On the deformation theory of structure constants for associative algebras

On the deformation theory of structure constan ts for asso ciativ e algebras B.G. Konop elc henk o Dipartimen to di Fisica, Univ ersita del Salen to and INFN, Se zione di Lecce, 73100 Lecce, Italy .e-mail:k onop el@le.infn.it Octob er 22, 2021 Abstract Algebraic sc heme for constructing deformations of structure constan ts for associative algebras generated by a deforma tion driving algebras (DDAs) is discussed. An ideal of l eft d ivisor s of zero plays a central role in th is construction. Deformations of associative three-dimensional alge bras with the D D A b eing a three-dimensional Lie algebra and t heir connection with integ rable systems a re studied. Mathematics Sub ject Classification. 16A58, 37K10 . Key w ords. Structure constants, deformations, divisors of zero, integrable systems. 1 In tro duction An idea to study deformations of structure constants for as sociative algebras go es back to the classical works of Gerstenhab er [1,2]. As one of the appr oac hes to deformatio n theory he sug g ested ” to take the p oint of view that the o b jects being deformed are not merely alge br as, but essentially a lgebra with a fixed basis” and to tr e a t ” the algebraic set of all structure consta n ts as parameter space for deformation theor y” [2]. Thu s, following this approa c h, o ne chooses the basis P 0 , P 1 , ..., P N for a given algebra A , takes the structure constants C n j k defined by the multiplication table P j P k = N X n =0 C n j k P n , j, k = 0 , 1 , ..., N (1) 1 and lo ok for their deforma tions C n j k ( x ) , where ( x ) = ( x 1 , ..., x M ) is the set of deformation parameter s , such that the ass o ciativity condition N X m =0 C m j k ( x ) C n ml ( x ) = N X m =0 C m kl ( x ) C n j m ( x ) (2) or simila r equation is satisfied. A r emark able ex ample of deformations of this type with M=N+1 has bee n discov ered by Witten [3] and Dijkgraaf-V erlinde-V erlinde [4]. They demo n- strated that the function F which defines the corr elation functions h Φ j Φ k Φ l i = ∂ 3 F ∂ x j ∂ x k ∂ x l etc in the defor med tw o-dimensional topolo gical field theory ob eys the asso ciativity equatio n (2) with the structure constants g iven by C l j k = N X m =0 η lm ∂ 3 F ∂ x j ∂ x k ∂ x m (3) where constants η lm = ( g − 1 ) lm and g lm = ∂ 3 F ∂ x 0 ∂ x l ∂ x m where the v ariable x 0 is asso ciated with the unite element. Each s olution of the WD VV equation (2 ), (3) desc r ibes a deformation of the structure constants o f the N+1- dimensional asso ciative algebra of primary fields Φ j . Int er pretation and for malization of the WD VV equa tion in terms of F rob e - nius manifolds pr opos e d by Dubrovin [5,6] provides us with a method to describ e class of defo r mations of the so-c alled F rob enius alge bras. An extension of this approach to genera l algebr a s and corres ponding F-manifolds has been given by Hertling and Manin [7]. Beautiful and rich theor y of F rob enius and F-ma nifo lds has v a r ious applica tions fro m the singula rit y theory to quant um cohomo lo gy (see e.g . [6,8,9] ). An alternative a pproach to the deformation theory o f the structure con- stants for commutativ e a ssocia tiv e algebr as has b een prop osed r ecen tly in [10- 14]. Within this metho d the deformatio ns o f the structure constants ar e g o v- erned b y the so-ca lled central system (CS) . Its concrete form depends on the class of deformations under consideration and CS con tains, as particu- lar reductions, many integrable s ystems like WD VV equa tion, oriented a sso- ciativity equation, integrable disp ersionless , disper siv e a nd discr e te equations (Kadomtsev-Petviash vili eq ua tion etc). The co mmon feature o f the coisotr o pic, quantum, discr ete deformations co nsidered in [10-1 4] is tha t for all of them el- ement s p j of the basis and deformation parameter s x j form a certain algebr a ( P o isson, Heisenberg etc). A general cla ss of deformations co nsidered in [13] is characterized by the condition that the ideal J = < f j k > g enerated by the elements f j k = − p j p k + P N l =0 C l j k ( x ) p l representing the multiplication table (1) is closed. It was shown that this cla ss co n tains a s ub class of so-c a lled integrable deformations for which the CS has a simple and nice geometr ical mea ning. In the present pap er we will discuss a pure alg ebraic formulation of such int eg rable deformations. W e will co nsider the case when the alg ebra generating deformations of the structure consta n ts, i.e. the algebra formed by the elements p j of the basis and deformation par ameters x k ( deforma tio n driving alg ebra 2 (DD A)), is a Lie alg ebra. The bas ic idea is to r e q uire that a ll elements f j k = − p j p k + P N l =0 C l j k ( x ) p l are left div is ors of zer o and that they gener ate the ideal J = < f j k > of left div is ors o f zero. This requir emen t gives ris e to the central system whic h governs deforma tions g enerated by DDA. Here we will study the defor mations of the structure consta nts for the three- dimensional algebra in the case when the DDA is given by one of the three- dimensional Lie algebras. Such defo rmations are par ametrized by a sing le de- formation v ar iable x . Depending on the choice of DDA and iden tification of p 1 , p 2 and x with the elements of DDA, the corresp onding CS takes the form of the s ystem of ordinary differential equations or the s ystem of discre te equa- tions (multi-dimensional mappings). In the first c ase the CS contains the third order ODEs from the Chazy-Bur e au list as the particular ex a mples. This ap- proach provides us also with the La x form of the a bov e equatio ns and their first int eg rals. The pap er is or ganized as follows. General formulation o f the de fo rmation theory for the structure co nstan ts is presented in sectio n 2. Quantum, discre te and coisotro pic defor mations ar e discussed in section 3 . Three-dimensiona l Lie algebras as DD As a re analy zed in section 4. Defor mations generated b y gen- eral DD As ar e studied in section 5. Defor mations dr iv en by the nilpotent and solv able DDAs a re c onsidered in sectio ns 6 and 7 , res pectively . 2 Deformations of the structure constan ts gen- erated b y DD A So, we consider a finite-dimensiona l nonco mm utative algebra A with ( or with- out ) unite element P 0 . W e will restrict ov er self to a clas s o f alge bras which po ssess a basis comp osed by pairwis e co mm uting elements P 0 , P 1 , ..., P N . The table of multiplication (1) defines the struc tur e cons tan ts C l j k . The commuta- tivit y of the bas is implies that C l j k = C l kj . In the presence of the unite element one has C l j 0 = δ l j where δ l j is the K r onek er s ym b ol. F ollowing the Gerstenhab er’s suggestion [1,2] we will trea t the structur e constants C l j k in a g iv en basis as the o b jects to de fo rm and will denote the deformation para meters by x 1 , x 2 , ..., x M . F or the undeformed structure con- stants the asso ciativity conditions (2) are nothing else than the co mpatibilit y conditions for the table of m ultiplication (1). In the construction of deforma- tions we sho uld first to sp ecify a ”defor med ” v er sion o f the mult iplica tio n table and then to r equire that this realiza tio n is selfco nsistence and meaningful. Thu s, to define deforma tions we 1) a ssocia te a s e t of elements p 0 , p 1 , ..., p N , x 1 , x 2 , ..., x M with the elemen ts of the bas is P 0 , P 1 , ..., P N and defo r mation para meters x 1 , x 2 , ..., x M , 2) cons ider the L ie algebr a B o f the dimension N+M+1 with the basis ele- men ts e 1 , ..., e N + M +1 ob eying the commutation rela tions 3 [ e α , e β ] = N + M +1 X γ =1 C αβ γ e γ , α, β = 1 , 2 , ..., N + M + 1 , 3) iden tify the elemen ts p 0 , p 1 , ..., p N , x 1 , x 2 , ..., x M with the elemen ts e 1 , ..., e N + M +1 th us defining the deformation driving alg e br a (DDA). Different ident ifica tio ns define differ en t DDAs. W e will a ssume that the element p 0 is always a central element of DDA . The commutativit y of the basis in the alg e bra A implies the commutativit y b et ween p j and in this pap er we assume the s a me prop erty for all x k . So, we will consider the DDAs defined by the commutation r elations of the t yp e [ p j , p k ] = 0 ,  x j , x k  = 0 , [ p 0 , p k ] = 0 ,  p 0 , x k  = 0 ,  p j , x k  = X l α k j l x l + X l β kl j p l (4) where α k j l and β kl j are so me constants, 4) consider the e lemen ts f j k = − p j p k + N X l =0 C l j k ( x ) p l , j, k = 0 , 1 , ..., N of the universal en veloping algebra U( B ) of the algebra DD A( B ). These f j k ”represe n t” the table (1) in U( B ). Note that f j 0 = f 0 j = 0 . 5) requir e the all f j k are non-zero left divisors of zer o and hav e a common right zero divisor. In this case f j k generate the left idea l J = < f j k > of left divisor s of zero. W e remind that non-zero elements a and b a re ca lled left and rig h t diviso rs of zero if ab = 0 (see e.g. [15 ]). Definition. The structure constants C l j k ( x ) ar e said to define deforma tions of the algebra A genera ted by given DDA if all f j k are left zero divisors w ith common right z ero divisor. T o justify this definition we first observe tha t the s implest p ossible r ealization of the multiplication ta ble (1) in U( B ) given by the equations f j k = 0 , j, k = 0 , 1 , ..., N is too restr ictiv e in general. Indeed, for instance, for the Heisenberg algrebr a B [12] such equations imply that [ p l , C m j k ( x )] = 0 and , hence, no deformations are allow ed. So, o ne sho uld lo ok for a weak er realization of the m ultiplicatio n table. A condition that a ll f j k are just non-zer o divisors of zero is a na tural candidate. Then, the co ndition of compatibility of the corr esponding equations f j k · Ψ j k = 0 , j, k = 1 , ..., N wher e Ψ j k are right zero div isors requir es that the l.h.s. of these equa tio ns and, hence, Ψ j k should hav e a common divisor (see e.g . [15] ). W e r estrict ours e lf to the case when Ψ j k = Ψ · Φ j k , j, k = 1 , ..., N where Φ j k are inv ertible elements of U(B). I n this case one has the co mpa tible set o f equations f j k · Ψ = 0 , j, k = 0 , 1 , ..., N (5) 4 that is a ll left zero divisors f j k hav e common r igh t zero divis o r Ψ. These conditions imp ose constraints on C m j k ( x ). T o cla rify these co nstrain ts we will use the basic pro perty of the a lg ebra A , i.e. its asso ciativity . Firs t we observe that due to the r elations (4) one has the iden tity [ p l , C m j k ( x )] = N X t =0 ∆ mt j k,l ( x ) p t where ∆ mt j k,l ( x ) are cer tain functions of x 1 , ..., x M only . Then, ta king in to ac- count (4 ), one obtains the identit y ( p j p k ) p l − p j ( p k p l ) = N X s,t =0 K st klj · f st + N X t =0 Ω t klj ( x ) · p t , j, k , l = 0 , 1 , ..., N (6) where K st klj = 1 2 ( δ s k δ t l + δ t k δ s l ) p j − 1 2 ( δ s k δ t j + δ t k δ s j ) p l + 1 2 ( δ s j C t kl + δ t j C s kl ) − 1 2 ( δ s l C t kj + δ t l C s kj )+∆ st kl,j − ∆ st kj,l and Ω t klj ( x ) = X s C s j k C t ls − X s C s lk C t j s + X s,n (∆ sn kj,l − ∆ sn kl,j ) C t sn . The ident ity (6) implies that for an asso ciative a lgebra N X s,t =0 K st klj · f st + N X t =0 Ω t klj ( x ) · p t = 0 , j, k, l = 0 , 1 , ..., N . (7) Due to the rela tions (5) equations (7) imply that N X t =0 Ω t klj ( x ) · p t ! Ψ = 0 . These equations are sa tisfied if Ω t klj ( x ) = X s C s j k C t ls − X s C s lk C t j s + X s,n (∆ sn kj,l − ∆ sn kl,j ) C t sn = 0 , j, k, l , t = 0 , 1 , .., N . (8) This system of equations plays a central role in our approach. If Ψ has no left zero divisors linear in p j and U(B) has no zero e lemen ts linear in p j then the relation (8) is the necessa ry condition for ex istence o f a commo n right zero divisor fo r f j k . 5 A t N ≥ 3 it is also a sufficien t condition. I ndee d, if C m j k ( x ) are suc h that equations (8 ) a re satisfied then N X s,t =0 K st klj · f st = 0 , j, k , l = 0 , 1 , ..., N . (9) Generically , it is the system of 1 2 N 2 ( N − 1 ) linear equations for N ( N +1) 2 unknowns f st with noncommuting co efficien ts K st klj . At N ≥ 3 for generic (non zeros, non zer o divisor s) K st klj ( x, p ) the sy stem (9) implies that α j k f j k = β lm f lm , j, k , l, m = 1 , ..., N (10) and γ j k f j k = 0 , j, k = 1 , ..., N (11) where α j k , β lm , γ j k are certain elements of U(B) ( see e.g. [16,17] ). Th us, all f j k are right zero divisors. They are also left zero divisors. Indeed, due to Ado’s theorem ( see e.g. [1 8] ) finite-dimensiona l Lie algebra B and, hence, U(B) ar e isomorphic to ma trix a lgebras. F or the matrix algebras zero diviso rs ( matrices with v anishing deter minan ts) ar e b oth right and left zero divis ors [15]. Then, under the a ssumption that all α j k and β lm are not ze ro divisor s , the relations (10) imply that the r igh t divisor of one of f j k is also the right zero divisor for the others. A t N=2 one has only t wo r e la tions of the type (10 ) and a rig h t z ero divisor of one o f f 11 , f 12 , f 22 is the right zer o divis o r o f the other s. W e note that it isn’t that easy to control as sumptions mentioned ab o ve. Nevertheless, the eq uations (5) a nd (8 ) cer tainly are fundament a l o ne for the whole approach. W e shall r efer to the sys tem (8) as the Central System (CS) governing defor- mations of the structure constants of the algebr a A generated by a given DD A. Its concrete form depends strongly on the form of the brack ets h p t , C l j k ( x ) i which are defined b y the relations (4) for the elements o f the basis of DD A. F or stationary so lutions (∆ t j k,l = 0) the CS (8) is reduced to the asso ciativity conditions (2). 3 Quan tum, discrete and coisotropic deforma- tions. Coisotro pic , quantum and discrete deformatio ns of asso ciative a lgebras cons id- ered in [10-1 4] repres en t particular rea lizations of the abov e genera l scheme asso ciated with different DDAs. F or the quan tum deformations of nonco mm utative algebra one has M = N and the deformation driving algebra is given by the Heisenberg algebr a [12]. 6 The elements of the basis of the alge bra A a nd defor ma tions parameters are ident ified with the elements of the Heisenberg alg e bra in such a way that [ p j , p 0 ] = 0 ,  p 0 , x k  = 0 , [ p j , p k ] = 0 ,  x j , x k  = 0 ,  p j , x k  = ~ δ k j p 0 , j, k = 1 , ..., N (12) where ~ is the (Pla nc k’s) co nstan t. F or the Heisenberg DDA ∆ mt j k,l = ~ δ t 0 ∂ C m j k ( x ) ∂ x l (13) and co nsequen tly Ω n klj ( x ) = ~ ∂ C n j k ∂ x l − ~ ∂ C n kl ∂ x j + N X m =0 ( C m j k C n ml − C m kl C n j m ) = 0 , j, k, l , n = 0 , 1 , ..., N . (14) Quantum CS (14 ) gov er ns deformations of structure constants for asso ciative algebra driven by the Heisenber g DD A. It has a simple geometrica l meaning of v anishing Riemann curv a ture tenso r for to rsionless Christoffel symbols Γ l j k ident ified with the structure consta n ts ( C l j k = ~ Γ l j k ) [12]. In the representation of the Heisenberg a lgebra (1 2) by op erators acting in a linear spa ce H left divisors of zero a re re a lized b y op erator s with nonempt y kernel. The ideal J is the left ideal gener ated b y op erators f j k which ha ve nontrivial c ommon kernel or , equiv alently , for which equations f j k | Ψ i = 0 , j, k = 1 , 2 , ..., N (15) hav e nontrivial common solutio ns | Ψ i ⊂ H . The compatibility condition for equations (15) is g iv en by the CS (1 4). The co mmon kernel of the op erators f j k form a subspace H Γ in the linear spa c e H . So , in the a ppr oach under consideratio n the multiplication table (1) is realized o nly o n H Γ , but not on the who le H . Suc h t yp e of rea lization of the constra in ts is well-kno wn in quantum theory a s the Dira c’s recip e for quantization of the first-cla ss co ns train ts [1 9]. In qua ntum theory co n text equatio ns (15) serve to sele c t the physical subspace in the w ho le Hilb ert spa ce. Within the deforma tio n theory o ne ma y re fer to the subspace H Γ as the ”structure consta n ts” subspace. In [12 ] the r ecipe (1 5 ) was the starting p oint for construction of the quantum deformations. Quantum CS (14) contains v ar io us classes of solutions which descr ibe differ- ent cla sses of deformations. An imp ortant sub class is given by iso- asso ciativ e deformations, i.e. by defor ma tions for which the asso ciativit y condition (2) is v a lid for all v a lues of deformation parameters . F or such qua n tum defor mations the structure constants sho uld ob ey the eq ua tions ∂ C n j k ∂ x l − ∂ C n kl ∂ x j = 0 , j, k , l, n = 0 , 1 , ..., N . (16) These equations imply that C n j k = ∂ 2 Φ n ∂ x j ∂ x k where Φ n are some functions while the asso ciativity condition (2) takes the form 7 N X m =0 ∂ 2 Φ m ∂ x j ∂ x k ∂ 2 Φ n ∂ x m ∂ x l = N X m =0 ∂ 2 Φ m ∂ x l ∂ x k ∂ 2 Φ n ∂ x m ∂ x j . (17) It is the oriented as socia tiv ity equatio n intro duced in [20 ,5 ]. Under the gr adi- ent reductio n Φ n = P N l =0 η nl ∂ F ∂ x l equation (18) b ecomes the WDVV equation (2),(3). Non iso-asso ciative deformations for whic h the condition (16) is not v alid are of interest to o. They a re des c r ibed by some well-known integrable soliton e qua- tions [12]. In particular , there are the B oussinesq equation among them for N=2 and the K adom tsev-Petviashvili (KP) hierar c hy for the infinite-dimensional al- gebra of polyno mials in the F aa’ de Bruno basis [12]. In the latter case the deformed structure constants are g iv en by C l j k = δ l j + k + H k j − l + H j k − l , j, k , l = 0 , 1 , 2 ... (18) with H j k = 1 ~ P k  − ~ e ∂  ∂ lo g τ ∂ x j , j, k = 1 , 2 , 3 , ... (19) where τ is the fa mo us tau-function for the K P hierarch y and P k  − ~ e ∂  + P k  − ~ ∂ ∂ x 1 , − 1 2 ~ ∂ ∂ x 2 , − 1 3 ~ ∂ ∂ x 3 , ...  where P k ( t 1 , t 2 , t 3 , ... ) are Schur po lynomials defined b y the genera ting for m ula exp  P ∞ k =1 λ k t k  = P ∞ k =0 λ k P k ( t ) . Discrete deformations of noncomm utative asso ciative alg e bras are g en- erated b y the DDA with M = N a nd co mm utation relations [ p j , p k ] = 0 ,  x j , x k  = 0 ,  p j , x k  = δ k j p j , j, k = 1 , ..., N . (20) In this ca s e ∆ mt j k,l = δ t l ( T l − 1) C m j k ( x ) , j, k , l, m, t = 0 , 1 , 2 , ..., N (21) where for an arbitr ary function ϕ ( x ) the action of T j is defined b y T j ϕ ( x 0 , ..., x j , ..., x N ) = ϕ ( x 0 , ..., x j + 1 , ...., x N ) . The corresp onding CS is of the form C l T l C j − C j T j C l = 0 , j, l = 0 , 1 , ..., N (22) where the matrices C j are defined as ( C j ) l k = C l j k , j, k , l = 0 , 1 , ..., N . The dis- crete CS (22 ) gov erns discr ete deformations of asso ciative a lgebras. The CS (22) contains, as par ticular cases, the dis c r ete versions of the or ie n ted asso ciativity equation, WD VV equa tion, Bo ussinesq e q uation, dis c r ete KP hier arch y and Hirota-Miwa bilinear equations for K P τ -function. F or coisotropi c deform ations of commutativ e alg e bras [10,11 ] aga in M = N , but the DDA is the Poisson algebra with p j and x k ident ified with the Darb oux co ordinates, i.e. 8 { p j , p k } = 0 , { x j , x k } = 0 , { p j , x k } = − δ k j , j, k = 0 , 1 , ..., N . (23) where { , } is the standard P ois son brack et. The algebra U( B ) is the commutativ e ring of functions a nd divis o rs of zer o ar e realiz e d by functions with zeros. So, the functions f j k should be functions with common set Γ of zer os. Thus, in the coisotro pic case the multiplication table (1) is realized by the set of equations [10] f j k = 0 , j, k = 0 , 1 , 2 , ..., N . (24) W ell-known co mpatibilit y conditon fo r these equations is { f j k , f nl } | Γ = 0 , j, k, l , n = 1 , 2 ..., N . (25) The set Γ is the co isotropic submanifold in R 2( N +1) . The condition (25) gives rise to the following system o f equations for the structure constants [ C, C ] m j klr + N X s =0 ( 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 (26) while the equa tions Ω n klj ( x ) = 0 have the form of as s ociativity conditions (2) Ω n klj ( x ) = N X m =0 ( C m j k ( x ) C n ml ( x ) − C m kl ( x ) C n j m ( x )) = 0 . (27) Equations (26) and (27) form the CS for coisotr opic defor ma tions [10]. In this ca se C l j k is tr ansformed as the tensor of the type (1,2 ) under the genera l tranformations o f co ordina tes x j and the whole CS (26), (27) is inv aria n t under these tranfor mations [14]. The bracket [ C, C ] m j klr has app eared for the fir st time in the pap er [21] where the co-called differential concomitants were studied. It was shown in [18 ] that this bra c ket is a tensor only if the tenso r C l j k ob eys the algebra ic constraint (27 ). In the pap er [7] the CS (26), (27) has a pp eared implicitly as the system of equations which characterizes the structure co nstan ts for F-manifolds. In [1 0] it has b een derived as the CS governing the coisotro pic deformations of asso ciative alg e br as. The CS (26 ), (27) contains the oriented asso ciativity equa tion, the WDVV equation, disp ersionless KP hierarch y a nd equatio ns from the genus zero uni- versal Whitham hier arch y as the pa rticular cases [10,11 ]. Y ano manifolds and Y ano alg ebroids asso ciated with the CS (26 ),(2 7 ) are s tudied in [14]. W e w o uld like to emphasize that for all deformations considered ab o ve the stationary solutions of the CSs o bey the global a ssocia tivit y conditio n (2). 9 4 Three-dimensional L ie algebras as DD A. In the rest o f the pap er w e will study deformations of asso ciative a lgebras gener- ated by three-dimensional r eal Lie algebra L . The complete list of such algebras contains 9 alge br as (see e.g. [1 8]). Denoting the basis elements by e 1 , e 2 , e 3 , one has the fo llo wing no nequiv alent cases: 1) ab elian alg ebra L 1 , 2) general algebra L 2 : [ e 1 , e 2 ] = e 1 , [ e 2 , e 3 ] = 0 , [ e 3 , e 1 ] = 0, 3) nilp o ten t algebra L 3 : [ e 1 , e 2 ] = 0 , [ e 2 , e 3 ] = e 1 , [ e 3 , e 1 ] = 0, 4)-7) four nonequiv a len t so lv able algebras : [ e 1 , e 2 ] = 0 , [ e 2 , e 3 ] = αe 1 + β e 2 , [ e 3 , e 1 ] = γ e 1 + δ e 2 with αδ − β γ 6 = 0, 8)-9) simple algebra s L 8 = s o(3) and L 9 =so(2,1). In vir tue of the o ne to one co rresp o ndence betw een the elemen ts of the bas is in DD A and the elements p j , x k an a lgebra L should has a n ab elian subalgebra and only one its element may play a role of the deformation pa rameter x . F or the original algebra A and the algebra B one has tw o options: 1) A is a tw o- dimensional algebra without unite element a nd B = L 2) A is a three-dimensional algebra with the unite element and B = L 0 ⊕ L where L 0 is the a lg ebra genera ted by the unite element p 0 . After the choice of B o ne should establis h a co r respo ndence b etw een p 1 , p 2 , x and e 1 , e 2 , e 3 defining DDA. F or ea c h algebra L k there are obviously , in gener al, six p o ssible identifi ca tions if one avoids linear s uperp ositions. Some o f them ar e equiv a len t. The incomplete list of no nequiv alent identifications is: 1) algebra L 1 : p 1 = e 1 , p 2 = e 2 , x = e 3 ; DD A is the commutativ e algebra with [ p 1 , p 2 ] = 0 , [ p 1 , x ] = 0 , [ p 2 , x ] = 0 . (28) 2) algebra L 2 : case a) p 1 = − e 2 , p 2 = e 3 , x = e 1 ; the co rresp onding DD A is the algebra L 2 a with the comm utatio n relations [ p 1 , p 2 ] = 0 , [ p 1 , x ] = x, [ p 2 , x ] = 0 , (29) case b) p 1 = e 1 , p 2 = e 3 , x = e 2 ; the corresp onding DD A L 2 b is defined by [ p 1 , p 2 ] = 0 , [ p 1 , x ] = p 1 , [ p 2 , x ] = 0 . (30) 3) algebra L 3 : p 1 = e 1 , p 2 = e 2 , x = e 3 ; DD A L 3 is [ p 1 , p 2 ] = 0 , [ p 1 , x ] = 0 , [ p 2 , x ] = p 1 , (31) 4) solv a ble algebr a L 4 with α = 0 , β = 1 , γ = − 1 , δ = 0 : p 1 = e 1 , p 2 = e 2 , x = e 3 ; DD A L 4 with [ p 1 , p 2 ] = 0 , [ p 1 , x ] = p 1 , [ p 2 , x ] = p 2 , (32) 10 5) s o lv able a lgebra L 5 at α = 1 , β = 0 , γ = 0 , δ = 1 : p 1 = e 1 , p 2 = e 2 , x = e 3 ; DDA L 5 is [ p 1 , p 2 ] = 0 , [ p 1 , x ] = p 1 , [ p 2 , x ] = − p 2 . (33) F or the s econd c ho ic e o f the algebr a B = L 0 ⊕ L mentioned above the table of m ultiplicatio n (1) consis ts from the trivial part P 0 P j = P j P 0 = P j , j = 0 , 1 , 2 and the no ntrivial part 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 , (34) P 2 2 = L P 0 + M P 1 + N P 2 . F or the first choice B = L the m ultiplication table is given by (34) with A=D=L=0. It is co n venien t also to arrange the structure constants A,B,...,N in to the matrices C 1 , C 2 defined b y ( C j ) l k = C l j k . One has C 1 =   0 A D 1 B E 0 C G   , C 2 =   0 D L 0 E M 1 G N   . (35) In terms o f these matr ic es the as socia tivity conditions (2) are written as C 1 C 2 = C 2 C 1 . (36) 5 Deformations ge nerated b y general DD As 1. Commutativ e DDA (28 ) obviously do es not generate any defo rmation. So, we b egin with the three-dimensional co mm utative algebra A and DD A L 2 a defined b y the comm utatio n relations (29). These re la tions imply that for an arbitrar y function ϕ ( x ) [ p j , ϕ ( x )] = ∆ j ϕ ( x ) , j = 1 , 2 (37) where ∆ 1 = x ∂ ∂ x , ∆ 2 = 0. Consequently , one has the fo llo wing CS Ω n klj ( x ) = ∆ l C n j k − ∆ j C n kl + 2 X m =0 ( C m j k C n lm − C m kl C n j m ) = 0 , j, k, l , n = 0 , 1 , 2 . (38) In terms o f the matrice s C 1 and C 2 defined ab ov e this CS ha s a form of the Lax equation x ∂ C 2 ∂ x = [ C 2 , C 1 ] . (39) 11 The CS (39) has a ll remark able standard pr operties of the Lax equations (see e.g . [20,21]): it has three indep endent first in teg r als I 1 = trC 2 , I 2 = 1 2 tr ( C 2 ) 2 , I 3 = 1 3 tr ( C 2 ) 3 (40) and it is eq uiv alent to the co mpatibilit y condition of the linear problems C 2 Φ = λ Φ , x ∂ Φ ∂ x = − C 1 Φ (41) where Φ is the column with three compo nen ts and λ is a spectra l parameter . Though the evolution in x desc r ibed b y the second linear problem (4 1) is too simple, nevertheless the CS (38) or (39) have the meaning of the iso-sp ectral deformations of the ma trix C 2 that is typical to the class of in tegr able systems (see e.g . [22,23]). CS (3 9 ) is the system of six equations for the structur e c onstan ts D,E,G,L,M,N with free A,B,C: D ′ = D B + L C − AE − D G, L ′ = D E + LG − AM − DN , E ′ = M C − E G − D , M ′ = E 2 + M G − B M − E N − L, (42) G ′ = GB + N C − C E − G 2 + A, N ′ = GE − C M + D where D ′ = x ∂ D ∂ x etc. Here we will consider only simple pa rticular cases of the CS (42). Fir st corre sponds to the cons tr ain t A=0 , B=0, C=0, i.e. to the nilpo ten t P 1 . The corre sponding solution is D = β ln x , E = − β + γ ln x , G = 1 ln x , L = αβ + 2 β 2 + δ ln x − β γ ln x , M = αγ + 3 β γ + µ ln x − δ (ln x ) 2 − γ 2 ln x , N = α + β − γ ln x (43) where α, β , γ , δ , µ a re arbitrary c onstan ts. The three integrals for this solution are I 1 = α , I 2 = 1 2 α 2 +3 β 2 +2 αβ + µ, I 3 = 1 3 (( α + β ) 3 − β 3 )+( α + β )( µ + β ( α +2 β )) − γ δ . (44) The second example is given by the constraint B=0 , C=1, G=0 for which the quantum CS (14) is equiv alent to the Boussinesq equation [12]. Under this constraint the CS (42 ) is reduced to the s ingle equation 12 E ′′ − 6 E 2 + 4 αE + β = 0 (45) and the o ther structure constants ar e given by A = 2 E − α, B = 0 , C = 1 , D = γ − 1 2 E ′ , G = 0 , L = − E 2 + αE + 1 2 β , M = γ + 1 2 E ′ , N = α − N (46) where α, β , γ are ar bitrary consta n ts. The cor respo nding fir st integrals are I 1 = α, I 2 = 1 2 ( β + α 2 ) , I 3 = 1 3 α 3 + γ 2 + 1 2 αβ − 1 4 ( E ′ ) 2 + E 3 − αE 2 − 1 2 β E . (4 7 ) Int eg ral I 3 repro duces the well-kno wn firs t integral of equation (45 ). Solutions of eq uation (45) a re given by elliptic integrals (see e.g. [24]). An y such solution together with the for m ulae (46) des c r ibes deforma tio n of the three-dimensiona l algebra A dr iv en b y DDA L 2 a . Now w e will co nsider deformations of the tw o- dimensional algebra A without unite elemen t according to the first o ption mentioned in the previo us s ection. In this ca s e the CS has the form (39 ) with the 2 × 2 matrices C 1 =  B E C G  , C 2 =  E M G N  (48) or in co mp onents E ′ = M C − E G, M ′ = E 2 + M G − B M − E N , G ′ = GB + N C − C E − G 2 , N ′ = GE − C M . (49) In this ca s e there are t wo indep endent integrals o f mo tion I 1 = E + N , I 2 = 1 2 ( E 2 + N 2 + 2 M G ) . (50) The cor respo nding sp ectral problem is given by (41). Eigenv alues of the matrix C 2 , i.e. λ 1 , 2 = 1 2 ( E + N ± p ( E − N ) 2 + 4 GM ) are in v aria nt under deformations and det C 2 = 1 2 I 2 1 − I 2 . W e note also an obviously inv aria nc e o f equations (4 2 ) and (49) under the rescaling of x . The system of equations (49) contains tw o arbitrary functions B and C. In virtue of the p ossible r escaling P 1 → µ 1 P 1 , P 2 → µ 2 P 2 of the basis for the alg ebra A with t wo arbitr a ry functions µ 1 , µ 2 , one has fo ur nonequiv alent choices 1 ) B=0 , C=0 , 2) B=1, C=0, 3) B= 0, C=1, 4) B=1, C=1. In the case B=0, C=0 ( nilp oten t P 1 ) the solution o f the system (49) is 13 B = 0 , C = 0 , E = β ln x , G = 1 ln x , M = γ ln x − β 2 ln x + αβ , N = − β ln x + α (51) where α , β , γ ar e ar bitrary constants. F or this solution the integrals are equal to I 1 = α, I 2 = γ + 1 2 α 2 and λ 1 , 2 = 1 2 ( α + p α 2 + 4 γ ). A t B=1 , C=0 the system (49) ha s the following solutio n B = 1 , C = 0 , E = γ x + β , G = x x + β , M = δ + ( αγ + β δ − γ 2 β ) 1 x + γ 2 β ( x + β ) , N = − γ x + β + α (52) where α, β , γ , δ are arbitra ry constants. The integrals a re I 1 = α, I 2 = δ + 1 2 α 2 . The formulae (51), (5 2 ) provide us with explicit defo r mations of the s tructure constants. In the last tw o cases the CS (49) is equiv alent to the s imple third order ordinary differential equations . At B=0, C=1 with additional constraint I 1 = 0 one gets G ′′′ + 2 G 2 G ′ + 4( G ′ ) 2 + 2 GG ′′ = 0 (53) while at B = 1,C=1 and I 1 = 0 the s ystem (49) be c omes G ′′′ + 2 G 2 G ′ + 4( G ′ ) 2 + 2 GG ′′ − G ′ = 0 . (54) The se c o nd integral for these ODEs is I 2 = − 1 2 G 4 + 1 2 ( G ′ ) 2 − 2 G 2 G ′ − GG ′′ + 1 2 B G 2 . (55) Equation (53 ) with G ′ = ∂ G ∂ y is the Chazy V equation fro m the well-known Chazy-Burea u list of the third order ODEs having Painlev e pr operty [25,26 ]. The integral (55) is known to o (see e.g. [27]). The app earance of the Chazy V equation among the par ticular cases o f the system (49) indicates that for other c hoic e s of B and C the CS (49) may b e equiv a len t to the other no table third order ODEs. It is really the case. Here we will consider o nly the reductio n C=1 with I 1 = N + E = 0 . In this case the system (49) is r educed to the following equation G ′′′ + 2 G 2 G ′ + 4( G ′ ) 2 + 2 GG ′′ − 2 G ′ Φ − G Φ ′ = 0 (56) where Φ = B ′ + 1 2 B 2 .The se c o nd integral is I 2 = − 1 2 G 4 + 1 2 ( G ′ ) 2 − 2 G 2 G ′ − GG ′′ + Φ G 2 . (57) and λ 1 , 2 = ± q I 2 2 . 14 Cho osing pa r ticular B or Φ, one gets equatio ns from the Chazy-Bureau list. Indeed, at Φ = 0 one has the Chazy V equa tion (53). Cho osing Φ = G ′ , one gets the C ha zy VI I equa tion G ′′′ + 2 G 2 G ′ + 2( G ′ ) 2 + GG ′′ = 0 . (58) A t B=2 G equa tion (56) b ecomes the Cha zy VI II equation G ′′′ − 6 G 2 G ′ = 0 . (59) Cho osing the function Φ such that  6Φ e 1 3 G  ′ = 2 G 2 G ′ + ( G ′ ) 2 + 4 GG ′′ , (6 0 ) one gets the Chazy II I equation G ′′′ − 2 GG ′′ + 3( G ′ ) 2 = 0 . (61) In the a bov e pa rticular cases the int eg ral I 2 (57) is r educed to those given in [27]. All Chazy equations pr esen ted ab ov e hav e the Lax repr esen tation (39) with E = − N = − 1 2 ( G ′ + G 2 + GB ) , M = − 1 2 ( G ′′ + 3 GG ′ + G 3 + G 2 B + ( GB ) ′ ) , C = 1 and the pr o per choice of B. Solutions of all these Cha zy equations provide us with the deformations o f the structure constants (48) for the t wo-dimensional a lgebra A genera ted by the DD A L 2 a . 2.Now we pa ss to the DD A L 2 b . The commutation re lations (3 0) imply that [ p 1 , ϕ ( x )] = ( T − 1) ϕ ( x ) · p 1 , [ p 2 , ϕ ( x )] = 0 (62) where ϕ ( x ) is an arbitr ary function a nd T ϕ ( x ) = ϕ ( x + 1) . Using (62), o ne finds the corresp onding CS 2 X m =0 ((∆ l + 1) C m j k ( x ) · C n lm ( x ) = = (∆ j + 1) C m kl ( x ) · C n j m ( x )) , j, k , l, n = 0 , 1 , 2 (63) where ∆ 1 = T − 1 , ∆ 2 = 0 . In ter ms of the matrices C 1 and C 2 this CS is C 1 T C 2 = C 2 C 1 . (64) F or no ndegenerate matrix C 1 one has T C 2 = C − 1 1 C 2 C 1 . (65) The CS (65) is the discrete version of the Lax equation (39) a nd has similar prop erties. It ha s three independent first integrals 15 I 1 = trC 2 , I 2 = 1 2 tr ( C 2 ) 2 , I 3 = 1 3 tr ( C 2 ) 3 (66) and r epresent s itself the c ompatibilit y condition for the linear problems Φ C 2 = λ Φ , T Φ = Φ C 1 . (67) Note that det C 2 is the fir s t int eg ral to o. The CS (64 ) is the discr e te dynamical sy s tem in the space of the s tructure constants. F or the tw o-dimensiona l algebra A with matrices (48) it is B T E + E T G = E B + M C, B T M + E T N = E 2 + M G, C T E + GT G = B G + C N , (68) C T M + GT N = E G + N G where B and C a r e arbitrary functions. F or no ndegenerate matrix C 1 , i.e. at B G − C E 6 = 0 , o ne has the resolved form (65), i.e . T E = GM − E N B G − C E C, T G = B + B N − C M B G − C E C, T M = GM − E N B G − C E G, T N = E + B N − C M B G − C E G. (69) This s y stem defines discrete deformations of the structure constants. 6 Nilp oten t DD A F or the ni lp oten t DD A L 3 , in virtue o f the defining rela tions (3 2), one has [ p 1 , ϕ ( x )] = 0 , [ p 2 , ϕ ( x )] = ∂ ϕ ∂ x · p 1 (70) or [ p j , ϕ ( x )] = ∂ ϕ ∂ x · 2 X k =1 a j k p k (71) where a 21 = 1 , a 11 = a 12 = a 22 = 0. Using (71), one g ets the following CS 16 2 X q =1 a lq 2 X m =0 C n qm ∂ C m j k ∂ x − 2 X q =1 a j q 2 X m =0 C n qm ∂ C m kl ∂ x + + 2 X m =0 ( C m j k C n lm − C m kl C n j m ) = 0 , j, k , l , n = 0 , 1 , 2 . (72) In the matrix form it is C 1 ∂ C 1 ∂ x = [ C 1 , C 2 ] . (73) F or invertible matrix C 1 ∂ C 1 ∂ x = C − 1 1 [ C 1 , C 2 ] . (74) This s y stem of ODEs has three indep enden t first int eg rals I 1 = tr C 1 , I 2 = 1 2 tr ( C 1 ) 2 , I 3 = 1 3 tr ( C 1 ) 3 . (75 ) and eq uiv alent to the compatibility condition for the linear system C 1 Φ = λ Φ , C 1 ∂ Φ ∂ x + C 2 Φ = 0 . (76) So, a s in the pr evious section the CS (73 ) describ es is o-spectr al defor mations of the matrix C 1 . This CS gov er ns deformations generated by L 3 . F or the t wo-dimensional algebra A without unite elemen t the CS is given by equation (73) with the matrices (48). First in tegr als in this case are I 1 = B + G, I 2 = 1 2 ( B 2 + G 2 + 2 C E ) and det C 1 = 1 2 I 2 1 − I 2 . Since det C 1 is a constant on the solutions o f the sys tem, then a t det C 1 6 = 0 one can alwa ys introduce the v a r iable y defined by x = y det C 1 such tha t CS (74 ) takes the form B ′ = E B G + E N C − GM C − C E 2 , E ′ = GB M + GE N − E C M − M G 2 , C ′ = B C E + B G 2 + M C 2 − C E G − B N C − GB 2 , G ′ = C M G + C E 2 − C E N − B GE (77) where B ′ = ∂ B ∂ y etc and M, N are ar bitrary functions. At det C 1 = B G − C E = 1 this system beco mes B ′ = E + C ( E N − GM ) , E ′ = M + G ( E N − GM ) , C ′ = G − B + C ( M C − B N ) , G ′ = − E − C ( E N − GM ) . (78) 17 Chosing M=N=0, one gets B ′ = E , E ′ = 0 , C ′ = G − B , G ′ = − E . (79) The so lution of this system is E = α, B = α y + β , G = − αy + γ , C = − y 2 + ( γ − β ) y + δ (80) where α , β , γ , δ are a rbitrary constants sub ject the constraint β γ − αδ = 1. First int eg rals for this solution are I 1 = β + γ , I 2 = 1 2 ( β 2 + γ 2 + 2 αδ ) . With the c hoice M=0 , N= 1 and under the cons train t I 1 = B + G = 0 the system (77) takes the fo r m B ′ = (1 + C ) E , E ′ = − B E , C ′ = − (2 + C ) B . (81) This system ca n b e written as a single equa tion in the different equiv ale n t forms. One of them is ( E ′ ) 2 + α E 4 − 2 E 3 + E 2 = 0 (82) where α is an a rbitrary constant and B 2 = − 1 − αE 2 + 2 E , C = αE − 2 , G = − B . (83) The se c o nd integral is equal to -1. Solutions of equation (82) can be expressed thro ugh the e lliptic integrals. Solution of (82) and the form ulae (83) define deformations o f the structure constants dr iv en b y DDA L 3 . 7 Solv able DD As 1. F or the sol v able DD A L 4 the relations (32) imply that [ p j , ϕ ( x )] = ( T − 1) ϕ ( x ) p j , j = 1 , 2 (84) where ϕ ( x ) is an arbitrar y function and T is the shift oper ator T ϕ ( x ) = ϕ ( x + 1). With the use of (84 ) o ne ar riv es a t the following CS C 1 T C 2 = C 2 T C 1 . (85) F or nondegenera te matr ix C 1 equation (8 5 ) is equiv alent to the e quation T ( C 2 C − 1 1 ) = C − 1 1 C 2 or T U = C − 1 1 U C 1 (86) where U + C 2 C − 1 1 . Using this form o f the CS, one promptly concludes that the CS (85 ) has three indep e nden t first integrals 18 I 1 = tr ( C 2 C − 1 1 ) , I 2 = 1 2 tr ( C 2 C − 1 1 ) 2 , I 3 = 1 3 tr ( C 2 C − 1 1 ) 3 (87) and is representable as the commut a tiv ity co ndition for the linear system Φ C 2 C − 1 1 = λ Φ , T Φ = Φ C 1 . (88) F or the t wo-dimensional alg ebra A one has the C S (85 ) with the matrices (48). It is the system o f four equations for six functions B T E + E T G = E T B + M T C, B T M + E T N = E T E + M T G, C T E + GT G = GT B + N T C, C T M + GT N = GT E + N T G. (89) Chosing B and C as free functions and as suming that BG-CE 6 = 0, one can easily res o lv e (89) with r espect to TE,TG,TM,TN. F or instance, with B =C=1 one gets the following four-dimensional mapping T E = M − E M − N E − G , T G = 1 + M − N E − G , T M = N + ( N − G ) M − N E − G − G  M − N E − G  2 , (9 0) T N = M + (1 − E ) M − N E − G +  M − N E − G  2 . 2. In a similar manner one finds the CS a ssocia ted with the s olv able DD A L 5 . Since in this ca se [ p 1 , ϕ ( x )] = ( T − 1) ϕ ( x ) p 1 , [ p 2 , ϕ ( x )] = ( T − 1 − 1) ϕ ( x ) p 2 (91) the CS takes the for m C 1 T C 2 = C 2 T − 1 C 1 . (92) F or no ndegenerate C 2 it is equiv a len t to T V = C 2 V C − 1 2 (93) where V + T − 1 C 1 · C 2 . Sim ila r to the previo us ca s e the CS has three first int eg rals I 1 = tr ( C 1 T C 2 ) , I 2 = 1 2 tr ( C 1 T C 2 ) 2 , I 3 = 1 3 tr ( C 1 T C 2 ) 3 (94) 19 and is equiv a len t to the compatibility condition for the linear sys tem ( T − 1 C 1 ) C 2 Φ = λ Φ , T Φ = C 2 Φ . (95) Note that the CS (92) is of the for m (22) with T 1 = T , T 2 = T − 1 . Thu s, the defor mations generated b y L 5 can be considered a s the r eductions of the discrete deformations (2 2) under the constra in t T 1 T 2 C n j k = C n j k . A class of so lutions of the CS (92) is g iv en by C j = g − 1 T j g (96) where g is 3 × 3 matrix and T 0 = 1 , T 1 = T , T 2 = T − 1 . Since C n j k = C n kj one has T j g m l = T l g m j and hence g m j = T j Φ m where Φ 0 , Φ 1 , Φ 2 are arbitrary functions. So, this s ubclass of deformations are defined by three arbitrary functions. 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