider here the three-dimensional 12 (N=3) algebr a with the unit y elemen t. The non triv ial part of the table of m ultiplica tion is of the for m P 2 1 = A P 0 + B P 1 + C P 2 , P 1 P 2 = D P 0 + E P 1 + G P 2 , (4.5) P 2 2 = L P 0 + M P 1 + N P 2 . As in the pap er [1 4] we consider the ”gaug e” B=0, C= 1 , G=0. The QCS system (2.13) in this case assumes the f o rm A + N − E = 0 , ~ A x 2 − ~ D x 1 + L − E A = 0 , − ~ E x 1 + M − D = 0 , (4.6) ~ D x 2 − ~ L x 1 + E D − M A − N D = 0 , ~ E x 2 − ~ M x 1 + E 2 − L − N E = 0 , − ~ N x 1 + D − M = 0 where A x j + ∂ A ∂ x j etc. T his system of equatio ns imp lie s that E = 1 2 A + 3 4 ǫ, N = − 1 2 A + 3 4 ε, L = 1 2 A 2 + 3 4 ǫA − ~ A x 2 + ~ D x 1 , (4.7) M = D + ~ 2 A x 1 and A x 2 − 4 3 D x 1 + ǫ x 2 − ~ 3 A x 1 x 1 = 0 , D x 2 − 3 4  A 2  x 1 − 3 4 ǫA x 1 + ~ A x 1 x 2 − ~ D x 1 x 1 = 0 . (4.8) where ǫ ( x 2 ) is a n arbitrary function. Eliminating D from the system (4.8) ,o ne gets the equatio n A x 2 x 2 − ǫA x 1 x 1 −  A 2  x 1 x 1 + ~ 2 3 A x 1 x 1 x 1 x 1 + ǫ x 2 x 2 = 0 . (4.9) A t ǫ =const it is the w e ll- known Boussinesq equation which des crib es surface wa ves (see e.g .[31] ). This equation is integrable by the inv erse scattering trans- form metho d [3 2 ] similar to the fa mo us Korteweg-de V ries a nd KP equa tions (see e.g. [33,3 4,35]). Equations (4.8) imply the existence o f the function F suc h that 13 A = − ǫ − 2 F x 1 x 1 , D = − 3 2 F x 1 x 2 + ~ 2 F x 1 x 1 x 1 . (4.10) In terms of the function F the sys tem (4.8) or equation (4.9) b ecome F x 2 x 2 − ǫF x 1 x 1 + 1 2 ( ǫ + 2 F x 1 x 1 ) 2 + ~ 2 3 F x 1 x 1 x 1 x 1 = 0 . (4.11) The function τ de fined by F = log τ is the τ - function for the Bo us sinesq equation (4.9) and equatio n (4.11) is the Hirota equation to it ( at ǫ = 0 see e.g. [3 6 ] ). An y solution of the B oussinesq equation (4.9) or the Hirota equation (4.11) provides us with the w ea kly (non)asso cia tive quantum de fo rmation o f the alge- bra (4.5) w ith the structure constants g iven by the formulae (4.7 ), (4.10). The quantum anomaly α q (4.3) for these Boussinesq deformations is of the form α q = ~   0 1 2 A x 2 + ~ 2 A x 1 x 1 1 2  A 2  x 1 0 − A x 1 − 1 2 A x 2 − ~ 2 A x 1 x 1 0 0 A x 1   dx 1 ∧ dx 2 . (4.12) T o present a simple co ncrete example of deformatio n for the algebra (4.5) we consider the following p olyno mial solution F = α  x 1  2 + β x 1 x 2 +  αǫ − 4 α 2   x 2  2 + γ  x 1  2 x 2 + 1 3 γ ( ǫ − 8 α )  x 2  3 − 2 3 γ 2  x 2  4 (4.13) of the Hir ota equation (4.11) with ǫ =const whe r e α, β , γ are arbitrar y constants. This s olution defines via (4.7) and (4.1 0) the following weakly (non)as so ciative deformation of the s tructure constan ts A = − 4 α − 4 γ x 2 , B = 0 , C = 1 , D = − 3 2 β − 3 γ x 1 , E = − 2 α − 2 γ x 2 + 3 4 ǫ, G = 0 , L = 8  α + γ x 2  2 − 3 ǫ  α + γ x 2  + ~ γ , M = − 3 2 β − 3 γ x 1 , N = 2 α + 2 γ x 2 + 3 4 ǫ. (4.14) F or this deformatio n the quant um anomaly is g iven by α q = 2 ~ γ   0 − 1 0 0 0 1 0 0 0   dx 1 ∧ dx 2 . (4.15) General for mulae (2.5 ) and (2.24 ) pro vide us with the auxiliar y linear pro b- lems for the Boussinesq equation. It is easy to show that equations (2.5) for the algebra (4.5) with B=G=0, C = 1 are equiv alent to the fo llowing t wo equatio ns 14  p 3 1 −  3 2 A + 3 4 ǫ  p 1 − ( D + ~ A x 1 ) p 0  | Ψ i = 0 , (4.16)  p 2 − p 2 1 + Ap 0  | Ψ i = 0 . (4.17) In the co o rdinate re presentation these equations look lik e ~ 3 Ψ x 1 x 1 x 1 + 3 2 ~  u − ǫ 2  Ψ x 1 + w Ψ = 0 , (4.18) ~ Ψ x 2 − ~ 2 Ψ x 1 x 1 − u Ψ = 0 (4.1 9) where u = − A and w = − D − ~ A x 1 . Eq ua tions (4.18)-(4.1 9) are the well- known auxiliary linea r pr oblems for the Boussinesq eq ua tion at zer o v alue of the spectral parameter [32-34]. Another set of linear problems one can obtains from the conditions (2.20). F or the Boussine s q alge bra (4.5 ) the matrices C 1 and C 2 are C 1 =   0 A D 1 0 1 2 A + 3 4 ǫ 0 1 0   , (4.20) C 2 =   0 D 1 2 A 2 − ~ A x 2 + ~ D x 1 + 3 4 ǫA 0 1 2 A + 3 4 ǫ D + ~ 2 A x 1 1 0 − 1 2 A + 3 4 ǫ   . (4.21 ) The c ommut a tivity condition (2 .2 0) for the connection ∇ j = ~ ∂ ∂ x j + C j , J = 1 , 2 represents the compatibilt y condition for the linear problems  ~ ∂ ∂ x 1 + C 1  ϕ = 0 ,  ~ ∂ ∂ x 2 + C 2  ϕ = 0 (4.22) where ϕ is the column ϕ =  ϕ 1 , ϕ 2 , ϕ 3  T . Equations (2.20 ) for C 1 and C 2 given by (4.20) and (4.21) are equiv a lent to equations (4.7), (4 .8), i.e. to the Boussinesq eq uation. So, equa tions (4.2 2) repr esent the a ux iliary matrix lin- ear problems for the Bo ussinesq equa tion. Equatio ns (4.2 2 ) imply the scala r equations ~ 3 ϕ 3 ,x 1 x 1 x 1 − ~  3 2 A + 3 4 ǫ  ϕ 3 ,x 1 +  D + ~ 2 A x 1  ϕ 3 = 0 , ~ ϕ 3 ,x 2 + ~ 2 ϕ 3 ,x 1 x 1 − Aϕ 3 = 0 . (4.23) Equations (4.2 3) are for mally adjoint to equations (4.18)-(4.19 ) and their compatibility condition giv es rise to the same B o ussinesq equation (4 .9 ). W e w o uld lik e to note that the ”zero curv a ture” representation  ∂ ∂ x 1 + U, ∂ ∂ x 2 + V  = 0 with the matrix-v alued functions U and V is quite common in the theory o f 15 the 1+1-dimensional in tegr able systems ( see e.g. [33-35]). The particular rep- resentation of the form (4.22) is of in terest a t least by tw o r e asons. First, for instance, for the Boussinesq equatio n the elements o f the matrices C 1 and C 2 (4.20), (4.21) really coincide with the comp o nents of the Chr is toffel symbol. Second, in such a r epresentation the elements o f C 1 and C 2 are nothing else than the structure constants of the defor med a sso ciative algebr a (4.5). All the ab ov e fo rmulae fo r the Boussines q quantum deformations in the formal limit ~ → 0 ( with Ψ = exp  S ~  ) a re reduced to those for the c oisotropic deformations of the s ame algebra (4.5 ) which a re describ ed by the stationa ry disp e rsionless KP equa tion [1 4]. Finally , w e note that eliminating F x 1 x 1 from the Hirota e quation (4.1 1) with the use of its differential consequences, one obtaines the equatio n F x 2 x 2 x 2 F x 1 x 1 x 1 − F x 2 x 2 x 1 F x 1 x 1 x 2 = ~ 2 3 ( F x 1 x 1 x 2 F x 1 x 1 x 1 x 1 x 1 − F x 1 x 1 x 1 F x 1 x 1 x 1 x 1 x 2 ) which represents the ”quant um” version of the Witten’s equation [3] F x 2 x 2 x 2 F x 1 x 1 x 1 − F x 2 x 2 x 1 F x 1 x 1 x 2 = 0 , i.e. equation (3.6) for the tw o -dimensional algebra without unit y element and the metric η =  0 1 1 0  . 5 Quan tum deformati ons of the infinite-dimensional algebra and KP hierarc hy . W e will c o nsider an infinite-dimensiona l algebr a of p olyno mials generated by a single element. In the so-c alled F aa’ di Bruno basis the structure co nstants of this algebra hav e the fo r m [1 2] C l j k = δ l j + k + H k j − l + H j k − l , j, k , l = 0 , 1 , 2 ... (5.1) where H k l = 0 at l ≤ 0 and H 0 l = 0 . Coiso tropic deformations o f the structure constants (5 .1) ha ve b een studied in [12]. It w as shown that they are descr ib e d by the dispersio nle s s KP hierarch y . Here we will discuss the quantum deforma tions of the s ame set (5.1) of the structure constan ts . Prop ositi o n 5 .1. F or the structure constants (5.1 ) the QCS (2.13) is equiv- alent to th e system ~ ∂ H k j ∂ x l + H l j + k + H k j + l − H k + l j + + j − 1 X n =1 H k j − n H l n − l − 1 X n =1 H k l − n H n j − k − 1 X n =1 H l k − n H n j = 0 , j, k , l = 0 , 1 , 2 , .... (5.2) 16 Pro of. Substitution of (5.1 ) into (2.13) gives ~ ∂ H k j − m ∂ x l + ~ ∂ H j k − m ∂ x l − ~ ∂ H k l − m ∂ x j − ~ ∂ H l k − m ∂ x j + + ∞ X n =0  δ n j + k + H k j − n + H j k − n   δ m n + l + H l n − m + H n l − m  − − ∞ X n =0  δ n l + k + H k l − n + H l k − n   δ m n + j + H j n − m + H n j − m  = 0 . (5.3) A t j > m,k < m,l < m using of the iden tity k − 1 X p = n − 1 H j k − p H m p − n = k − 1 X p = n − 1 H m k − p H j p − n , one gets equation (5.2) with the subs titution j → j-m. At m > j,m > l and m < k equation (5.3) is reduced to ∂ H j k ∂ x l − ∂ H l k ∂ x j = 0 . (5.4) It is eas y to see that equation (5.2) directly implies (5.4) due to the symmetry of the nondifferential pa rt in the indices k and l. An analysis of all other choices of indices in (5.3) shows that the resulting equations are all equiv alent to (5.2)  . Solutions of the QCS (5.2 ) pr ovide us with the q uantum deformation o f the po lynomial algebra in the F aa’ di B r uno basis. In general, these deformations are weakly (non)asso ciative o ne and the quan tum anomaly is given by α n klj = ~ ∂ H k l − n ∂ x j − ∂ H k j − n ∂ x l ! or α q = − ~ dA (5.5) where the matrix-v alued one -form A has elemen ts ( A ) n k = P ∞ l = n +1 H k l − n dx l . At ~ → 0 the QCS (5.2) conv erts into the cla ssical ass o ciativity condition for the structure constan ts (5.1) [12]. W e note that for the first time the sys tem (5.2) has b een derived in [37] within a different context as the comp onent-wise v ersio n of the central system for the currents asso ciated with the KP hierar ch y . It was sho wn in [3 7] that it enco des a complete algebraic informatio n a bo ut the KP hierarch y . W e will demonstrate t his in a little bit differe n t manner. Similar to the coisotropic case [12 ] there are, at least, t wo w ays to deco de information contained in the QCS (5.2 ). Fir st approa ch is to c ho o se first an appropria te parametriza tio n of H j k . As in the classical case [1 2] we in tro duce the functions u,v and w defined by the formulae 17 H 1 1 = − 1 2 u, H 1 2 = − 1 3 v , H 1 3 = − 1 4 w + 1 8 u 2 . (5.6) F rom the QCS (5.2) one gets H 2 1 = 2 H 1 2 + ~ ∂ H 1 1 ∂ x 1 , H 3 1 = 3 H 1 3 + ~ ∂  H 1 2 + H 2 1  ∂ x 1 , H 2 2 = − ~ ∂ H 1 1 ∂ x 2 − H 1 3 + H 3 1 +  H 1 1  2 . Hence H 2 1 = − 2 3 v − ~ 2 u x 1 , H 3 1 = − 3 4 w + 3 8 u 2 − ~ v x 1 − ~ 2 2 u x 1 x 1 , H 2 2 = 1 2 u 2 − 1 2 w + ~ 2 u x 2 − ~ v x 1 − ~ 2 2 u x 1 x 1 (5.7) Substituting these expr essions in to the first exactness conditions (5.4),i.e. ∂ H 1 1 ∂ x 2 − ∂ H 2 1 ∂ x 1 = 0 , ∂ H 1 2 ∂ x 2 − ∂ H 2 2 ∂ x 1 = 0 , ∂ H 1 1 ∂ x 3 − ∂ H 3 1 ∂ x 1 = 0 (5.8 ) and eliminating w, one gets the equations u x 3 − ~ 2 4 u x 1 x 1 x 1 − 3 4  u 2  x 1 − ϕ x 2 = 0 , u x 2 − 4 3 ϕ x 1 = 0 (5.9) where ϕ = v + 3 4 ~ u x 1 . It is the famous Kado mtsev-Petviashvili equation (see e.g. [33-35]). Using higher equations (5.2 ) and (5.4), o ne in a similar manner obtains the higher KP eq uations and the whole KP hier arch y . In the limit ~ → 0 equatio n (5.9) is reduced to the disp ersionless KP equation while at the stationaly case u x 3 = 0 one gets the Boussinesq equation (4.9 ) at ǫ = 0 with A = − u, ϕ = − D − ~ 4 A x 1 . Another wa y to de a l with the QCS (5.2) is to solv e first all exactness condi- tions. O ne of them is given b y (5 .4). It implies the existence of the functions F k such that H j k = ∂ F k ∂ x j , j, k = 1 , 2 , 3 ... (5.10) The system (5.2) , in addition, c o ntains another exactness type condition. In- deed, as it was sho wn in [3 7], equations (5.2) lea d to the following cons traint 18 system ~ ∂ ∂ x j n − 1 X k =1 H k n − k ! + nH j n = H j n + n − 1 X k =1  H j + n − k k − H k j + n − k  + + j − 1 X l =1 n − 1 X k =1 H n − k l H j − l k . (5.11) The r .h.s. of (5 .11) is sy mmetr ic in the indices j and n. Hence ~ ∂ ∂ x j n − 1 X k =1 H k n − k ! + nH j n = ~ ∂ ∂ x n j − 1 X k =1 H k j − k ! + j H n j . (5.1 2) Substitution of (5.1 0) into equations (5.12 ) gives the exac tnes s conditions ∂ ∂ x j nF n + ~ n − 1 X k =1 ∂ F n − k ∂ x k ! = ∂ ∂ x n j F j + ~ j − 1 X k =1 ∂ F j − k ∂ x k ! , j, n = 1 , 2 , 3 ... (5.13) Prop ositi o n 5.2 H j k = 1 ~ P k  − ~ e ∂  F x j , j, k = 1 , 2 , 3 , ... (5.14) where P k  − ~ e ∂  + P k  − ~ ∂ ∂ x 1 , − 1 2 ~ ∂ ∂ x 2 , − 1 3 ~ ∂ ∂ x 3 , ...  and P k ( t 1 , t 2 , t 3 , ... ) are Sch ur polynomials . Pro of. Equations (5.13) imply the existence of a function F such that j F j + ~ j − 1 X k =1 ∂ F j − k ∂ x k = − F x j . (5.15) Resolving (5.15) recur rently , one gets F 1 = − F x 1 , 2 F 2 = − F x 2 + ~ F x 1 x 1 , 3 F 3 = − F x 3 + 3 2 ~ F x 1 x 2 − 1 2 ~ 2 F x 1 x 1 x 1 and so on. The compact form of these relations is F k = 1 ~ P k  − ~ e ∂  F where Sch ur polynomia ls ar e defined , as usual, by the generating form ula exp  P ∞ k =1 λ k t k  = P ∞ k =0 λ k P k ( t ) . Then, in virtue of (5.10) , one has (5.14 )  . Substitution of the expres sions (5.14) for H j k int o the QCS (5 .2) gives the infinite system of different ia l e q uations,bilinear in F. The simplest o f them is 4 3 F x 1 x 3 − ~ 2 3 F x 1 x 1 x 1 x 1 − 2 ( F x 1 x 1 ) 2 − F x 2 x 2 = 0 . (5.16) 19 In terms of the function τ = exp F the a bove equation and equations (5.2) with H j k = 1 ~ P k  − ~ e ∂  τ x j τ , j, k = 1 , 2 , 3 , ... (5.17) are nothing but the famous bilinear Hirota equations for the KP hierarch y ( see e.g. [3 4 -36]). Hence, the function τ is the celebrated KP τ -function. Thu s, an y K P τ -function defines weakly (no n)asso ciative quantum defo r - mations of the structure constants (5.1) for the infinite-dimensiona l algebra by the formula (5.17) and Hirota bilinear equations. Quantum anomaly for these deformations is given by (5 .5) with ( A ) n k = 1 ~ ∞ X l = n +1 P l − n  − ~ e ∂  τ x k τ dx l . (5.18) A t last, in the limit ~ → 0 all the ab ove formulae are reduced to those for coisotr opic deformations [12]. W e e mpna size that quantum and isotropic deformations repres ent different deformations of the same structure constants (5.1). F or concrete solutions of the KP hierar ch y certain s tructure constants ma y remain undeformed and comp onents o f q ua ntu m anomaly may v anish. F or example, the function F giv e n by (4.14) with ǫ = 0 is the solution o f equa tion (5.16) to o. F or this solution a ll H j k with j ≥ 3 v anish a s well as A n k = 0 for n,k ≥ 3 . One soliton solution of the K P equation corres p o nds to τ = 1 + exp  k  x 1 + px 2 + q x 3  where q = ~ 2 4 k 2 + 3 4 p 2 and k,p are a rbitrary c onstants ( see e.g . [33- 36]). F or this soliton deformation H j k = 0 at j ≥ 4 and A n k = 0 for k,n ≥ 4 . A qua si-triangula r structure of the cons ta nt s (5.1) allows us to rewrite e qua- tions (2.5) in the eq uiv a lent form p n − p n 1 − n − 2 X m =1 u nm ( x ) p m 1 − u n 0 p 0 ! | Ψ i = 0 , n = 1 , 2 , 3 , ... (5.19) where the co efficients u nm are the certain functions of H j k . F or example, u 20 = − 2 H 1 1 , u 31 = − 3 H 1 1 , u 30 = H 1 2 + H 2 1 + 2 ∂ H 1 1 ∂ x 1 . In the co or dinate re pr esentation equations (2.24) due to (2.2 5) take the form − ~ 2 ∂ 2 e Ψ ∂ x j ∂ x k + ~ ∂ e Ψ ∂ x j + k + ~ j − 1 X l =1 H k j − l ∂ e Ψ ∂ x l + ~ k − 1 X l =1 H j k − l ∂ e Ψ ∂ x l +  H j k + H k j  e Ψ = 0 , j, k = 1 , 2 , 3 , ... (5.20) The system o f linea r equa tions (5.2 0) is equiv alent to the s tandard set of auxiliary linear pro blems f or the KP hierarch y 20 ~ ∂ e Ψ ∂ x n = ~ n ∂ n e Ψ ( ∂ x 1 ) n + n − 2 X m =0 ~ m u nm ( x ) ∂ m e Ψ ( ∂ x 1 ) m (5.21) that is the coo rdinate representation o f equations (5.1 9). T o g et a standa rd for m of the ab ov e formulae with a sp ectral para meter z one considers a formal Laurent series H ( j ) ( x, z ) + P ∞ 1 z − k H j k . In virtue of (5.15) one has H ( j ) = 1 ~ ∂ ∂ x j ( exp − ~ ∞ X n =1 z − n n ∂ ∂ x n ! − 1 ! F ) = 1 ~ ∂ ∂ x j log τ  x −  z − 1  τ ( x ) ! (5.22) where x −  z − 1  +  x 1 − 1 z , x 2 − 1 2 z 2 , x 3 − 1 3 z 3 , ...  is the Miw a shift [36]. Int ro- ducing th e wa ve function χ b y H ( j ) + 1 ~ ∂ log χ ∂ x j ,one obtains χ ( x, z ) = τ  x −  z − 1  τ ( x ) (5.23) that repro duces the standard form of the dr essed KP wav e-function e Ψ ( x, z ) = exp ∞ X n =1 z n x n ! χ ( x, z ) = exp ∞ X n =1 z n x n ! τ  x −  z − 1  τ ( x ) (5.24) in terms of the τ − function [36]. F or more details see [3 7]. It is w ell-known that the stationary r eductions of the KP hiera r ch y give rise to the Gelfand-Dick ey hier archies ( see e.g . [33 -35]). A t the same time one can s how that the stationa rity constra int ∂ H j k ∂ x N = 0 conv er ts the infinite- dimensional po lynomial algebra int o the finite-dimensiona l one. So, statio n- ary solutions of the K P hierarchy provide us with the weakly (non)asso cia tive quantum deformatio ns of the finite-dimensional algebras . The Bo us sinesq de- formation (4.8)-(4.12) is the simplest example. F or the g eneral Gelfand-Dic key case s e e also [18]. Finally , we would like to no te tha t the qua nt um deformations of algebra s obtained by the pro cess of gluening [1 2 ] of N algebr a s of the type (5 .1) are describ ed b y the N-comp onent KP hierarchy . A t last, in the limit ~ → 0 all the ab ove formulae are reduced to those for coisotr opic deformations [12]. W e e mpna size that quantum and isotropic deformations represent differen t defor mations of the same structure co ns tants (5.1). 6. Co nclusion The appro ach presented in the paper can b e e x tended in differen t directions. F or instance, the basic idea of identification of the ele men ts P j of the basis a nd deformations parameter s x j with the elements of the Heisenberg algebr a ca n be applied to other t yp e of algebras. 21 W e note also that the formula (2.10) gives a simple r ealization for previously discussed idea of geometrical interpretation of the a sso ciator for a algebra as a curv ature tensor (see e.g . [38] ). 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