Noncommutative Toda Chains, Hankel Quasideterminants And Painleve II Equation

NONCOMMUT A TIVE TOD A C H AINS, HA N KEL QUASIDETERMINANTS AND P AINLEV ´ E I I EQUA TION Vladimir Ret akh and Vladimir Rubtso v Abstra ct. W e construct so lutions of an infinite T o da system and an analogue of the Painlev ´ e II eq uation ov er noncom mutativ e differential divisi on rings in te rm s of quasideterminants of Hankel matrices. Intr oduction Let R b e an asso ciative algebra o v er a field wit h a deriv ation D . Set D f = f ′ for an y f ∈ R . Assume that R is a divi sion ring. In this pap er we construct solutions for the system of equations (0.1) o v er al g ebra R (0.1-n) ( θ ′ n θ − 1 n ) ′ = θ n +1 θ − 1 n − θ n θ − 1 n − 1 , n ≥ 1 assuming that θ 1 = φ, θ 0 = ψ − 1 , φ, ψ ∈ R and its “negative” coun terpart (0.1’) (0.1’-m) ( η − 1 − m η ′ − m ) ′ = η − 1 − m η − m − 1 − η − 1 − m +1 η − m , m ≥ 1 where η 0 = φ − 1 , η − 1 = ψ . Note that θ ′ θ − 1 and θ − 1 θ ′ are noncomm utativ e analogues of t he logarithmic deriv ative (log θ ) ′ . W e u se t hen the solutio ns of the T o da equati ons under a certain anzatz for constructing solutions of the nonc ommutative Pai nlev´ e II e quation P I I ( u, β ) : u ′′ = 2 u 3 − 2 xu − 2 ux + 4( β + 1 2 ) where u, x ∈ R , x ′ = 1 and β is a scala r parameter, β ′ = 0. 2000 Ma thematics Subje ct Classific ation . 37K10; 16B99; 16W25. Key wor ds and ph r ases. noncommutative Pa inlev´ e equation, quasideterminants, a lmost Hank el matrices,. T yp eset by A M S -T E X 1 2 VLADIMIR RET AKH AND VLADIMIR RUBTSO V Unlike pap ers [NGR] and [N] we consider here a “pure noncomm utat iv e” vers ion of the Painlev ´ e equation without an y addit i onal assumption for our algebra R . In fact a noncomm utative (”matrix”) version of Painlev ´ e I I P I I ( u, β ) : u ′′ = 2 u 3 + xu + β I . w as considered in the first time in the pap ers of V. Sokolo v with differen t coau- thors: W e men tion here f.e. [BS]. But t heir form of this equation, sati sfying the P ainlev ´ e test, in the same time can not b e obt a i ned as a reduction of some matr i x analog of mKDV system. Our equation is simi lar to this noncomm utative P ainlev´ e I I but there i s an es- sen ti al difference: w e write the second term in the R.H. S. in the symmetric or ”an ticomm utator” form. This spli t ting form is muc h more adapta bl e t o some gen- eralizations of t he usual comm utative P ainlev ´ e I I. Our motiv at i on is the follo wing. In the commutativ e case one can consider an infinite T o da syst em (see, for example [KMNO Y, JKM]): (0.2-n) τ ′′ n τ n − ( τ ′ n ) 2 = τ n +1 τ n − 1 − φψ τ 2 n with t he conditions τ 1 = φ, τ 0 = 1 , τ − 1 = ψ . Let n ≥ 1. By setting θ n = τ n /τ n − 1 the system can b e written as (log τ n ) ′′ = θ n +1 θ − 1 n − φψ . F or n = 1 we ha v e the eq uat ion ( 0.1-1)with θ 1 = φ, θ 0 = ψ − 1 . By subtracting equation (2.2 -n) from (2.2 -(n+1)) and replacing the difference log τ n +1 − log τ n b y log τ n +1 τ n one can get (0.1-n). Similarly , the system (0.2-m) for p osit i v e m implies the system (0 . 1 ′ − m ) for θ − m = τ − m /τ − m +1 . By goi ng from τ n ’s to their consequtive relati o ns w e are cutting the system of equations parametrized b y −∞ < n < ∞ to its “p ositive” and “negative” part. A sp ecial case of the semi-infinite system (0.1) ov er noncomm utativ e algebra with θ − 1 0 formally equal to zero was treated in [GR2]. In this pap er solutions of the T o da system (0.1) wi th θ − 1 0 = 0 w ere constructed as quasi determinants of certa i n Hankel matrices. It was the first application of quasideterminants i n t ro duced in [GR1] to noncomm ut a tive in tegrable systems. This line w as con tin ued b y sev eral reseache rs, see, for example, [EGR1, E GR2], pap ers by Gla sgow sc ho ol [GN, GNO, GNS] and a recen t [DFK]. In this pap er w e generalize the result of [GR2] for θ 0 = ψ − 1 and extend it to t he infinite T o da system. The soluti ons a r e also giv en in terms of quasideterminan ts of Ha nkel matri ces but the computations are mu c h harde r. W e foll o w here the comm utativ e a pproach dev elop ed in [ KMNO Y , JKM] with some adjustmen ts but NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 3 our pro ofs are far from a stra i gh tforw ard generalizatio n. In particular, for our pro of w e hav e to in tro duce and in v estigate almost Hank el matri c es (see Section 2.2). F rom solutions of the systems (0. 1) and (0.1’ ) under certain anzatz w e deduce solutions for the noncomm utat i v e equation P I I ( u, β ) for v arious parameters β (The- orem 3. 2 ). This i s a noncomm utative dev elopmen t o f an idea from [KM]. W e start this pap er b y a r emi nder of basic propert ies of quasideterminan ts, then construct soluti ons of t he systems (0.1) and (0.1’ ), then apply our results to non- comm utativ e Painlev ´ e I I equations following the approach b y [KM]. Our pap er shows that a theory of “pure” noncommutativ e P ainlev ´ e equations and the related ta u-functions can b e rather rich and interesting. The Painlev ´ e I I t yp e w as c hosen a s a mo del and w e are going t o in v estigate other t y p es of P ainlev ´ e equations. Ac kno wledgements . Both authors are thank ful to MA TPYL pro ject “Noncom- mm utative I ntegrable Systems” (2008-201 0 ) for a support o f visits of V . Retak h to A ngers. W e are grateful to the F ederation of Mathemati cs of Loire Region and to LAREMA for t he help a nd w arm hosp itality . V. Rubtso v was partially sup- p orted during the p erio d of this work by PICS Pro ject “Probl` emes de Ph ysique Math ´ emati ques” (F r a nce-Ukraine). He enjo yed in 2 009-2010 a CNRS delegati on at LPTM of Cergy-P on toise Univers ity and he ackno wledges a warm hospital it y and stim ulating atmosphere of LPTM. He is thankful to M. Kon tsevic h and to S. Duzhin who inspired his in terest in noncommutativ e integrable systems and to B . Enriquez for a long collab oration and discussions whic h triggered him to study application of quasideterminan ts in quan tum integrabilit y . Vladimi r Retakh would li k e to thank IHES for i ts hospitality during his visit in 201 0. W e thank the referee for the careful reading of the man uscript a nd helpful re- marks. 1. Quasideterminants The not ion of quasideterminan ts was introduced i n [GR1], see also [GR2-3, GGR W]. Let A = || a ij || , i, j = 1 , 2 , . . . , n be a matrix ov er an asso ciati v e unital ring. Denote b y A pq the ( n − 1) × ( n − 1 ) submatrix of A obtained b y deleting the p -th ro w and q -th column. Let r i b e the row ma t rix ( a i 1 , a i 2 , . . . ˆ a ij , . . . , a in ) and c j b e the column matri x with en tries ( a 1 j , a 2 j , . . . ˆ a ij , . . . , a nj ). F or n = 1, | A | 11 = a 11 . F or n > 1 the quasideterminant | A | ij is defined if the matrix A ij is i nv erti ble. In this case | A | ij = a ij − r i ( A ij ) − 1 c j . If the inv erse matri x A − 1 = || b pq || exists then b pq = | A | − 1 q p pro vided t hat the quasideterminan t is i n v ertible. If R i s commu tative then | A | ij = ( − 1) i + j det A/ det A ij for any i and j . 4 VLADIMIR RET AKH AND VLADIMIR RUBTSO V Examples. ( a) F or the generic 2 × 2-matrix A = ( a ij ), i, j = 1 , 2, there are four quasideterminan ts: | A | 11 = a 11 − a 12 a − 1 22 a 21 , | A | 12 = a 12 − a 11 a − 1 21 a 22 , | A | 21 = a 21 − a 22 a − 1 12 a 11 , | A | 22 = a 22 − a 21 a − 1 11 a 12 . (b) F or the g eneric 3 × 3-matrix A = ( a ij ), i, j = 1 , 2 , 3, there are 9 quasideter- minan ts. One of them is | A | 11 = a 11 − a 12 ( a 22 − a 23 a − 1 33 a 32 ) − 1 a 21 − a 12 ( a 32 − a 33 a − 1 23 a 22 ) − 1 a 31 − a 13 ( a 23 − a 22 a − 1 32 a 33 ) − 1 a 21 − a 13 ( a 33 − a 32 · a − 1 22 a 23 ) − 1 a 31 . Here are the transformation prop erties of q uasideterminan ts. Let A = || a ij || b e a square ma t rix of order n o v er a ring R . (i) The q uasideterminan t | A | pq do es not depend on p erm utations of ro ws and columns in the mat ri x A that do not inv ol v e the p -th ro w and the q -th col umn. (ii) The multiplic ation of r ows and c olumns. Let the matrix B = || b ij || b e ob- tained from the matrix A by m ultiplying the i -th ro w by λ ∈ R from the left, i.e., b ij = λa ij and b kj = a kj for k 6 = i . Then | B | kj =  λ | A | ij if k = i, | A | kj if k 6 = i and λ i s inv ertible. Let the ma trix C = || c ij || b e obtained from the matrix A by multiplying the j -th column b y µ ∈ R from t he right, i .e. c ij = a ij µ and c il = a il for all i and l 6 = j . Then | C | iℓ =  | A | ij µ if l = j, | A | iℓ if l 6 = j and µ is in v ertible. (iii) The ad d i tion of r ows and c olumns. Let the matrix B b e obtained from A b y replacing the k -th row of A wit h the sum of the k -th and l -th rows, i.e., b kj = a kj + a lj , b ij = a ij for i 6 = k . The n | A | ij = | B | ij , i = 1 , . . . k − 1 , k + 1 , . . . n, j = 1 , . . . , n. W e will need the foll o wing prop erty of quasideterminan ts sometimes call ed the nonc ommutative L ewis Carr ol l identity . It is a sp ecial case of the no nc ommutative Sylvester identity from [GR1-2] or her e dity princi p le form ulated i n [GR3]. Let A = || a ij || , i, j = 1 , 2 , . . . , n . Consider the foll o wng ( n − 1) × ( n − 1 ) - submatrices X = || x pq || , p, q = 1 , 2 , . . . , n − 1 of A : matrix A 0 = || a pq || obtained from A b y deleting its n -th row and n -th column; matri x B = || b pq || obtained from A b y deleting its ( n − 1)-th ro w and n -th col umn; matrix C = || c pq || obtained from A by deleting its n -th row and ( n − 1)-th column; ma t rix D = || d pq || obtai ned from A by deleting its ( n − 1)-th row and ( n − 1)-th column. Then (1.1.) | A | nn = | D | n − 1 ,n − 1 − | B | n − 1 ,n − 1 | A 0 | − 1 n − 1 ,n − 1 | C | n − 1 ,n − 1 NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 5 2. Quasideterminant solutions of noncommut a tive Toda e qua tions 2.1. Noncomm utat iv e T o da equations in bi linear form. Let F b e a com- m utative field and R b e an asso ciat i v e ring containing F -al gebra. Let D : R → R b e a deriv ation o v er F , i.e. an F -li near map satisfying the Leibniz rule D ( ab ) = D ( a ) · b + a · D ( b ) for an y a, b ∈ R . Also, D ( α ) = 0 for an y α ∈ F . A s usual, w e set u ′ = D ( u ) , u ′′ = D ( D ( u )) , . . . . Recall that D ( v − 1 ) = − v − 1 v ′ v − 1 for any inv ertible v ∈ R . Let φ, ψ ∈ R and R b e a div ision ring. W e construct no w solutions for the noncomm ut a tive T o da eq uations (0.1) and (0 .1’) assuming that θ 0 = ψ − 1 , θ 1 = φ and η 0 = φ − 1 , η − 1 = ψ . Set (cf. [KMNO Y, JKM] for t he commutativ e case) a 0 = φ, b 0 = ψ and (2.1) a n = a ′ n − 1 + X i + j = n − 2 ,i,j ≥ 0 a i ψ a j , b n = b ′ n − 1 + X i + j = n − 2 ,i,j ≥ 0 b i φb j , n ≥ 1 . Construct Hankel matrices A n = || a i + j || , B n = || b i + j || , i, j = 0 , 1 , 2 . . . , n . Theorem 2.1. Set θ p +1 = | A p | p,p , η − q − 1 = | B q | q ,q . The elements θ n for n ≥ 1 satisfy the sys te m (0. 1) and the elements η − m , m ≥ 1 sati s fy the system (0.1’) . This theorem can b e view ed as a noncommu tative generalization of Theorem 2.1 from [KMNO Y]. In [ KMNOY] it w as pro v ed that in the commutativ e case the Hank el determinan t s τ n +1 = det A n , n ≥ 0 , τ 0 = 1 , τ − n − 1 = det B n , n ≤ 0 satisfy the system ( 0 .2). Example. T he (noncomm utative ) logari thmic deriv at ive θ ′ 1 θ − 1 1 satisfies the non- comm utativ e T o da eq uat ion (0.1-1) : ( θ ′ 1 θ − 1 1 ) ′ = θ 2 θ − 1 1 − φψ . In fact, ( θ ′ 1 θ − 1 1 ) ′ = ( a 1 a − 1 0 ) ′ = ( a 2 − a 0 ψ a 0 ) a − 1 0 − ( a 1 a − 1 0 ) 2 = ( a 2 − a 1 a − 1 0 a 1 ) a − 1 0 − a 0 ψ = θ 2 θ − 1 1 − φψ . Our pro of of Theorem 2.1. in the general case is based on prop erties o f quaside- terminan ts of almo st Hankel matrices. 2.2. Almost Hankel matrices and their quasidetermina n ts. W e define a l- most Hankel matrices H n ( i, j ) = || a st || , s, t = 0 , 1 , . . . , n , i, j ≥ 0 for a sequence a 0 , a 1 , a 2 , . . . as follows. Set a nn = a i + j and for s, t < n a s,t = a s + t , a n,t = a i + t , a s,n = a s + j . and a nn = a i + j . Note that H n ( n, n ) is a Ha nk el matri x. Denote b y h n ( i, j ) t he quasideterminan t | H n ( i, j ) | nn . Th en h n ( i, j ) = 0 i f at least o ne of the inequali t ies i < n , j < n holds. 6 VLADIMIR RET AKH AND VLADIMIR RUBTSO V Lemma 2.2. (2.2) h n ( i, j ) ′ = κ n ( i, j ) − i X p =1 a p − 1 ψ h n ( i − p, j ) − j X q =1 h n ( i, j − q ) ψ a q − 1 wher e (2.3a) κ n ( i, j ) = h n ( i + 1 , j ) − h n − 1 ( i, n − 1) h − 1 n − 1 ( n − 1 , n − 1) h n ( n, j ) . A lso, (2.3b) = h n ( i, j + 1) − h n ( i, n ) h − 1 n − 1 ( n − 1 , n − 1) h n − 1 ( n − 1 , j ) . Note that some summands h n ( i − p, j ), h n ( i, j − q ) in form ula (2.2) can b e equal to zero. Since h n ( i, j ) = 0 when i < n or j < n we ha v e the following corollary . Corollary 2.3. h n ( n, n ) ′ = κ n ( n, n ) , h n ( i, n ) ′ = κ n ( i, n ) − i X s =1 a s − 1 ψ h n ( i − s, n ) , h n ( n, j ) ′ = κ n ( n, j ) − j X v =1 h n ( n, j − v ) ψ a v − 1 . Pr o of of L emma 2.2. W e pro v e Lemma 2.2 b y induction. By definition, h 1 ( i, j ) ′ = a i + j +1 − i + j − 1 X k =0 a k ψ a i + j − 1 − k − ( a i +1 − i − 1 X s =0 a s ψ a i − 1 − s ) a − 1 0 a j + a i a − 1 0 a 1 a − 1 0 a j − a i a − 1 0 ( a j +1 − j − 1 X t =0 a j − 1 − t ψ a t ) . Set κ 1 ( i, j ) = a i + j +1 − a i +1 a − 1 0 a j + a i a − 1 0 a 1 a − 1 0 a j − a i a − 1 0 a j +1 , w e can c hec k formu las (2.3a) a nd (2 .3b). The rest of the pro of for n = 1 is easy . Assume no w that form ula (2. 2) is true for n ≥ 1 and prov e it for n + 1. By t he noncomm ut a tive Sylvester iden t it y (1.1) (2.4) h n +1 ( i, j ) = h n ( i, j ) − h n ( i, n ) h − 1 n ( n, n ) h n ( n, j ) . NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 7 Set h n +1 ( i, j ) ′ = κ n +1 ( i, j ) + r n +1 ( i, j ) where κ n +1 con tains all terms without ψ . Then κ n +1 ( i, j ) = κ n ( i, j ) − κ n ( i, n ) h − 1 n ( n, n ) h n ( n, j ) + h n ( i, n ) h − 1 n ( n, n ) κ n ( n, n ) h − 1 n ( n, n ) h n ( n, j ) − h n ( i, n ) h − 1 n ( n, n ) κ n ( n, j ) . By induction, the first tw o terms can b e written as h n ( i + 1 , j ) − h n − 1 ( i, n − 1) h − 1 n − 1 ( n − 1 , n − 1) h n ( n, j ) +[ h n ( i + 1 , n ) − h n − 1 ( i, n − 1) h − 1 n − 1 ( n − 1 , n − 1) h n ( n, n )] h − 1 n ( n, n ) h n ( n, j ) = h n ( i + 1 , j ) − h n ( i + 1 , n ) h − 1 n ( n, n ) h n ( n, j ) . This expression equals t o h n +1 ( i + 1 , j ) by the Sylvester iden t i t y . The last tw o terms in κ n +1 ( i, j ) can b e writt en as h n ( i, n ) h − 1 n ( n, n )[ h n ( n +1 , n ) − h n − 1 ( n, n − 1) h − 1 n − 1 ( n − 1 , n − 1) h n ( n, n )] h − 1 n ( n, n ) h n ( n, j ) − h n ( i, n ) h − 1 n ( n, n )[ h n ( n + 1 , j ) − h n − 1 ( n, n − 1) h − 1 n − 1 ( n − 1 , n − 1) h n ( n, j )] = h n ( i, n ) h − 1 n ( n, n )[ − h n ( n + 1 , n ) + h n ( i, n ) h − 1 n ( n, n ) h n ( n + 1 , j )] = − h n ( i, n ) h − 1 n ( n, n ) h n +1 ( n + 1 , j ) also by the Sylvester iden tit y . Therefore, κ n +1 ( i, j ) sat i sfies formula (2.3a). F orm ula ( 2.3b) can b e obtained in a simi lar wa y . Let us lo ok at the terms con taining ψ . According to the inductiv e assumption h n ( i, j ) ′ = κ n ( i, j ) − i X k =1 a k − 1 ψ h n ( i − k , j ) − j X ℓ =1 h n ( i, j − ℓ ) ψ a ℓ − 1 . Using t he Co r o llary 2.3 and form ula ( 2.2) for n one can write r n +1 ( i, j ) as − i X k =1 a k − 1 ψ h n ( i − k , j ) − j X ℓ =1 h n ( i, j − ℓ ) ψ a ℓ − 1 + i X k =1 a k − 1 ψ h n ( i − k , n )] h − 1 n ( n, n ) h n ( n, j ) + h n ( i, n ) h − 1 n ( n, n ) j X ℓ =1 h n ( n, j − ℓ ) ψ a ℓ − 1 8 VLADIMIR RET AKH AND VLADIMIR RUBTSO V = − i X k =1 a k − 1 ψ [ h n ( i − k, j ) − h n ( i − k, n ) h − 1 n ( n, n ) h n ( n, j )] − j X ℓ =1 [ h n ( i, j − ℓ ) − h n ( i, n ) h − 1 n ( n, n ) h n ( n, j − ℓ )] ψ a ℓ − 1 . Our lemma follo ws now from the Sy lv ester identit y applied to each expression in square brac k ets. Corollary 2. 3 and form ula (2.3a) immediately imply Corollary 2.4. F or n > 1 h n ( n, n ) ′ h − 1 n ( n, n ) = h n ( n + 1 , n ) h − 1 n ( n, n ) − h n − 1 ( n, n − 1) h − 1 n − 1 ( n − 1 , n − 1) . Note in the right hand side w e hav e a difference of left quasi-Pl ¨ uc ker co ordinates (see [GR3]). 2.3. Pro of of Theorem 2.1. Our solution of the T o da sy stem (0.1) fol lo ws from Corol lary 2.4 and the following lemma. Lemma 2.5. F or k > 0 [ h k ( k + 1 , k ) h − 1 k ( k , k )] ′ = h k +1 ( k + 1 , k + 1) h − 1 k ( k , k ) − a 0 ψ . Pr o of . Corol lary 2. 3 and form ula (2.3b) imply h k ( k +1 , k ) ′ = h k ( k +1 , k +1) − h k ( k +1 , k ) h − 1 k − 1 ( k − 1 , k − 1) h k − 1 ( k − 1 , k ) − a 0 ψ h k ( k , k ) b ecause h k ( k + 1 − s , k ) = 0 for s > 1. Then, using agai n formu la (2.3b) one has [ h k ( k + 1 , k ) ′ h − 1 k ( k , k )] ′ = [ h k ( k + 1 , k + 1) − h k ( k + 1 , k ) h − 1 k − 1 ( k − 1 , k − 1) h k − 1 ( k − 1 , k ) − a 0 ψ h k ( k , k )] h − 1 k ( k , k ) − h k ( k + 1 , k ) h − 1 k ( k , k )][ h k ( k , k + 1) − h k ( k , k ) h − 1 k − 1 ( k − 1 , k − 1) h k − 1 ( k − 1 , k )] h − 1 k ( k , k ) = [ h k ( k + 1 , k + 1) − h k ( k + 1 , k ) h − 1 k ( k , k ) h k ( k , k + 1) ] h − 1 k ( k , k ) − a 0 ψ = h k +1 ( k + 1 , k + 1) h − 1 k ( k , k ) − a 0 ψ b y the Sylvester form ula. Theorem 2.1. now foll o ws from Corollary 2.4 and Lemma 2.5. The sta temen t for η − m , m ≥ 1 can b e pro v ed in a similar w a y . NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 9 3. Noncommut a tive P ainlev ` e I I 3.1 Comm utative Painlev ` e I I and Hankel d eterminants: motiv ation. The P ainlev ` e I I ( P I I ) eq uat ion ( w i th comm utative v ariables) u ′′ = 2 u 3 − 4 xu + 4( β + 1 2 ) admits unique rati onal soluti on for a half-in teger v al ue of the parameter β . These solutions can b e expressed in terms of logarit hmic deriv ative s of ratios of Hank el- t yp e determinants. Namely , if β = N + 1 2 then u = d dx log det A N +1 ( x ) det A N ( x ) , where A N ( x ) = || a i + j || where i, j = 0 , 1 , . . . , n − 1. The en tries of the matrix are p olynomial s a n ( x ) sub jected to t he recurrence relat ions: a 0 = x, a 1 = 1 , a n = a ′ n − 1 + n − 2 X i =0 a i a n − 2 − i . (see [JKM]) 3.2 Noncomm ut ativ e and “quantum” Painlev` e I I. W e will consider here a nonc ommutative vers ion of P I I whic h we will denote nc − P I I ( x, β ): u ′′ = 2 u 3 − 2 xu − 2 ux + 4( β + 1 2 ) , where x, u ∈ R , x ′ = 1 and β is a central scalar parameter ( β ∈ F , β ′ = 0). This equat ion i s a sp eciali zation of a general noncomm utati v e Painlev ´ e I I system with r esp ect t o t hree dep enden t noncomm utative v ariables u 0 , u 1 , u 2 : u ′ 0 = u 0 u 2 + u 2 u 0 + α 0 u ′ 1 = − u 1 u 2 − u 2 u 1 + α 1 u ′ 2 = u 1 − u 0 . Indeed, ta k ing the deriv ative of the third and using the first and second, w e get u ′′ 2 = − ( u 0 + u 1 ) u 2 − u 2 ( u 0 + u 1 ) + α 1 − α 0 . Then we hav e: ( u 0 + u 1 ) ′ = − u ′ 2 u 2 − u 2 u ′ 2 + α 0 + α 1 and, immediat ely − ( u 0 + u 1 ) = u 2 2 − ( α 0 + α 1 ) x − γ , γ ∈ F . Compare wit h u ′′ 2 w e obt a i n the following nc − P I I : u ′′ 2 = 2 u 3 2 − ( α 0 + α 1 ) xu 2 − ( α 0 + α 1 ) u 2 x − 2 γ u 2 + α 1 − α 0 . Our eq uat ion corresp onds the choice γ = 0 , α 1 = 2( β + 1) , α 0 = − 2 β . 10 VLADIMIR RET AKH AND VLADIMIR RUBTSO V Remark. The nonc ommutative Painlev ´ e II s yste m ab ove i s the str aightforwar d gen- er alization of the a na lo gues system in [NGR] when the vari ables u i , i = 0 , 1 , 2 ar e sub or dinate d to some c ommutation r elations. Her e we don ’t assume that the “i nd e - p endent” v ariable x c ommutes with u i . Going further with this analogy w e will write a “fully non-comm utati ve” Hamil- tonian of the system H = 1 2 ( u 0 u 1 + u 1 u 0 ) + α 1 u 2 and in tro duce the “canonical” v ariabl es p := u 2 , q := u 1 , x := 1 2 ( u 0 + u 1 + u 2 2 ) . Prop osition 3.1. L et a triple ( x, p, q ) b e a “solution” of the “Hamiltoni a n system” with the Hamiltonian H and α 1 = 2( β + 1) . p x = − H q q x = H p . Then p satis fies the nc − P I I : p xx = 2 p 3 − 2 px − 2 xp + 4( β + 1 2 ) . Pr o of . Strai gh tforw a r d computat ion gi v es that: p x = p 2 + 2 q − 2 x q x = α 1 − ( q p + pq ) . T aking p xx = p x p + pp x + 2 q x − 2 and substituting p x and q x w e obtain the result. W e give (for t he sake of completeness) the explicit expression of the Painlev ´ e Hamiltonia n H in the ”canonical” co ordinates: H ( x, p, q ) = q x + xq − q 2 − 1 2 ( q p 2 + p 2 q ) + 2( β + 1) p. 3.3 Solution s of the noncomm utative P ainlev´ e an d of the T o da system. Theorem 3.2. L et φ and ψ sati sfy the fol lowing identitie s: (3.1) ψ − 1 ψ ′′ = φ ′′ φ − 1 = 2 x − 2 φψ , (3.2) ψ φ ′ − ψ ′ φ = 2 β . Then for n ∈ N 1) u n = θ ′ n θ − 1 n satisfies n c − P I I ( x, β + n − 1) ; 2) u − n = η ′ − n η − 1 − n satisfies n c − P I I ( x, β − n ) . Let us start with the follo wing useful (t hough sli gh tly tech nical) lemma NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 11 Lemma 3.3. Under the c onditions of the The or em 3.1 we have the chain of iden- tities ( n ≥ 0 ): 1) θ ′ n θ − 1 n + θ ′ n − 1 θ − 1 n − 1 = 2( β + n − 1) θ n − 1 θ − 1 n 2) θ ′′ n θ − 1 n = 2( x − θ n θ − 1 n − 1 ) and also, for n ≥ 1 3) η − 1 − n η ′ − n + η − n +1 η ′ − n +1 = − 2( β − n + 1) η − 1 − n η − 1 − n η − n +1 4) η − 1 − n η ′′ − n = 2( x − η − 1 − n +1 η − n . Pr o of . Remark that the first step in the ch ain ( n = 1 ) directly follows from our assumption: θ 1 = φ, θ 0 = ψ − 1 : φ ′ φ − 1 + ( ψ − 1 ) ′ ψ = 2 β ψ − 1 φ − 1 . Indeed, we hav e φ ′ φ − 1 − ψ − 1 ψ ′ = 2 β ψ − 1 φ − 1 where t he result: ψ φ ′ − ψ ′ φ = 2 β . The second step ( n = 2) is a littl e bit tric ky . W e consider the T o da eq uat ion ( φ ′ φ − 1 ) ′ = θ 2 φ − 1 − φψ and find easily θ 2 (using φ ′′ φ − 1 = 2 x − 2 φψ ): θ 2 = 2 xφ − φψ φ − ( φ ′ φ − 1 ) φ ′ . T aking t he deriv at ion and using the same T o da and the first step iden tity , we get θ ′ 2 = 2 φ ( β + 1) − φ ′ φ − 1 θ 2 . The second ( n = 2) identit y is rather strai g h t forw ard: θ ′′ 2 + ( θ ′ 1 θ − 1 1 ) ′ θ 2 + ( θ ′ 1 θ − 1 1 ) θ ′ 2 = 2( β + 1) θ ′ 1 . Again using the T o da and the first i den ti t y we obtai n finally: θ ′′ 2 θ − 1 2 + θ 2 φ − 1 − φψ − ( φ ′ φ − 1 ) 2 = 0 and then θ ′′ 2 θ − 1 2 + 2 x − 2( φψ + ( φ ′ φ − 1 ) 2 ) = θ ′′ 2 θ − 1 2 − 2( x − θ 2 φ − 1 ) = 0 . W e will discuss one more step, namely the passage from n = 2 to n = 3 (then the recurrence w i ll b e clear) . W e wan t t o sho w that: 1) θ ′ 3 θ − 1 3 + θ ′ 2 θ − 1 2 = 2( β + 2) θ 2 θ − 1 3 ; 2) θ ′′ 3 θ − 1 3 = 2( x − θ 3 θ − 1 2 ) . 12 VLADIMIR RET AKH AND VLADIMIR RUBTSO V F rom the second T o da and second i dentit y we get θ 3 = 2 xθ 2 − θ 2 θ − 1 1 θ 2 − θ ′ 2 θ − 1 2 θ ′ 2 . It i mplies θ ′ 3 = 2 θ 2 + 2 xθ ′ 2 − θ ′ 2 θ − 1 1 θ 2 + θ 2 θ − 1 1 θ ′ 1 θ − 1 1 θ 2 − θ 2 θ − 1 1 θ ′ 2 − − 2( x − θ 2 θ − 1 1 ) θ ′ 2 + ( θ ′ 2 θ − 1 2 ) 2 θ ′ 2 − θ ′ 2 θ − 1 2 (2 x − 2 θ 2 θ − 1 1 ) θ 2 . W e simplify and obtain from this θ ′ 3 = 2 θ 2 + θ 2 θ − 1 1 ( θ ′ 2 + θ ′ 1 θ − 1 1 θ 2 ) + θ ′ 2 θ − 1 1 θ 2 + ( θ ′ 2 θ − 1 2 ) 2 θ ′ 2 − 2 θ ′ 2 θ − 1 2 xθ 2 . By the iden tit y for θ ′ 2 w e ha v e θ ′ 3 = 2 θ 2 + θ 2 θ − 1 1 · 2(1 + β ) θ 1 + θ ′ 2 θ − 1 1 θ 2 + θ ′ 2 θ − 1 2 ( − θ 3 − θ 2 θ − 1 1 θ 2 ) . whic h assure the first identit y for n = 3. No w w e pro v e t he second. Set a = 2( β + 2) . W e hav e θ ′ 3 = aθ 2 − ( θ ′ 2 θ − 1 2 ) θ 3 . T ak e the second deriv ation: θ ′′ 3 = aθ ′ 2 − ( θ ′ 2 θ − 1 2 ) ′ θ 3 − θ ′ 2 θ − 1 2 θ ′ 3 . By using the form ula for θ ′ 3 w e hav e θ ′′ 3 = aθ ′ 2 − ( θ ′ 2 θ − 1 2 ) ′ θ 3 − θ ′ 2 θ − 1 2 ( aθ 2 − θ ′ 2 θ − 1 2 θ 3 ) . The terms wit h a are cancelled a nd we hav e θ ′′ 3 = − ( θ ′ 2 θ − 1 2 ) ′ θ 3 + ( θ ′ 2 θ − 1 2 ) 2 θ 3 . Note that − ( θ ′ 2 θ − 1 2 ) ′ + ( θ ′ 2 θ − 1 2 ) 2 = θ ′′ 2 θ − 1 2 − 2( θ ′ 2 θ − 1 2 ) ′ . W e already know that the first summand in the righ t hand side equals 2( x − θ 2 θ − 1 1 ) and b y our T o da system ( θ ′ 2 θ − 1 2 ) ′ = θ 3 θ − 1 2 − θ 2 θ − 1 1 NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 13 w e obt a i n the second iden tit y for θ 3 . The n − th step of the recurrence go es as follows: from n − th T o da and recurrence conjecture w e hav e θ n +1 = 2 xθ n − θ n θ − 1 n − 1 θ n − θ ′ n θ − 1 n θ ′ n . It i mplies θ ′ n +1 = 2 θ n + 2 xθ ′ n − θ ′ n θ − 1 n − 1 θ n + θ n θ − 1 n − 1 θ ′ n − 1 θ − 1 n − 1 θ n − θ n θ − 1 n − 1 θ ′ n − − 2( x − θ n θ − 1 n − 1 ) θ ′ n + ( θ ′ n θ − 1 n ) 2 θ ′ n − θ ′ n θ − 1 n (2 x − 2 θ n θ − 1 n − 1 ) θ n . Then, after some simplifications we get θ ′ n +1 = 2 θ n + θ n θ − 1 n − 1 ( θ ′ n + θ ′ n − 1 θ − 1 n − 1 θ n ) + θ ′ n θ − 1 n − 1 θ n + ( θ ′ n θ − 1 n ) 2 θ ′ n − 2 θ ′ n θ − 1 n xθ n . By the recurren t formu la for θ ′ n , we hav e θ ′ n + θ ′ n − 1 θ − 1 n − 1 θ n = 2( β + 1 − n ) θ n − 1 and θ ′ n +1 = 2 θ n + 2( β + n − 1) θ n + θ ′ n θ − 1 n − 1 θ n + ( θ ′ n θ − 1 n ) 2 θ ′ n − 2 θ ′ n θ − 1 n xθ n = = 2( β + n ) θ n + θ ′ n θ − 1 n − 1 θ n + θ ′ n θ − 1 n ( θ ′ n θ − 1 n − 1 θ ′ n − 2 xθ n ) = = 2( β + n ) θ n + θ ′ n θ − 1 n − 1 θ n + θ ′ n θ − 1 n ( − θ n +1 − θ n θ − 1 n − 1 θ n ) = = 2( β + n ) θ n + θ ′ n θ − 1 n − 1 θ n − θ ′ n θ − 1 n θ n +1 − θ ′ n θ − 1 n − 1 θ n . whic h assure the first identit y for n + 1. W e leav e the pro of of the second iden tit y for an y n as an easy (though a bit length y) ex ercise sim i lar to the case n = 3 ab ov e. The iden tities 3) and 4) can b e prov ed in a simila r w a y . Lemma 3.4. F or n = 1 the lef t lo ga ri thmic derivative φ ′ φ − 1 =: u 1 satisfies to nc − P I I ( x, β ) . Pr o of . F rom the previous lemma w e ha v e from the first T o da equation: ( φ ′ φ − 1 ) ′ = θ 2 φ − 1 − φψ = φ ′′ φ − 1 − ( φ ′ φ − 1 ) 2 = 2( x − φψ ) − u 2 1 and hence θ 2 φ − 1 = 2 x − φψ − u 2 1 . 14 VLADIMIR RET AKH AND VLADIMIR RUBTSO V In other hand, taking the deriv ative of the first T o da, we get u ′′ 1 = ( θ 2 φ − 1 − φψ ) ′ = θ ′ 2 φ − 1 − θ 2 φ − 1 u 1 − ( φ ′ ψ + φψ ′ ) . W e replace θ ′ 2 φ − 1 b y 2( β + 1) − u 1 θ 2 φ − 1 = 2( β + 1) − u 1 (2 x − φψ − u 2 1 ) . Finally w e obta in u ′′ 1 = 2 u 3 1 − 2 u 1 x − 2 xu 1 + 2( β + 1) + u 1 φψ + φψ u 1 − ( φ ′ ψ + φψ ′ ) , but u 1 φψ + φψ u 1 − ( φ ′ ψ + φψ ′ ) = φψ φ ′ φ − 1 − φψ ′ = 2 β whic h gi v es the desired result. Our pro of of Theorem 3.2 in the general case al most ve rb atim rep eats the pro of of the Lemma 3.4 . Pr o of of The or em 3.2. Let u n := θ ′ n θ − 1 n . No w the same argumen ts, from the Lemma 3.4, show that: a) θ n +1 θ − 1 n = 2 x − θ n θ − 1 n − 1 − u 2 n ; b) θ ′ n +1 θ − 1 n = 2( β + n ) − θ ′ n θ − 1 n θ n +1 θ − 1 n ; c) u ′′ n = 2 u 3 n − 2 xu n − 2 u n x + 2( β + n ) + θ n θ − 1 n − 1 ( θ ′ n θ − 1 n + θ ′ n − 1 θ − 1 n − 1 ). This implies that u ′′ n = 2 u 3 n − 2 xu n − 2 u n x + 4( β + n − 1 2 ) . Remark. Usi ng identitie s 3) and 4) fr om L emma 3. 3 we c an pr ove the se c ond statement of the The or e m 3. 2. 4. Discussion and pe rspectives W e ha ve developed an approac h to in tegrabil it y of a fully noncomm utative analog of the Painlev ´ e equation. W e construct soluti o ns of this equati on relat ed to the “fully noncomm utative ” T o da chain, generalizing t he results of [ GR2, EGR1]. This solutions a dmi t an explicit description in terms of Hank el quasideterminan ts. W e consider here only t he noncomm utative generalization of P ainlev ´ e I I but it is not difficult to write do wn some noncomm utative analogs of other Painlev ´ e tran- scendan ts. It is in teresting to study their solutions, noncommu tative τ − functions, etc. W e hop e that o ur equation ( like its “commutativ e” protot yp e) is a part of a whole noncommutativ e Painlev ´ e hierarc h y whic h relat es (vi a a noncomm utative Miura transform) to the noncommu tative m-KdV and m-KP hierarc hies (see i .e. [EGR1-2],[GN] ,[GNS]). Another in teresting problem i s to study a noncomm uta- tive v ersion of isomono dromic tra nsformati ons problem for our P ainlev´ e equation. NONCOMMUT A TIVE P AINLEV ´ E EQUA TION 15 The natural approach to this problem is a noncomm utativ e generali zat ion of gen- erating functions, constructed in [JKM]. T he noncommutativ e “non-autonomous” Hamiltonia n should b e studied more extensively . 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Se c ond and F orth Painlev ´ e Equa- tions, PII and PIV , Math. Ann. 275 (1986), 221–255. V. Ret akh: Dep ar t ment of Ma thema tics, Rutgers University, Pisca t a w a y, NJ 08854- 16 VLADIMIR RET AKH AND VLADIMIR RUBTSO V 8019 USA V. Rubtso v: D ´ ep ar tement d e Ma t ´ ema tiques, Univ ersit ´ e d ’Angers, LAREMA UMR 6093 du CNRS, 2, bd. La v oisier, 49045, Angers, Cedex 01, France and Theor y Division, ITEP, 25, B. Tcheremushkin ska y a, 117259, Moscow, R ussia E-mail addr ess : vretakh @math.ru tgers.edu Volodya .Roubtso v@univ-angers.fr