B"acklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations

Pro c.Ro y .So c.A 465 (2009) 2613 Decem ber, 2008 B¨ ac klund T ra nsformations and the A tiy a h-W ard Ansatz for Noncomm utativ e An ti-Self-Dual Y ang-Mills Equations Clai re R. Gilson † , Masashi Hamana k a ‡ and Jonathan J. C. Nimmo † † Dep artmen t o f Mathematics, University of Glas g o w Glasgow G12 8QW, UK ‡ Dep artmen t o f Mathematics, University of Nagoya, Nagoya, 464-8602, JAP AN Abstract W e presen t B¨ ac klund transformations for the noncommutativ e anti-self-dual Y ang-Mills equation where the gauge group is G = GL (2) and use it to generate a series of exact solutions from a simple seed solutio n. The solutions generated b y this appro ac h are repre- sen t ed in terms of quasideterminan ts. W e also explain the origins of all of the ingredien ts of the B¨ ac klund tr ansformations within the framew ork of noncommutativ e t wistor theory . In pa rticular we sho w that the generated solutions b elong to a noncomm utative v ersion of the Atiy ah-W a rd ansatz. Key W ords noncomm utativ e in tegrable systems, anti-self-dual Y ang-Mills equation, quasideterminan t solutions, A tiy ah-W ard ansatz, P enrose-W ard tr a nsformation 1. In tro duction In b oth mathematics and phys ics, a noncomm utativ e extension is a natural generalization of a comm utativ e theory that sometimes leads to a new and deep er understanding of that theory . While matrix (or more gen eral, non- Abelian) generalizations ha v e b een s tudied for a long time, the generalization to noncomm utativ e fla t spaces, trigg ered b y dev elopmen ts in string theory (see e.g. D ouglas and Nekrasov (200 2 ); Szab o (20 0 3)), has b ecome a hot topic recen tly . These generalizations are realized b y replacing all pr o ducts in the comm u- tativ e theory with asso ciativ e but noncomm utativ e Mo y al pro ducts. In gauge theories, suc h a noncomm utativ e extension is equiv alen t to the presence of a bac kground magnetic field. Man y successful a pplicatio ns of the analysis o f noncommu tativ e solitons to D -brane dynamics ha v e b een made. (Here w e use the w ord “soliton” as a stable config ura tion whic h possesses lo calized energy densities and hence includes static configurations suc h as instan to ns.) In tegrable systems and soliton theory can also b e extended to a noncommutativ e setting and yield intere sting results and a pplications. (F or reviews, see e.g. D imakis and M ¨ uller-Hoissen ( 2 004); Hamanak a (2003 ); Hamanak a (2005 a); Hamanak a and T o da (2003); Kup ershmidt (2000); Lec h tenfeld (2008); Mazzan t i (2007 ); T amassia (200 5 ).) Among them, the noncommu tativ e anti-self-dual Y a ng-Mills (ASDYM ) equation in 4- dimensions is imp ortan t b ecause in the Euclidean signature (+++ + ) , the ADHM con- struction can b e used to find all exact instanton solutions and giv es rise to new phy sical ob jects suc h as U (1) instan tons (Nekraso v and Sc h w arz (1 9 98)). In the split signature (++ −− ), many noncomm utativ e in tegrable equations can b e deriv ed f r o m the noncom- m utativ e ASD YM equation by a reduction pro cess (See Hamanak a (2005b); Hamanak a (2006) and references therein). Integrable asp ects o f the noncomm utative ASDYM equa- tion can b e understo o d in the geometrical framework of noncomm utativ e t wistor theory (Brain ( 2005); Brain and Ma jid (2008 ) ; Ha nnabuss (2 001); Horv´ ath, Lec h tenfeld and W olf ( 2 002); Ihl and Uhlmann (200 3); Kapustin, Kuznetsov a nd Orlov (2001 ) ; Lec hte n- feld and P op ov (2 002); T ak asaki (2 001)). Therefore it is w orth studying the in tegrable asp ects o f the noncomm utativ e ASD YM equation in detail b oth for the applications to lo w er-dimensional in tegrable equations and to the corresp onding ph ysical situations in the framew ork of a noncomm utativ e a na logue of N = 2 string theory (Lec h tenfeld, Popov and Sp endig (2001a ); Lec h tenfeld, P op ov and Sp endig (200 1 b)). Here solitons are not, in general, static a nd suggest the existence of some kinds of new configuratio ns. F o r these purp oses, B¨ ac klund transformations play an imp ortant role in constructing exact solu- tions and rev ealing an (infinite-dimensional) symmetry of the solutio n space in terms of the transformation group. Also, a t wistor description is us eful for a discu ssion of the origin of the tra nsfor ma t io ns a nd for c hec king whether or not the group action is transitiv e. In the presen t pap er, w e giv e B¨ ac klund transformations for the noncomm utativ e AS- D YM equation where the ga uge group is G = GL (2) a nd use them to generate a se- 1 quence of e xact solutions from a s imple se ed solution. This approac h give s both finite action solutions (instan tons) and infinite action solutions (suc h as nonlinear plane wa ves ). The solutions obta ined are written in terms of quasideterminan ts (Gelfa nd and Retakh (1991); Gelfand and Retakh (1992)) whic h app ear also in the construction o f exact soli- ton solutions in lo w er-dimensional noncomm utativ e in tegrable equations suc h as the T o da equation (Etingo f, Gelfa nd and Retakh (1997); Etingof, Gelfand and Retakh (1998); Li and Nimmo (2008); Li a nd Nimmo (2009 ) ), the KP and KdV equations (Dimakis and M ¨ ulle r-Hoissen (2007); Etingof, Gelfand and Retakh (1 997); Gilson and Nimmo (200 7); Hamanak a (2007)), the Hirota-Miw a equation (Gilson, Nimmo and Oh ta (200 7); Li, Nimmo and T amizhmani (2 0 09); Nimmo (2006)), the mKP equation (Gilson, Nimmo and So oman (2008a ); Gilson, Nimmo and So oman (200 8b)), the Sc hr¨ odinger equation (Gon- c harenk o and V eselo v (1998); Samsono v and P eche ritsin (2 004)), the Da v ey-Stew artson equation (Gilson and Macfarlane (2009)), t he dispersionless equation (Hassan (2009)), and the chiral mo del (Haider and Hassan (2 0 08)), where they pla y the role that determi- nan ts do in the corresp onding comm utativ e integrable systems. W e also clarif y the orig in of the results from the view p oin t of noncommutativ e t wistor theory b y us ing noncommu- tativ e P enrose-W ar d corresp ondence or b y solving a noncommutativ e Riemann-Hilb ert problem. It is sho wn that the solutions g enerated b elong t o a noncomm utative v ersion of the A tiyah-War d an s a tz (Atiy ah and W ard (1977)). The discuss ion and strategy used in this pa p er are simple noncomm utativ e general- izations of those used in the comm utat ive case (Corrigan, F airlie, Y ates and Go ddard (1978); Mason, Chakra v art y and New man (198 8 ); Mason and W o o dhouse (1996)). In the comm uta t ive limit, our results coincide in part with the kno wn results but in the noncomm utativ e case, there are sev eral non trivial po in t s. Firstly , in Section 3.2 w e show that quasideterminan ts are ideally suited to the noncomm utativ e extension of the kn ow n results and greatly simplify the pro ofs of the B¨ ac klund tra nsfor ma t io ns eve n in the com- m utativ e limit. The simple quasideterminan t represen tatio ns of Y ang’s J -matrix are new and imply the imp ortan t result that the B¨ ac klund transformation is not just a ga uge transformation. It is p ossible for the noncomm utativ e t wistor desc ription to w ork as it do es in the comm utativ e setting b ecause one of the three lo cal co ordinates can b e tak en to be a comm utat ive v ariable. In o ur treatmen t, all dep enden t v ariables b elong to a ring, whic h has an asso ciativ e but not necessarily comm utativ e pro duct. Hence t he results w e obtain are av ailable in an y noncommutativ e settings suc h as the Mo y al-deformed, matrix or quaternion-v alued ASD YM equations. 2. The noncomm utativ e A SD YM equation Let us consider noncomm utativ e Y ang-Mills theories in 4- dimensions where the gauge group is GL ( N ) and t he real co ordinates are x µ , µ = 0 , 1 , 2 , 3. In the rest o f the pap er, 2 w e follo w the con v en t io ns of notation giv en in Mason and W oo dho use (1996). (a) T h e nonc omm utative ASDYM e q uation The no ncomm utative ASDYM equation is deriv ed from the compatibilit y condition of the linear syste m Lψ := ( D w − ζ D ˜ z ) ψ = ( ∂ w + A w − ζ ( ∂ ˜ z + A ˜ z )) ψ = 0 , M ψ := ( D z − ζ D ˜ w ) ψ = ( ∂ z + A z − ζ ( ∂ ˜ w + A ˜ w )) ψ = 0 , (1) where A z , A w , A ˜ z , A ˜ w and D z , D w , D ˜ z , D ˜ w denote g auge fields and co v ariant deriv ativ es in Y ang-Mills theory , resp ectiv ely . The (commutativ e) v ariable ζ is a lo cal coordina t e of a one-dimensional complex pro jectiv e space C P 1 , and is called the sp e ctr al p ar ameter . W e note that ψ is not regular at ζ = ∞ b ecause if it w ere regular, b y Liouville’s theorem it w ould b e a constan t fuction and the gauge fields w ould be fla t (see e.g. Mason and W o o dhouse (1996)). The four complex co ordina t es z , ˜ z , w , ˜ w are double null co o r dinates (Mason and W o o d- house (1996)). By imp osing the corresp onding reality c onditions, we c an realize real spaces with differen t signatures, that is, • the Euclidean space, o btained b y putting ¯ w = − ˜ w ; ¯ z = ˜ z , for example,  ˜ z w ˜ w z  = 1 √ 2  x 0 + ix 1 − ( x 2 − ix 3 ) x 2 + ix 3 x 0 − ix 1  , (2) • the Ultrah yp erbo lic space, obtained b y putting ¯ w = ˜ w ; ¯ z = ˜ z , for examp le,  ˜ z w ˜ w z  = 1 √ 2  x 0 + ix 1 x 2 − ix 3 x 2 + ix 3 x 0 − ix 1  or z , w , ˜ z , ˜ w ∈ R . (3) The compatibilit y condition [ L, M ] = 0, giv es rise to a quadratic polynomial in ζ and eac h co efficien t yields the noncomm utative ASDYM equations, with explicit represen ta- tions F w z = ∂ w A z − ∂ z A w + [ A w , A z ] = 0 , F ˜ w ˜ z = ∂ ˜ w A ˜ z − ∂ ˜ z A ˜ w + [ A ˜ w , A ˜ z ] = 0 , F z ˜ z − F w ˜ w = ∂ z A ˜ z − ∂ ˜ z A z + ∂ ˜ w A w − ∂ w A ˜ w + [ A z , A ˜ z ] − [ A w , A ˜ w ] = 0 , (4) whic h is equiv alen t to the ASD condition fo r gauge fields F µν = − ∗ F µν in the real represen tatio n where the sym b ol ∗ is the Ho dge dual op erator. When the compatibility conditions are satisfied, the linear system (1) has N indepen- den t solutions. Hence the solution ψ ( x ; ζ ) can b e in terpreted as an N × N matrix whose columns are the N indep enden t solutions. Gauge transformations act o n the linear system (1) a s L 7→ g − 1 Lg , M 7→ g − 1 M g , ψ 7→ g − 1 ψ , g ∈ G. (5) 3 (b) T h e nonc ommutative Y ang e quation and J, K -ma tric es Here we discuss the differen t p oten tial forms of t he noncomm utativ e ASDYM equations suc h as the noncommutativ e J, K -matrix forma lisms and the noncomm utativ e Y a ng equa- tion, whic h were a lready presen ted by e.g. T ak asaki ( 2001). Let us first discu ss the J -m atrix formalism of the noncommutativ e ASD YM equation (4). The first and second equations of (4) are the compatibilit y conditions of D z h = 0 , D w h = 0 , and D ˜ z ˜ h = 0 , D ˜ w ˜ h = 0 , (6) resp ectiv ely . Here h a nd ˜ h are N × N matr ices, whose N columns of h and ˜ h are inde- p enden t solutions o f the linear systems. The existence of them is formally prov ed in the case of the Mo y al deformation by T a k asaki (2001) and presumed here. These equations can be satisfied b y c ho osing A z = − ( ∂ z h ) h − 1 , A w = − ( ∂ w h ) h − 1 , A ˜ z = − ( ∂ ˜ z ˜ h ) ˜ h − 1 , A ˜ w = − ( ∂ ˜ w ˜ h ) ˜ h − 1 . (7) By defining J = ˜ h − 1 h , the t hir d equation of (4) b ecomes ∂ z ( J − 1 ∂ ˜ z J ) − ∂ w ( J − 1 ∂ ˜ w J ) = 0 . (8) This equation is called the nonc ommutative Y ang e quation and the matrix J is called Y ang’s J -matrix . Gauge transformations act o n h and ˜ h a s h 7→ g − 1 h, ˜ h 7→ g − 1 ˜ h, g ∈ G. (9) Hence Y ang’s J - matrix is gauge in v a rian t. Gauge fields are obtained from a solution J of the noncomm utativ e Y ang’s equation via a decomp osition J = ˜ h − 1 h , and (7). The differen t decompo sitions corresp ond to differen t c hoices of gauge. There is another p oten tial form of the noncommu tativ e ASD YM equation, know n as the K -matrix formalism . In the gauge in whic h A w = A z = 0 , the third equation of (4) becomes ∂ z A ˜ z − ∂ w A ˜ w = 0. This implies the existence of a potential K suc h that A ˜ z = ∂ w K, A ˜ w = ∂ z K . Then the second equation o f (4) b ecomes ∂ z ∂ ˜ z K − ∂ w ∂ ˜ w K + [ ∂ w K, ∂ z K ] = 0 . (10) This gauge is suitable for the discussion of (binary) D arb oux transfor ma t io ns for (non- comm utativ e) ASD YM equation (G ilson, Nimmo and Oh ta (1998 ) ; Nimmo, G ilson and Oh ta (2000); Salee m, Hassan and Siddiq (2007)). 3. B¨ ac klun d transformation for the noncomm utativ e ASD YM equation In this se ction, w e presen t t w o kind of B¨ ac klund tra nsformations whic h lea v e the non- comm utativ e Y ang equation for G = GL (2) in v ariant. This is a noncommutativ e v ersion 4 of the Corrigan- F airlie-Y ates-Go ddar d transformatio n (Corrigan, F a ir lie, Y ates and Go d- dard (1978)). This transformation generates a class of exact solutions whic h belong to a noncomm utativ e vers ion of the Atiy ah-War d ansatz (A tiyah and W ard (1978)) lab eled b y a nonnegativ e integer l ∈ Z ≥ 0 . The o rigin of these results will b e clarified in the next section. In order to discuss B¨ ac klund t r ansformations for the noncomm utativ e Y ang equation, w e parameterize the 2 × 2 matrix J as J =  f − g b − 1 e − g b − 1 b − 1 e b − 1  . (11) This parameterization is alw a ys p ossible when f and b are in v ertible. In con trast with the commutativ e case, whe re only f app ears, in t he noncomm utativ e setting, w e need to in tro duce another v ariable b . In t he comm utativ e limit w e ma y c ho ose b = f . Then the noncomm utative Y ang equation (8) is decompo sed as ∂ z ( f − 1 g ˜ z b − 1 ) − ∂ w ( f − 1 g ˜ w b − 1 ) = 0 , ∂ ˜ z ( b − 1 e z f − 1 ) − ∂ ˜ w ( b − 1 e w f − 1 ) = 0 , ∂ z ( b ˜ z b − 1 ) − ∂ w ( b ˜ w b − 1 ) − e z f − 1 g ˜ z b − 1 + e w f − 1 g ˜ w b − 1 = 0 , ∂ z ( f − 1 f ˜ z ) − ∂ w ( f − 1 f ˜ w ) − f − 1 g ˜ z b − 1 e z + f − 1 g ˜ w b − 1 e w = 0 , (12) where subsc ripts denote partial deriv ativ es. (a) T h e nonc omm utative C o rrigan-F airli e -Y ates-Go ddar d tr ansfo rmation The noncomm utativ e Corrigan-F airlie-Y ates-Go ddard transformation is a comp osition of the following t w o B¨ a c klund tra nsforma t ions for the noncomm utativ e Y ang equations (12). • β -tra nsformation (Mason and W o o dhouse (1996)): e new w = − f − 1 g ˜ z b − 1 , e new z = − f − 1 g ˜ w b − 1 , g new ˜ z = − b − 1 e w f − 1 , g new ˜ w = − b − 1 e z f − 1 , f new = b − 1 , b new = f − 1 . (13) The first four equations can be in terpreted as in tegrabilit y conditions for the first t w o equations in (12). W e can easily che c k that the last tw o equations in (12) are in v arian t under this transformation. • γ 0 -transformation (Gilson, Hamanak a and Nimmo (200 9)):  f new g new e new b new  =  b e g f  − 1 =  ( b − ef − 1 g ) − 1 ( g − f e − 1 b ) − 1 ( e − bg − 1 f ) − 1 ( f − g b − 1 e ) − 1  . (14) This follo ws from the fact that the transformation γ 0 : J 7→ J new is equiv alent to the simple conjugation J new = C − 1 0 J C 0 , C 0 =  0 1 1 0  , whic h clearly lea v es the noncommutativ e Y a ng equation (8) in v arian t. The relatio n (14) is deriv ed by comparing elemen ts in t his tra nsformation. It is easy to see t ha t β ◦ β = γ 0 ◦ γ 0 = id , the identit y transforma t io n. 5 (b) Exact nonc om mutative A tiyah-War d a nsatz solutions No w w e construct exact solutions b y using a chain of B¨ ac klund transformatio ns from a seed solution. Let us consider b = e = f = g = ∆ − 1 0 . W e can easily find that the decomposed noncomm utativ e Y ang equation is r educed to a noncomm uta tiv e linear equation ( ∂ z ∂ ˜ z − ∂ w ∂ ˜ w )∆ 0 = 0. (W e note that fo r the Euclidean space, this is the noncomm utativ e Laplace equation b ecause of the reality condition ¯ w = − ˜ w .) Hence we can generate t w o series of exact solutions R l and R ′ l b y iterating the β - and γ 0 -transformations one af t er the other as follo ws: R 0 α / / ` ` β A A A A A A A A R 1 α / / ` ` β A A A A A A A A R 2 α / / ` ` β A A A A A A A A R 3 α / / ` ` β A A A A A A A A R 4 / / · · · R ′ 1 α ′ / /   γ 0 O O R ′ 2 α ′ / /   γ 0 O O R ′ 3 α ′ / /   γ 0 O O R ′ 4 / /   γ 0 O O · · · where α = γ 0 ◦ β : R l → R l +1 and α ′ = β ◦ γ 0 : R ′ l → R ′ l +1 . These t w o kind of series of solutions in fact arise from some class of noncomm utativ e A tiy ah-W ard ansatz. The explicit form of the solutions R l or R ′ l can b e r epresen ted in terms of quasideterminan t s whose elemen ts ∆ i ( i = − l + 1 , − l + 2 , · · · , l − 1) satisfy ∂ ∆ i ∂ z = ∂ ∆ i +1 ∂ ˜ w , ∂ ∆ i ∂ w = ∂ ∆ i +1 ∂ ˜ z , − l + 1 ≤ i ≤ l − 2 ( l ≥ 2) , (15) whic h imply t ha t ev ery elem en t ∆ i is a solution of the noncomm utative linear equation ( ∂ z ∂ ˜ z − ∂ w ∂ ˜ w )∆ i = 0. A brief in tro duction of quasideterminan ts is giv en in App endix A. The results are as follo ws: • noncomm utativ e A tiy ah-W ard ansatz solutions R l The elem en ts in J l are giv en explicitly in terms of quaside terminan ts of the same ( l + 1) × ( l + 1) matrix: b l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , f l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , e l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , g l = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 . In the comm utativ e limit, w e can easily c hec k by using (50) that b l = f l . The ansatz R 0 leads again to the so called the Corrigan-F airlie-’t Ho oft-Wilcze k ansatz (Corrigan and F airlie (19 7 7); t’Ho o ft (19 76), Wilczek (1977)) . 6 • noncomm utativ e A tiy ah-W ard ansatz solutions R ′ l The elemen ts in J ′ l are giv en explicitly in terms of quasideterminan ts of the l × l matrices: b ′ l = ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ 1 ∆ 0 · · · ∆ 2 − l . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 , f ′ l = ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ 1 ∆ 0 · · · ∆ 2 − l . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 , e ′ l = ∆ − 1 ∆ − 2 · · · ∆ − l ∆ 0 ∆ − 1 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l − 2 ∆ l − 3 · · · ∆ − 1 , g ′ l = ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 2 ∆ 1 · · · ∆ 3 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 1 . In t he comm utativ e case, b ′ l = f ′ l also holds. F or l = 1, we get b ′ 1 = f ′ 1 = ∆ 0 , e ′ 1 = ∆ − 1 , g ′ 1 = ∆ 1 and then the relation (1 5) implies that e ′ 1 ,z = f ′ 1 , ˜ w , e ′ 1 ,w = f ′ 1 , ˜ z , b ′ 1 ,z = g ′ 1 , ˜ w , b ′ 1 ,w = g ′ 1 , ˜ z , and leads to the Corrig an-F airlie-’t Ho oft-Wilczek ansatz a s first p oin ted out b y Y ang (1977). The γ 0 -transformation is prov ed simply using the no ncomm utative Jacobi identit y (53 ) applied to the four corner elemen ts. F or example, b − 1 l = ∆ 0 · · · ∆ 1 − l . . . . . . . . . ∆ l − 1 · · · ∆ 0 − ∆ 1 · · · ∆ 2 − l . . . . . . . . . ∆ l · · · ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . ∆ l − 1 · · · ∆ 0 − 1 ∆ − 1 · · · ∆ − l . . . . . . . . . ∆ l − 2 · · · ∆ − 1 = ( f ′ l − g ′ l b ′− 1 l e ′ l ) . The pro of of the β -transformatio n uses b oth the noncomm utativ e Jacobi iden tit y (53 ) and also the homological relations (54). W e will consider the first equation in the β - transformation: e ′ l,w = f − 1 l − 1 g l − 1 , ˜ z b − 1 l − 1 . (16) The RHS is equal to − b ′ l g l − 1 ( g − 1 l − 1 ) ˜ z g l − 1 f ′ l . (17) In this, it follows from (54) that the first t w o and last tw o factors are b ′ l g l − 1 =            0 ∆ − 1 · · · ∆ 1 − l 0 ∆ 0 · · · ∆ 2 − l . . . . . . . . . 0 ∆ l − 3 · · · ∆ − 1 1 ∆ l − 2 · · · ∆ 0            , g l − 1 f ′ l =          ∆ 0 ∆ − 1 · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . ∆ l − 2 ∆ l − 3 · · · ∆ 0 ∆ − 1 1 0 · · · 0 0          . (18) 7 Next, from (55), w e ha v e ( g − 1 l − 1 ) ˜ z =            ∆ 0 , ˜ z ∆ − 1 · · · ∆ 1 − l ∆ 1 , ˜ z ∆ 0 · · · ∆ 2 − l . . . . . . . . . ∆ l − 2 , ˜ z ∆ l − 3 · · · ∆ − 1 ∆ l − 1 , ˜ z ∆ l − 2 · · · ∆ 0            + l − 1 X k =1            ∆ − k , ˜ z ∆ − 1 · · · ∆ 1 − l ∆ 1 − k , ˜ z ∆ 0 · · · ∆ 2 − l . . . . . . . . . ∆ l − 2 − k , ˜ z ∆ l − 3 · · · ∆ − 1 ∆ l − 1 − k , ˜ z ∆ l − 2 · · · ∆ 0                     ∆ 0 ∆ − 1 · · · ∆ − k · · · ∆ 1 − l . . . . . . . . . . . . ∆ l − 2 ∆ l − 3 · · · ∆ l − 2 − k · · · ∆ − 1 0 0 · · · 1 · · · 0          . The effect o f the left and righ t factors on this expression is t o mo v e expansion p oin ts as sp ecified in (54 ) , obtaining f − 1 l − 1 g l − 1 , ˜ z b − 1 l − 1 = −            ∆ − 1 · · · ∆ 1 − l ∆ 1 − l, ˜ z ∆ 0 · · · ∆ 2 − l ∆ 2 − l, ˜ z . . . . . . . . . ∆ l − 3 · · · ∆ − 1 ∆ − 1 , ˜ z ∆ l − 2 · · · ∆ 0 ∆ 0 , ˜ z            − l − 2 X k =0          ∆ − 1 · · · ∆ 1 − l ∆ − k , ˜ z ∆ 0 · · · ∆ 2 − l ∆ 1 − k , ˜ z . . . . . . . . . ∆ l − 2 · · · ∆ 0 ∆ l − 1 − k , ˜ z                   0 · · · 1 · · · 0 0 ∆ 0 · · · ∆ − k · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . ∆ l − 2 · · · ∆ l − 2 − k · · · ∆ 0 ∆ − 1          . On the other hand, e ′ l,w =            ∆ − 1 · · · ∆ 1 − l ∆ − l,w ∆ 0 · · · ∆ 2 − l ∆ 1 − l,w . . . . . . . . . ∆ l − 3 · · · ∆ − 1 ∆ − 2 ,w ∆ l − 2 · · · ∆ 0 ∆ − 1 ,w            + l − 2 X k =0          ∆ − 1 · · · ∆ 1 − l ∆ − k − 1 ,w ∆ 0 · · · ∆ 2 − l ∆ − k ,w . . . . . . . . . ∆ l − 2 · · · ∆ 0 ∆ l − 2 − k ,w                   0 · · · 1 · · · 0 0 ∆ 0 · · · ∆ − k · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . ∆ l − 2 · · · ∆ l − 2 − k · · · ∆ 0 ∆ − 1          , and then the result follo ws immediately from ∆ i,w = ∆ i +1 , ˜ z in ( 1 5). W e can find that the proo f of these results relies on using quaside terminan t iden ti- ties alone. Th us w e can conclude that nonc omm utative B¨ acklund tr ansfo rmations ar e identities of quasideterminants . 8 W e can also presen t a compact form o f the w hole of Y ang’s J - matrix in terms of a single quasideterminan t expanded b y a 2 × 2 submatrix:        a b c d e f g h i        :=         a b d e         a c d f         a b g h         a c g i         . The solutions for the J -matrix can be presen ted as follo ws: • noncomm utativ e A tiy ah-W ard ansatz solutions R l J l =                 0 − 1 0 · · · 0 0 1 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0                 , J − 1 l =                 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 1 0 0 · · · 0 − 1 0                 . • noncomm utativ e A tiy ah-W ard ansatz solutions R ′ l J ′ l =                 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l − 1 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l 0 . . . . . . . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 0 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0 0 1 0 · · · 0 0 0                 , J ′− 1 l =                 0 0 0 · · · 0 1 0 ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ − l 0 ∆ 1 ∆ 0 · · · ∆ 2 − l ∆ 1 − l . . . . . . . . . . . . . . . . . . 0 ∆ l − 1 ∆ l − 2 · · · ∆ 0 ∆ − 1 − 1 ∆ l ∆ l − 1 · · · ∆ 1 ∆ 0                 . Because J is ga ug e in v ariant, this shows that the presen t B¨ ack lund transformation is not just a gauge transformation but a nontrivial transformation. The pro of of these repres en tations is given b y using the no ncommutativ e Jacobi iden- tit y , homological relations and the in v ersion form ula for J : J − 1 =  f − 1 f − 1 g − ef − 1 b − ef − 1 g  , (19) or simply using the form ula (56). (F or a detailed pro of, see App endix A in Gilson, Hamanak a and Nimmo (2 009).) 9 (c) Some Ex p licit Examples By solving the noncomm utativ e linear equation ( ∂ z ∂ ˜ z − ∂ w ∂ ˜ w )∆ 0 = 0 for the seed solutio n of the B¨ ac klund transformations, w e can obtain exact solutions ex plicitly . F or example, in Euclidean space, the noncommutativ e linear equation is j ust the 4- dimensional noncommutativ e Laplace equation whose solutions include a noncommutativ e v ersion of the fundamen tal solutio n: ∆ 0 = 1 + P k p =1 ( a p / ( z ˜ z − w ˜ w )) ( a p are constants ), whic h leads to noncommutativ e instan ton solutions whose instan ton num b er is k (Correa, Lozano, Moreno and Sc hap osnik (200 1 ); Lec h tenfeld and P op o v (2002); Nekraso v and Sc h w arz (1998) ) . The B¨ ac klund transformations do not increase the instan ton n um b er. There is a lso a simple new solution: ∆ 0 = c exp ( az + b ˜ z + aw + b ˜ w ) , (20) where a, b and c are constan ts. This leads to a noncomm utativ e v ersion of no nlinear plane w a v e s olutions (de V ega (1988)). These solutions b eha v e as standard solitons in low er- dimension and do not deca y at infinit y , whic h implies that this giv es an infinite v alue for the Y ang-Mills action. By follo wing the analysis used in Hamanak a (2007) for the noncomm utativ e KP equation, the asymptotic b eha viour of these solutions can be shown to be the same a s the corresp onding comm utativ e ones. Other solutions are also easily obtained and a more detailed discu ssion on this t o pic will be rep orted elsewhe re. 4. Twistor descriptions of the noncomm utativ e ASDYM equations In this section, we explain the o rigin o f the B¨ acklund transfor mat ions for the noncomm u- tativ e ASD YM equations and noncomm utativ e A tiy ah-W ard ansatz solutions from the geometrical viewpoint of noncommu tativ e tw istor theory . Here w e just need a one-to- one correspondence b etw een a solution of the noncomm utativ e ASD YM equations and a patc hing matrix P = P ( ζ w + ˜ z , ζ z + ˜ w, ζ ) of a noncomm utativ e holomorphic ve ctor bundle on a noncomm utativ e 3- dimensional pro jectiv e space, whic h is called the noncomm utativ e P enrose-W ard corresp ondence. This corresp ondence is established in the Mo y al-deformed case b y Brain (2005); Bra in and Ma jid (2008); Kapustin, Kuzn etso v and Orlo v (2001 ); Lec h t enfeld and P op ov (2002); T ak asaki (2001 ) and here we apply their formal pro cedure to general noncomm utative situations. Suc h twis tor treatmen ts are useful not only for constructing exact solutions but a lso for c hec king whether the B¨ a cklund transformation act on the solution spaces transitiv ely . In order to review this corresp o ndence briefly , w e in tro duce ano t her linear system defined on anot her lo cal patc h whose (comm utative) coordinat e is ˜ ζ = 1 /ζ , ( ˜ ζ D w − D ˜ z ) ˜ ψ = 0 , ( ˜ ζ D z − D ˜ w ) ˜ ψ = 0 . (21) 10 A nontrivial solution ˜ ψ ( N × N matrix) of the linear system (21) is supp o sed to exist and is not regular at ˜ ζ = ∞ (or equiv alen tly ζ = 0) as disc ussed earlier for ψ . An y solution of the noncommu tativ e ASDYM equation determines solution ψ and ˜ ψ that are unique up to gauge transformatio n, and then the corresp onding patc hing matrix is giv en by P ( x ; ζ ) = ˜ ψ − 1 ( x ; ζ ) ψ ( x ; ζ ) . (22) Con v ersely , if a patc hing matrix P = P ( ζ w + ˜ z , ζ z + ˜ w , ζ ) is factorized as P ( ζ w + ˜ z , ζ z + ˜ w , ζ ) = ˜ ψ − 1 ( x ; ζ ) ψ ( x ; ζ ) , (23) where ψ and ˜ ψ a re regular near ζ = 0 and ζ = ∞ , resp ectiv ely , then ψ and ˜ ψ are solutions of the linear syste m ( 1) for the noncomm utativ e ASDYM equation. Then w e can reco v er the ASDYM gauge fields in terms o f h and ˜ h by using ( 7 ) and the fact that h ( x ) = ψ ( x, ζ = 0) , ˜ h ( x ) = ˜ ψ ( x, ζ = ∞ ). (W e can easily understand this b y comparing the linear s ystems (1) and (21) with (6).) In the comm utativ e case, if P is holomorphic w.r.t. ζ , then the factorization is guar- an teed b y the Bir khoff factorizatio n theorem. In the case o f the Moy al deformatio n, this is formally pro v ed by T ak asaki (20 01). Here w e will see that under the A tiy ah-W ard ansatz f o r the patc hing matrix, the factorization problem (t he Riemann-Hilb ert pr oblem ) is solv ed. In this section, w e fix the ga uge to b e, what w e call in this pap er, the Mason-Wo o dhouse gauge J =  f − g b − 1 e − g b − 1 b − 1 e b − 1  =  1 g 0 b  − 1  f 0 e 1  = ˜ h − 1 MW h MW . (24) W e not e that the gauge transforma t io n g = diag( f 1 / 2 , b 1 / 2 ) connects the Mason-W o o dhouse gauge with a noncomm utativ e v ersion of Y ang’s R -gauge (Y ang ( 1 977)): J =  f − g b − 1 e − g b − 1 b − 1 e b − 1  =  f − 1 / 2 f − 1 / 2 g 0 b 1 / 2  − 1  f 1 / 2 0 b − 1 / 2 e b − 1 / 2  = ˜ h − 1 R h R , (25) where the square ro o t is considered as any quan tit y whic h satisfies f 1 / 2 f 1 / 2 = f , f − 1 / 2 := ( f 1 / 2 ) − 1 and whenev er this notation is used, it is assumed to exis t. The w av e functions ψ and ˜ ψ can be expanded b y ζ and ˜ ζ = 1 /ζ , resp ectiv ely: ψ = h + O ( ζ ) =  h 11 + P ∞ i =1 a i ζ i h 12 + P ∞ i =1 b i ζ i h 21 + P ∞ i =1 c i ζ i h 22 + P ∞ i =1 d i ζ i .  , ˜ ψ = ˜ h + O ( ˜ ζ ) =  ˜ h 11 + P ∞ i =1 ˜ a i ˜ ζ i ˜ h 12 + P ∞ i =1 ˜ b i ˜ ζ i ˜ h 21 + P ∞ i =1 ˜ c i ˜ ζ i ˜ h 22 + P ∞ i =1 ˜ d i ˜ ζ i .  . (26) 11 (a) Riemann-Hilb ert pr oblem for nonc ommutative A tiyah-War d Ansatz F rom now o n, we restrict ourselv es to G = GL (2). In this case, we can tak e a simple ansatz for the patc hing matrix P , whic h is called the A tiyah-W ard ansatz in the comm utativ e case (A tiy ah a nd W ard (1978)). The noncomm utative generalization of this ansatz is straigh tforw ard and a ctually leads to a solution of the factor izat io n problem. The l -th order noncommu tativ e A tiy ah-W ard ansatz ( l = 0 , 1 , 2 , . . . ) is sp ecified b y choosing the patc hing matrix to b e: P l ( x ; ζ ) =  0 ζ − l ζ l ∆( x ; ζ )  . (27) (The standard r epresen tation of t he a nsatz is not P l but C 0 P l . Both represen tations a re essen tially the same.) W e note that the co ordinate dep endence of P l = P l ( ζ w + ˜ z , ζ z + ˜ w , ζ ) implies that ( ∂ w − ζ ∂ ˜ z )∆ = 0 , ( ∂ z − ζ ∂ ˜ w )∆ = 0. Hence , the Lauren t expansion of ∆ w.r.t. ζ ∆( x ; ζ ) = ∞ X i = −∞ ∆ i ( x ) ζ − i , (28) giv es rise to the follow ing relationships amongst t he co efficien ts, ∂ ∆ i ∂ z = ∂ ∆ i +1 ∂ ˜ w , ∂ ∆ i ∂ w = ∂ ∆ i +1 ∂ ˜ z , (29) whic h coincide with the recurrence relation ( 15). W e will soo n see that the co efficien ts ∆ i ( x ) are the scalar functions in the solutions generated by the B¨ acklund tr a nsformations in the previous section. W e will now solv e the factorizatio n problem ˜ ψ P l = ψ for the noncomm utativ e A tiy ah- W ard ansatz. In explicit form this is  ˜ ψ 11 ˜ ψ 12 ˜ ψ 21 ˜ ψ 22   0 ζ − l ζ l ∆( x ; ζ )  =  ψ 11 ψ 12 ψ 21 ψ 22  , (30) where ψ ij is the ( i, j )- th elemen t of ψ , and so, ˜ ψ 12 ζ l = ψ 11 , ˜ ψ 22 ζ l = ψ 21 , (31) ˜ ψ 11 ζ − l + ˜ ψ 12 ∆ = ψ 12 , ˜ ψ 21 ζ − l + ˜ ψ 22 ∆ = ψ 22 . (32) F rom (26) and (31), we find that some en tries in ψ and ˜ ψ are p olynomials w.r.t. ζ and ˜ ζ = ζ − 1 : ψ 11 = h 11 + a 1 ζ + a 2 ζ 2 + · · · + a l − 1 ζ l − 1 + ˜ h 12 ζ l , ψ 21 = h 21 + c 1 ζ + c 2 ζ 2 + · · · + c l − 1 ζ l − 1 + ˜ h 22 ζ l , ˜ ψ 12 = ˜ h 12 + a l − 1 ζ − 1 + a l − 2 ζ − 2 + · · · + a 1 ζ 1 − l + h 11 ζ − l , ˜ ψ 22 = ˜ h 22 + c l − 1 ζ − 1 + c l − 2 ζ − 2 + · · · + c 1 ζ 1 − l + h 21 ζ − l . (33) 12 By substituting these formulae in to (32), w e get the following sets of equations for h a nd ˜ h in the co efficien ts of ζ 0 , ζ − 1 , · · · , ζ − l : ( h 11 , a 1 , · · · , a l − 1 , ˜ h 12 ) D l +1 = ( − ˜ h 11 , 0 , · · · , 0 , h 12 ) , ( h 21 , c 1 , · · · , c l − 1 , ˜ h 22 ) D l +1 = ( − ˜ h 21 , 0 , · · · , 0 , h 22 ) , (34) where D l :=      ∆ 0 ∆ − 1 · · · ∆ 1 − l ∆ 1 ∆ 0 · · · ∆ 2 − l . . . . . . . . . . . . ∆ l − 1 ∆ l − 2 · · · ∆ 0      . (35) These noncomm utativ e linear equations can b e solv ed in terms of quasideterminan ts (cf. (49)) as h 11 = h 12 | D l +1 | − 1 1 ,l +1 − ˜ h 11 | D l +1 | − 1 1 , 1 , h 21 = h 22 | D l +1 | − 1 1 ,l +1 − ˜ h 21 | D l +1 | − 1 1 , 1 , ˜ h 12 = h 12 | D l +1 | − 1 l +1 ,l +1 − ˜ h 11 | D l +1 | − 1 l +1 , 1 , ˜ h 22 = h 22 | D l +1 | − 1 l +1 ,l +1 − ˜ h 21 | D l +1 | − 1 l +1 , 1 . (36) In (36) there are four equations but eigh t unkno wns a nd so in order to solve them it is necessary to imp ose four conditions cor r esponding to a choice of gauge. In particular, t he Mason-W o o dhouse gauge ( h 12 = ˜ h 21 = 0 , ˜ h 11 = h 22 = 1) leads to simple represen tations: h 11 = − ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , h 21 = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , ˜ h 12 = − ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 2 · · · ∆ 0 − 1 , ˜ h 22 = ∆ 0 ∆ − 1 · · · ∆ − l ∆ 1 ∆ 0 · · · ∆ 1 − l . . . . . . . . . . . . ∆ l ∆ l − 1 · · · ∆ 0 − 1 , (37) whic h coincides exactly with the solutions R l generated b y the B¨ ack lund tra nsfor ma t io n in t he previous section except for signs in f l and g l . (The mismatc h of the signs is not essen tial b ecause it can b e absorb ed into the reflection s ymmetry f 7→ − f , g 7→ − g of the noncomm utativ e Y ang equation (12).) That is wh y we call t hem the noncomm utativ e A tiy ah-W ard ansatz solutio ns. The class of solutions R ′ l is also obtained in the same w ay b y starting with the A tiy ah-W ard ansatz C − 1 0 P l C 0 . 13 (b) Origin of the nonc ommutative Corrigan-F airlie-Y ates-Go ddar d tr ansformation Finally let us discuss the or ig in of the noncomm utativ e Corrigan-F airlie-Y ates-G o ddard transformation, constructed from the β -transformation and the γ 0 -transformation and giv e a generalization of it. Suc h geometrical understanding is useful when discussing whether the group action o f the B¨ ac klund tra nsformations is tr a nsitiv e and hence to find the symmetry of the noncommutativ e ASD YM equation. The presen t results are essen- tially due to Mason, Chakra v art y and Newman (198 8 ), Mason and W o o dhouse (1996). These transformations can b e view ed as adjoin t actions of the patc hing matrix P : β : P 7→ P new = B − 1 P B , γ 0 : P 7→ P new = C − 1 0 P C 0 , (38) where B =  0 1 ζ − 1 0  , C 0 =  0 1 1 0  . (39) It is o b vious that β ◦ β = id, γ 0 ◦ γ 0 = id . The comp osition of these tr a nsformations actually maps the l -t h Atiy ah- W ard ansatz to the ( l + 1)- t h one: P l 7→ C − 1 0 B − 1  0 ζ − l ζ l ∆  B C 0 =  0 ζ − ( l +1) ζ l +1 ∆  = P l +1 . (40) The action of C 0 leads to h 7→ hC 0 , ˜ h 7→ ˜ hC 0 and hence to the γ 0 -transformation, and the action of B is defined at the lev el of ψ and ˜ ψ as follo ws: ψ new = g − 1 ψ B , ˜ ψ new = g − 1 ˜ ψ B , (41) where g − 1 =  0 ζ b − 1 f − 1 0  . (42) The gauge transformatio n g is needed to maintain the regularit y of ψ and ˜ ψ , w.r.t ζ and ˜ ζ resp ectiv ely , in the factorization of P . The explicit calculation g iv es ψ new =  b − 1 ψ 22 ζ b − 1 ψ 21 ζ − 1 f − 1 ψ 12 f − 1 ψ 11  . (43) In the ζ → 0 limit, this reduces to h new =  f new 0 e new 1  =  b − 1 0 f − 1 k 12 1  , (44) where ψ = h + k ζ + O ( ζ 2 ). 14 Here w e note that the linear system (1) can b e repres en ted in terms of b, f , e, g as Lψ = ( ∂ w − ζ ∂ ˜ z ) ψ +  − f w f − 1 ζ g ˜ z b − 1 − e w f − 1 ζ b ˜ z b − 1  ψ = 0 , M ψ = ( ∂ z − ζ ∂ ˜ w ) ψ +  − f z f − 1 ζ g ˜ w b − 1 − e z f − 1 ζ b ˜ w b − 1  ψ = 0 . (45) By considering the first order term of ζ in the (1,2 ) comp onen t of the first equation, w e find that ∂ w ( f − 1 k 12 ) = − f − 1 g ˜ z b − 1 . (46) Hence from the (1,1) and (2,1) componen ts of (44), w e hav e f new = b − 1 , ∂ w e new = ∂ w ( f − 1 k 12 ) = − f − 1 g ˜ z b − 1 , (47) whic h are just parts of the β - t ransformation (13). In similar w a y , w e can get the other ones. Therefore the β -transformat io n (13) can be in terpreted as the transformation of the pat ching matrix P 7→ B − 1 P B together with the gauge transformation g . The results presen ted in the section and the previous one lead to a simpler pro of of the results in Section 3. W e note that the γ 0 -transformation can b e generalized to the following transformatio n (the γ -tr ansformation ): γ : P 7→ P new = C − 1 P C , (48) where C is an arbitr a ry constan t matrix. The actions of β - and γ - t r a nsformations gen- erate the a ction of the lo op g r o up LGL (2) on P b y conjugation. Therefore t he symmetry group of the noncomm utative ASD YM equation includes the lo op group LGL (2) as a subgroup. 5. Conclusion and Discussion In this pap er, w e ha v e presen ted B¨ a c klund transformations for the noncomm utativ e AS- D YM equation with G = GL (2) and constructed from a simple seed solution a series of exact noncomm utativ e A tiyah-W ard ansatz solutions express ed explicitly in t erms of quasideterminan ts. W e ha ve found that the B¨ ack lund transformations generate a wide class of new solutions. W e ha v e also giv en the orig in of the B¨ ac klund transformatio n and the generated solutions in the framework of noncommutativ e twis tor theory and general- ized them. The pr esen t results could b e t a k en as the starting p oint to rev eal an infinite-dimensional symmetry of the noncommutativ e ASDYM equation in terms of some infinite-dimensional algebra. W e hav e to prov e that the A tiy ah-W ard ansatz co vers a ll solutions of noncom- m utativ e ASD YM equation and generalize the B¨ acklund transformat ions β and γ so that they should act on the solution space tra nsitiv ely . 15 F urthermore inv estigation of the noncomm utativ e extension o f a bilinear f orm a p- proac h to the ASD YM equation (G ilson, Nimmo and Oh t a (19 98); Sa sa, Oh ta and Mat - sukidaira (1998); W ang and W adati (2004)) w ould be beneficial b ecause many asp ects in these pap er are close t o ours. The relationship with noncomm utativ e Darb oux and noncomm utativ e bina r y Darb oux tra nsformations (Salaam, Ha ssan and Siddiq (2007)) is also in teresting. A cknow le dgments MH would lik e to tha nk L. Mason and JJCN for hospitality (and man y helpful commen ts from LM) during sta y at Mathematical Institute, Univ ersit y of Oxford and at Departmen t of Mathematics, Univ ersity of Glasgo w, respectiv ely . The w ork of MH w as supported by Gran t-in-Aid for Y oung Scien tists (#18740142), the Nishina Memorial F oundation and the Daik o F oundation. A. Brief review of quasideterminan ts In this section, w e give a brief intro duction to quasideterminan ts, in tro duced b y Gelfand and Retakh (2001), in whic h a few of the k ey prop erties whic h play imp o rtan t roles in Section 3 are describ ed. More detailed discussion is seen in the surv ey (Gelfand, Gelfand, Retakh and Wils on (2005)). Quasideterminan ts are defined in terms of inv erse matrices and we supp ose the exis- tence of all matrix inv erses r eferred to. Let A = ( a ij ) b e an n × n matrix and B = ( b ij ) b e the in v erse matrix of A , tha t is, AB = B A = 1. Here the matrix en tries b elong to a noncomm utativ e ring. Quasideterminan ts o f A are defined f ormally as the in v erses of the en tries in B : | A | ij := b − 1 j i . (49) In the case that v ariables comm ute, this is reduced to | A | ij = ( − 1) i + j det A det A ij , (50) where A ij is the matrix obtained from A by deleting the i -th ro w and the j -th column. W e can a lso write dow n a more explic it definition of quasideterminan ts. In order to see this, let us rec all the follo wing fo r mula for the in v erse a square 2 × 2 blo ck square matrix:  A B C d  − 1 =  A − 1 + A − 1 B S − 1 C A − 1 − A − 1 B S − 1 − S − 1 C A − 1 S − 1  , where A is a square matrix, d is a single elemen t and B and C are column and ro w v ectors of appropr ia te length and S = d − C A − 1 B is called a Schur c omplem ent . In fact this formula is v alid fo r A , B , C and d in any ring not just for matrices. Th us the 16 quasideterminan t a ssociated with the b ottom right elemen t is simply S . By choosing an appropriate partitioning, an y en try in the in v erse of a square matrix can be express ed as the in ve rse of a Sc h ur complemen t and hence quaside terminan ts can also b e defined recursiv ely b y: | A | ij = a ij − X i ′ ( 6 = i ) ,j ′ ( 6 = j ) a ii ′ (( A ij ) − 1 ) i ′ j ′ a j ′ j = a ij − X i ′ ( 6 = i ) ,j ′ ( 6 = j ) a ii ′ ( | A ij | j ′ i ′ ) − 1 a j ′ j . (51) It is sometimes con v enient to use the following alternativ e notation in whic h a b o x is dra wn abo ut the corresp onding en try in the matrix: | A | ij = a 11 · · · a 1 j · · · a 1 n . . . . . . . . . a i 1 a ij a in . . . . . . . . . a n 1 · · · a nj · · · a nn . (52) Quasideterminan ts ha v e v arious inte resting prop erties similar t o those of determinan t s. Among them, the follo wing ones play imp ortant roles in this pa per. In the blo c k matrices giv en in these results, low er case letters denote single en tries and upp er case letters denote matrices of compatible dimensions so that the ov erall mat r ix is square. • noncomm utativ e Jacobi iden tity A simple and useful sp ecial case of the noncomm utativ e Sylv ester’s Theorem (Gelfand and Retakh (1991)) is       A B C D f g E h i       =     A C E i     −     A B E h         A B D f     − 1     A C D g     . ( 5 3) • Homological relations (Gelfand and R etakh (1 991))       A B C D f g E h i       =       A B C D f g E h i             A B C D f g 0 0 1       ,       A B C D f g E h i       =       A B 0 D f 0 E h 1             A B C D f g E h i       (54) • A deriv ativ e form ula for quasideterminan ts (G ilson a nd Nimmo (2009))     A B C d     ′ =     A B ′ C d ′     + n X k =1      A ( A k ) ′ C ( C k ) ′          A B e t k 0     , (55) 17 where A k is the k th column of a matrix A and e k is the column n - vector ( δ ik ) (i.e. 1 in the k th ro w and 0 elsewhere). • A sp ecial form ula of in ve rse of a quasideterminan t (Gilson, Hamanak a and Nimmo (2007))         a B c α D E F 0 g H i 0 β 0 0 0         − 1 =         0 0 0 γ 0 a B c 0 D E F δ g H i         , (56) with α β = γ δ = − 1 , α + γ = 0, where lo w er case letters denote single en tries, upp er case letters denote matrices of compatible dimensions and Greek letters are scalars (i.e. comm ute with ev erything). 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