Geometric algebra and quadrilateral lattices

GEOMETRIC ALGEBRA AND QUADRILA TERAL LA TTICES ADAM DOLIW A Abstract. Motiv ated b y the fundamen tal results of the geometric algebra we study quadrila- teral lat tices in pro jectiv e spaces ov er divi sion rings. After giving the noncomm utative discrete Darb ou x equations we discuss difference s and si milarities with the comm utativ e case. Then we consider the fundamen tal transformation of suc h lattices in th e v ectorial setting and w e sho w the corresponding p erm utability theorems. W e discuss also the p ossibility of obtaining in a simil ar spirit a nonc ommut ativ e ve rsion of the B-(Moutard) qua drilateral lattices. 1. Introduction 1.1. In tegrable discrete geo metry. In the co urse of las t ten years many results of the class ical geometric appr oac h to integrable partial differential equations [47, 41] has b een transfer ed to the discrete s etting (see [25] and r eferences ther ein). The key role in the theory ha s been a ttributed to the multidimensional qua drilateral lattice [2 3], which is the discr ete analo g [43] of a conjugate net [13, 29]. It turns out that integrability of the quadrila teral lattice is enco ded in a very s imple geometric sta temen t, visualized on Fig ure 1. Geometric In tegrability Scheme. Consider p oints x 0 , x 1 , x 2 and x 3 in gener al p osition in the pr oje ctive sp ac e P M , M ≥ 3 . On the plane h x 0 , x i , x j i , 1 ≤ i < j ≤ 3 cho ose a p oint x ij not on the li nes h x 0 , x i i , h x 0 , x j i and h x i , x j i . Then ther e exists the un ique p oint x 123 which b elongs simultane ously to t he thr e e planes h x 3 , x 13 , x 23 i , h x 2 , x 12 , x 23 i and h x 1 , x 12 , x 13 i . x x x x x x x x 0 1 3 2 12 13 23 123 Figure 1. The Geometric Integrabilit y Scheme Despite of (or, thanks to) extremely simple form ula tion of the Geometric Integrability Sc heme, the corres ponding no nlinear discr ete sys tem (the discrete Darboux equations) t urns out to be the generic discrete system integrable by the nonlo cal ¯ ∂ -dressing metho d [9]. Also the finite-gap integration scheme, a s tandard to ol in the integrable systems theo ry [8], can be applied to t hat system in its pure form [1]. W e men tion that the differ ential Darb oux equations, whic h ha ve app eared first in pro jective differential geometry of mult idimensional conjugate nets [13], play an imp ortant role [10, 28] in the Date : Se ptem b er 12, 2021. Key wor ds and phr ases. i n tegrable discrete geometry; i nciden ce ge ometry , Darb oux transformations. 1 2 ADAM DOLIW A m ulticomp onen t Ka dom tsev –P etviashvilii (KP) hiera rc h y , which is commonly considered [14, 32] as the fundamental system of equatio ns in integrability theory . Int egrable r eductions of the quadrila teral lattice (and thus o f the discr ete Da rboux equations) arise from additional constraints whic h are compatible with the geometric integrability scheme. In [20, 21, 2 2 ] we isolated the incidence geometry theor ems whic h are resp onsible for the ba sic reductions of the quadrila teral lattice: B- and C- reductions pr o viding geometric interpretation for BKP and CKP hierarchies [1 5], and the s o called q uadratic reduction. On the geometric level there is no essential difference b et ween the quadrilater al lattice construc- tion and b et ween its Darb oux-t ype transfo rmations [36, 33, 27, 35]. In particula r, a ll classical transformatio ns of conjugate nets [29, 28] hav e found their quadrilater al la ttice ana logs and hav e bee n shown to b e reductions o f the discrete a nalog of the fundamental tra nsformation of Jonas. Although the geometric in tegrability scheme w a s initially conside red for real pro jective spaces, it is v alid to pro jective spaces ov er o ther fields. In pa rticular, finite field v ersion toge ther with the a lgebro- geometric method of construction of so lutions to the corresp onding discr ete equa tions has been given in [26, 6]. The main idea of the pres en t pap er is tha t the geometric integrabilit y scheme remains v alid in pro jective spaces o ver divis ion rings (called a lso skew fields, for details s ee [11]), whose simplest example are quaternions. W e would like to mention that divis ion ring s app ear na turally in a generaliza tion of the notio n of determinant to ma trices with nonc omm utative en tries [30]. The so ca lled quasideterminants ha ve been effectiv e in many areas including noncommutativ e s ymmetric functions, noncommutativ e int egrable sy stems, quantum a lgebras a nd Y angians, noncommutativ e algebraic geometry . Last but not least, the division ring of formal pseudo differen tial oper ators lies in the hear t of the Sato approa c h [42] to integrable systems, s ee a lso [40]. W e should warn the Reader that a ring of square matrices usua lly is not a division ring (the sum of tw o in vertible matrices do es not hav e to be in v ertible or the zero matrix). Also, by the W edder burn theorem, finite division rings are commut ative. The sub ject of no ncomm utative versions o f integrable s ystems was s tudied in the literature in many pap ers, see, for example [18, 34] and r eferences therein; we would like to stress that in the present pap er nonco mm utativity is considered only on the level of dependent v ar iables, i.e., the independent v a riables ar e still commutativ e ones . In relatio n to our work we w ould like to mention the paper [38] wher e the noncomm utative discre te KP equation was cons idered, and the pap ers [7, 46]. Mo reo v er, in [44] a quantization of the discrete Dar boux equatio ns was inv estiga ted. It should b e also mentioned that already in the pap er [9] the discrete Darb oux eq uations, together with some of their tra nsformations, were co nsidered in the matrix version within the non-lo cal ¯ ∂ - dressing method, th us in the noncommut ative setting (for the differential matr ix Dar boux-Mana k ov– Zakharov equations see [49]). The pap er is constructed as follows. In Section 2 we study the multidimensional qua drilateral lattices in pro jectiv e spaces ov er divisio n rings. In par ticular, we co nsider the corres ponding dis- crete nonco mm utative Darb oux equations tog ether with the corres ponding linear problem, and we discuss differe nces and similarities with the commutativ e c ase. In section 3 we give the vectorial fundamen tal tr ansformation for such lattices. Finally , in Section 4 we study p ossibility of the geo- metric generalizatio n of the B-quadrilater al lattices (and thus of the dis crete BKP equations) to the noncommutativ e setting. W e show that the additiona l incidence g eometry structure s which imply int egrability of the B-q uadrilateral lattices [2 1] force the divis ion ring to b e commu tative. The main results of the paper w ere pres en ted in m y talk Ge ometric algebr a and qu adrilater al lattic es during the ISLAND 3 (Integrable Systems: Linear and Nonlinear Dynamics) co nference Alge br aic Asp e cts of Int e gr able Systems , Port Ellen, Isle of Islay , Scotland (July , 20 07). 1.2. Some basi c facts from geom etric algebra. Beca use the in tended target of the pap er con- sists of s pecialists from in tegrable systems theory we start from prese n ting some basic facts on the int erplay betw e en incidence geometry axioms a nd the corresp onding algebraic structures (for details see [3 1 , 2, 3, 5, 4]). A pr oje ctive pl ane is a s et, whos e elements are called p oints and a set of subsets, ca lled lines , satisfying the following four axio ms: GEOMETRIC ALGEBRA AND QUADRILA TERAL LA TTICES 3 P1 Two distinct po in ts lie on one and exactly one line; P2 Two distinct lines meet in precisely one point; P3 There exist four p oin ts with no three collinear. l B’ B A A’ C C’ O Figure 2. The Desargues configur ation: the triangles △ AB C and △ A ′ B ′ C ′ are per spective from the p oint O , and ar e per spective from the line l . It is k no wn that axioms P 1-P3 make p ossible to in tro duce on the plane coordinates from an a lgebraic structure called the ternary ring . If, in addition to P1-P 3, the Desar gues axiom holds: (see Figure 2 ): P4 If t wo triangles ar e pe rspective from a p oint then they ar e pe rspective from a line; then a xioms P1-P 4 imply p ossibilit y of co ordinatization o f the plane in ter ms of a divisio n ring. If, instead, o ne adds to the axioms P1-P 3 the so called Pappus’ axiom : P4’ If the s ix v er tices of a hexagon lie alterna tely on t wo lines, then the three p oin ts of intersection of pair s of opp osite sides a re collinear ; then o ne has co ordinates in co mm utative division ring, i.e. in a field. F or mo re dimensio nal pro jective spaces the basic incidence axioms, analo gous to P1– P3, are enough to show that the spac es are actua lly co ordinatized by div ision rings, i.e., there is no need A E K L M C D F B l k Figure 3. The Pappus c onfiguration: t he vertices of the hexago n AB C DE F lie alternately on tw o (copla nar) lines k and l , and the three p oint s K , L , M of inter- section o f pairs o f opp osite sides are collinear 4 ADAM DOLIW A for the Desa rgues a xiom (whic h beco mes a theorem). In order to hav e a pr o jective geometry over a field one has to add the Pappus ax iom (or its equiv alent formulations). 2. Quadrila teral la ttice in sp a ces over division rings (affine description) Because the Geometric Integrabilit y Scheme is v a lid in pr o jective spac es over division ring s, this motiv ates us to consider quadrila teral lattices in such spaces. 2.1. The Laplace and Darbo ux equations. Consider a multidimensional qua drilateral lattice, i. e., a ma pping x : Z N → P M ( D ) with all the elementary quadrila terals planar [2 3]; here Z N is N ≥ 3 dimensional integer lattice, and P M ( D ) is M ≥ N dimensional rig h t pro jectiv e space ov e r division ring D (we multiply vectors by scala rs fro m rig h t). It turns out that the theo ry of quadrilater al lattices in spaces over divisio n rings do es not differ conside rably fr om the sta ndard case where D was assumed to b e commutativ e. One should b e only careful with the order of co efficien ts. Below w e will us e the affine descr iption o f the quadrila teral la ttice. Recall that the a ffine space A M = P M \ H ∞ is the pro jectiv e space with removed a fixed hyperplane H ∞ ⊂ P M (called the hyperplane at infinit y; s ee, for example [1 2]). Two lines of A M called par allel if they intersect in a po in t of H ∞ . In the affine gauge the lattice is repre sen ted b y a mapping x : Z N → D M , the planar it y conditio n can b e formulated in terms of the Laplace equa tions (2.1) T i T j x − x = ( T i x − x ) A ij + ( T j x − x ) A j i , i 6 = j, i, j = 1 , . . . , N , where T i is the trans lation operator in the i -th dir ection. Then the co efficients A ij : Z N → D satisfy , by compatibilit y of the system (2.1), (2.2) A j k T k A j i = 1 + ( A j i − 1) T j A ik + ( A j k − 1) T j A ki , i, j, k distinct . The i ↔ k symmetry of RHS of (2.2) implies existence of the p oten tials H i , i = 1 , . . . , N , (called the L am ´ e co efficien ts) such that (2.3) A ij = T i  H − 1 i T j H i  , i 6 = j. If we introduce the suitably s caled tangent vectors X i : Z N → D M , i = 1 , ..., N , by equations (2.4) ∆ i x = X i T i H i , (here D i = T i − id) and the r otation coe fficien ts Q ij : Z N → D , i 6 = j , by (2.5) ∆ i H j = Q ij T i H i , i 6 = j, then e quations (2.1) can b e r ewritten as a first or der s ystem (2.6) ∆ j X i = X j T j Q ij , i 6 = j. The compatibility co ndition for the system (2 .6 ) (o r its adjoint (2.5)) gives the following form of the MQL (or discre te Darbo ux) equations (2.7) ∆ k Q ij = Q kj T k Q ik , i, j, k distinct . R emark. The a bov e equa tions (up to small mo dification which, in our languag e, r esults from con- sidering left-vector spaces) app eared first in the matrix setting in [9]. An imp ortan t geometric fact, which lies in the hea rt of int egrability of the qua drilateral lattice, is the multidimensional consistency of the g eometric in tegrability sc heme. Its four dimensio nal version reads as follows (see Fig. 4). F our Di mensional Cons istency o f the Geometric In tegrability Scheme. Given p oints x 0 , x 1 , x 2 , x 3 and x 4 in gener al p osition in P M , M ≥ 4 , cho ose generic p oints x ij ∈ h x 0 , x i , x j i , 1 ≤ i < j ≤ 4 , on the c orr esp onding planes. Using the planarity c ondition c onstruct the p oints x ij k ∈ h x 0 , x i , x j , x k i , 1 ≤ i < j < k ≤ 4 – t he re maining vertic es of the four (c ombinatorial) cu b es. Then ther e a r e four differ ent ways to c onst r u ct the p oint x 1234 , which is the last vertex of the (c ombi- natorial) hyp er cub e. However al l of them give t he same r esu lt due t o t he fact that the p oint x 1234 is the GEOMETRIC ALGEBRA AND QUADRILA TERAL LA TTICES 5 x 1 x 3 x 12 x 4 x 0 2 x x x 234 x 134 x 123 x 23 13 x 124 x 1234 Figure 4 . The fo ur dimensional co nsistency of the geometric integrability sc heme: Starting from the initial quadrilater als ( solid lines) in the fist step of the constr uction (dashed lines) one obta ins four hexa hedra shearing the vertex x 0 . The seco nd step of the constr uction (dotted lines) g iv es another four hexa hedra exhausting this way all the hexahedr a o f the hypercub e. unique interse ction p oint of the four thr e e dimensional subsp ac es h x 1 , x 12 , x 13 , x 14 i , h x 2 , x 12 , x 23 , x 24 i , h x 3 , x 13 , x 23 , x 34 i , and h x 4 , x 14 , x 24 , x 34 i of t he four dimensional subsp ac e h x 0 , x 1 , x 2 , x 3 , x 4 i . 2.2. The bac kwa rd data and the connection factors. The backw a rd tangent vectors ˜ X i , the backw a rd La m ´ e co efficien ts ˜ H i , i = 1 , . . . , N and the backw a rd rota tion co efficien ts ˜ Q ij are defined with the help of the backward shifts T − 1 i . They are again c ho sen in such a wa y that the T − 1 i v ariation o f ˜ X j is pro portiona l to ˜ X i only: (2.8) ∆ i ˜ X j = ( T i ˜ X i ) ˜ Q ij , i 6 = j . Then (2.9) ∆ i x = ( T i ˜ X i ) ˜ H i , and (2.10) ∆ j ˜ H i = ( T j ˜ Q ij ) ˜ H j , i 6 = j. The new functions ˜ Q ij satisfy the backward Darbo ux (MQL) equations (2.11) ∆ k ˜ Q ij = ( T k ˜ Q ik ) ˜ Q kj , i, j, k distinct . R emark. No tice that, opp osite to the commutativ e case [3 3, 24 ], the backward Dar boux equa tions are not the same like the for w ar d Darb oux eq uations (2.7). The connec tion fa ctors ρ i : Z N → D ar e the pro portionality co efficients betw een X i and T i ˜ X i (bo th vectors are prop ortional to ∆ i x ): (2.12) X i = − ( T i ˜ X i ) ρ i , T i H i = − ρ − 1 i ˜ H i , i = 1 , . . . , N . Going a round an elemen tary quadr ilateral it is no t difficult to show that (2.13) ρ j T j Q ij = ( T i ˜ Q j i ) ρ i , and (2.14) T j ρ i = ρ i (1 − ( T i Q j i )( T j Q ij )) = (1 − ( T j ˜ Q ij )( T i ˜ Q j i )) ρ i , i 6 = j . 6 ADAM DOLIW A R emark. In the commutativ e case there exis ts yet ano ther p oten tial (the τ -function of the quadri- lateral la tt ice) such that ρ i = T i τ τ , which is a n immediate co nsequence of (2.15) T i ρ j ρ j = T j ρ i ρ i . The la st equa tion do es not hold in the noncommutative case b ecause, in g eneral ( T i Q j i )( T j Q ij ) 6 = ( T j Q ij )( T i Q j i ). 3. Transf orma tions of the q uadrila teral l a ttice Due to its v ecto rial character, the theory of tr ansformations of quadrilateral lattices tra nsfers to the nonco mm utative ca se a lmost without changes. There fore mos tly w e just state the re lev a n t formulas (the pro ofs are by direct verification along lines given in [3 6, 27, 35]). Given the so lution Y i : Z N → D K , of the linear sy stem (2.6), and given the solution Y ∗ i : Z N → ( D K ) ∗ , of the linear system (2.5); we recall that elements o f D K we r epresen t by column vectors, and element s of its dual ( D K ) ∗ as r o w vectors. These a llo w to construct the linear op erator v a lued po ten tial Ω ( Y , Y ∗ ) : Z N → M K K ( D ), defined by (3.1) ∆ i Ω ( Y , Y ∗ ) = Y i ⊗ T i Y ∗ i , i = 1 , . . . , N ; similarly , one defines Ω ( X , Y ∗ ) : Z N → M M K ( D ) and Ω ( Y , H ) : Z N → D K by ∆ i Ω ( X , Y ∗ ) = X i ⊗ T i Y ∗ i , (3.2) ∆ i Ω ( Y , H ) = Y i ⊗ T i H i . (3.3) W e re mark that b ecause we multiply v ectors from the rig h t then cov ector s a re multiplied from the left. This makes the tensor pro ducts ab o ve well defined. Prop osition 1. If Ω ( Y , Y ∗ ) is invertible t hen the ve ctor function x ′ : Z N → D M given by (3.4) x ′ = x − Ω ( X , Y ∗ ) Ω ( Y , Y ∗ ) − 1 Ω ( Y , H ) , r epr esent a quadrilater al latt ic e (the ve ctorial fundamental tra nsform of x ), whose L am´ e c o efficients H ′ i , normalize d tangent ve ctors X ′ i and r otation c o efficients Q ′ ij ar e given by H ′ i = H i − Y ∗ i Ω ( Y , Y ∗ ) − 1 Ω ( Y , H ) , (3.5) X ′ i = X i − Ω ( X , Y ∗ ) Ω ( Y , Y ∗ ) − 1 Y i , (3.6) Q ′ ij = Q ij − Y ∗ j Ω ( Y , Y ∗ ) − 1 Y i . (3.7) Mor e over, the b ackwar d data and the c onne ction c o efficients tr ansform ac c or ding to ˜ H ′ i = ˜ H i + ρ i Y ∗ i Ω ( Y , Y ∗ ) − 1 Ω ( Y , H ) , (3.8) ˜ X ′ i = ˜ X i + Ω ( X , Y ∗ ) Ω ( Y , Y ∗ ) − 1 Y i ρ − 1 i , (3.9) ˜ Q ′ ij = ˜ Q ij − ρ i Y ∗ i Ω ( Y , Y ∗ ) − 1 Y j ρ − 1 j , (3.10) ρ ′ i = ρ i (1 + T i Y ∗ i Ω ( Y , Y ∗ ) − 1 Y i ) . (3.11) R emark. W e would like to mention that the ab ov e formulas can b e put int o a form using the so called q uasideterminan ts [30], like it w as done, for example, in [34] for a no n-Abelian T o da la ttice. R emark. As it was shown in [27] for the comm utative case, o ther Darb oux-type transformations of the q uadrilateral lattice, like the Laplace, Combescure, L ´ evy , adjoint L´ evy or the radial transforma- tions, can be obtained as reductions of the fundamental trans formation. There a re no obstructions which would prevent the g eometric re asoning applied in [2 7 ] to transfer such a statement to the noncommutativ e case. GEOMETRIC ALGEBRA AND QUADRILA TERAL LA TTICES 7 The vectorial fundamental transfor mation ca n b e considered a s sup erpositio n o f dim V (sc alar) fundamen tal transformatio ns; on in termediate stages the rest of the trans formation data should b e suitably transformed as well. Such a description contains already the principle of per m utability of such transformations, whic h fo llo ws from the following obser v a tion [2 7 ]. Prop osition 2. Assume t he fol lowing splitting of the data of the ve ctorial fundamental tr ansforma- tion (3.12) Y i =  Y a i Y b i  , Y ∗ i =  Y ∗ ai Y ∗ bi  , asso ciate d with the p artition D K = D K a ⊕ D K b , which implies the fol lowing splitting of t he p otentials (3.13) Ω ( Y , H ) =  Ω ( Y a , H ) Ω ( Y b , H )  , Ω ( Y , Y ∗ ) =  Ω ( Y a , Y ∗ a ) Ω ( Y a , Y ∗ b ) Ω ( Y b , Y ∗ a ) Ω ( Y b , Y ∗ b )  , (3.14) Ω ( X , Y ∗ ) =  Ω ( X , Y ∗ a ) , Ω ( X , Y ∗ b )  . Then the ve ctorial fundamental tr ansformation is e quivalent to the fo l lowing sup erp osition of ve ctorial fundamental tr ansformations: 1) T r ansformation x → x { a } with the data Y a i , Y ∗ ai and the c orr esp onding p otentials Ω ( Y a , H ) , Ω ( Y a , Y ∗ a ) , Ω ( X , Y ∗ a ) x { a } = x − Ω ( X , Y ∗ a ) Ω ( Y a , Y ∗ a ) − 1 Ω ( Y a , H ) , (3.15) X { a } i = X i − Ω ( X , Y ∗ a ) Ω ( Y a , Y ∗ a ) − 1 Y a i , (3.16) H { a } i = H i − Y ∗ ia Ω ( Y a , Y ∗ a ) − 1 Ω ( Y a , H ) . (3.1 7) 2) Applic ation on t he r esult t he ve ctorial fundamental tr ansformation with the tr ansforme d data Y b { a } i = Y b i − Ω ( Y b , Y ∗ a ) Ω ( Y a , Y ∗ a ) − 1 Y a i , (3.18) Y ∗{ a } ib = Y ∗ ib − Y ∗ ia Ω ( Y a , Y ∗ a ) − 1 Ω ( Y a , Y ∗ b ) , (3.19) and p otentials Ω ( Y b , H ) { a } = Ω ( Y b , H ) − Ω ( Y b , Y ∗ a ) Ω ( Y a , Y ∗ a ) − 1 Ω ( Y a , H ) = Ω ( Y b { a } , H { a } ) , (3.20) Ω ( Y b , Y ∗ b ) { a } = Ω ( Y b , Y ∗ b ) − Ω ( Y b , Y ∗ a ) Ω ( Y a , Y ∗ a ) − 1 Ω ( Y a , Y ∗ b ) = Ω ( Y b { a } , Y ∗{ a } b ) , (3.21) Ω ( X , Y ∗ b ) { a } = Ω ( X , Y ∗ b ) − Ω ( X , Y ∗ a ) Ω ( Y a , Y ∗ a ) − 1 Ω ( Y a , Y ∗ b ) = Ω ( X { a } , Y ∗{ a } b ) , (3.22) i.e., (3.23) x ′ = x { a,b } = x { a } − Ω ( X , Y ∗ b ) { a } [ Ω ( Y b , Y ∗ b ) { a } ] − 1 Ω ( Y b , H ) { a } . Pr o of. The tr ansformation rules for the in termediate data and po ten tials a re cons equence of prop o- sition 1 . Denote Ω ( Y , Y ∗ ) = Ω =  Ω a a Ω a b Ω b a Ω b b  , Ω ( Y , H ) =  Ω a Ω b  , Ω ( X , Y ∗ ) =  Ω a , Ω b  , and notice that (3.24) Ω =  1 a 0 Ω b a ( Ω a a ) − 1 1 b   Ω a a Ω a b 0 ( Ω b b ) { a }  , which gives (3.25) Ω − 1 =  ( Ω a a ) − 1 − ( Ω a a ) − 1 Ω a b (( Ω b b ) { a } ) − 1 0 (( Ω b b ) { a } ) − 1   1 a 0 − Ω b a ( Ω a a ) − 1 1 b  . Inserting such Ω − 1 int o for m ula (3.4) we obtain (3.26) x ′ = x −  Ω a ( Ω a a ) − 1 , ( Ω b ) { a } (( Ω b b ) { a } ) − 1   Ω a ( Ω b ) { a }  , 8 ADAM DOLIW A th us equation (3 .23).  R emark. The sa me res ult x ′ = x { a,b } = x { b,a } is obta ined exchanging the order of tr ansformations, exchanging also the indices a and b in formulas (3.15)-(3.23). R emark. The sca lar, i.e. K = 1, fundamental trans formation pre serv es in the nonco mm utative cas e its geometric mea ning as a tra nsformation be t ween tw o quadrilater al lattices such that x , x ′ T i x , T i x ′ are coplanar. Ther efore a lso in the noncommutativ e case the fundamental transformatio n can be co nsidered as a construction of a new level (in the new dimension direction) o f the quadr ilateral lattice. In particular, in the case K = 2, K a = K b = 1, any p oint x o f the lattice a nd its tra nsforms x { a } , x { b } and x { a,b } are copla nar. 4. The B-(Mout ard) quadril a teral la ttice W e will concentrate b elo w o n the B-(Moutard) quadrila teral lattice whic h pr o vides geometric int erpretation of the discrete BKP equations. W e will study implications of the co rrespo nding addi- tional (apart fro m the Geometric In teg rabilit y Sc heme) incidence geometric str uctures, whic h assure int egrability of the a bov e men tioned re duction, on the pos sibilit y of der iving their noncommutativ e versions. The main result of this Section is that the mult idimensional co nsistency of the reduction holds if and only if the divisio n ring is co mm utative. In the geometric consideratio ns b elow we assume generality o f configur ations, i.e ., only thos e explicitly stated (and their cons equences) ho ld. In particular, the subspac e h x 0 , x 1 , x 2 , x 3 , x 4 i = h x 1234 , x 123 , x 124 , x 134 , x 234 i of the hyperc ube in the F our Dimensional Co nsistency o f the Geometr ic In tegrability Scheme has dimension four . 4.1. The B-quadrilateral lattice. The B -quadrilateral lattice was defined geo metrically in [2 1] in the comm utative case (we consider for a moment the pro jectiv e spa ce over a (commutativ e) field F ) as follows. Definition 1. A quadr ilateral lattice x : Z N → P M ( F ) is called the B-quadrilater al lattic e if for any triple of different indices i, j, k the p oin ts x , T i T j x , T i T k x a nd T j T k x a re coplanar . In [21] it w as a lso shown that the homo geneous coor dinates x : Z N → F M +1 ∗ satisfy (in appropriate gauge) the system of discrete Mo utard equations [17, 39] (4.1) T i T j x − x = ( T i τ ) T j τ τ T i T j τ ( T i x − T i x ) , 1 ≤ i < j ≤ N , where the τ -function ab ov e is the square ro ot o f the τ -function of the quadrilateral lattice men tioned in the last remar k of section 2.2. The compa tibilit y condition of the linear system (4.1) is Miwa’s discrete BKP s ystem [37] (4.2) τ T i T j T k τ = ( T i T j τ ) T k τ − ( T i T k τ ) T j τ + ( T j T k τ ) T i τ , 1 ≤ i < j < k ≤ N . Because the B-reductio n condition is imp osed on the elementary hex ahedra level, to show inte- grability o f the B-qua drilateral lattice it is impo rtan t to chec k its fo ur dimensional co mpatibilit y with the Geometric Integrability Sch eme. The four dimensional co nsistence of the BQ L-constraint was proved algebraica lly in [21] in the commutativ e cas e. W e will show geo metrically that, in con- trary to the quadr ilateral lattice cas e, one cannot obtain dir ectly the noncommutativ e integrable B-quadrila teral lattice. Theorem 3. Multidi mensional c onsistency of the B-quad rilater al lattic e c onst r aint holds if and only if t he division ring D is c ommutative. Pr o of. It is an immediate c onsequence of t w o Lemmas b elo w.  Lemma 4. The B-c onstr aint is multidimensional ly c onsistent if a nd only if for any triple of differ ent indic es i, j, k the p oints T i x , T j x , T k x and T i T j T k x ar e c oplanar as wel l. GEOMETRIC ALGEBRA AND QUADRILA TERAL LA TTICES 9 x x 123 0 2 x x 12 x 3 x 23 13 x x 1 Figure 5. Elementary hex ahedron of the B-quadr ilateral lattice Lemma 5. Under hyp otheses of t he Ge ometric In t e gr ability Scheme, assume that x 0 , x 12 , x 13 and x 23 ar e c oplanar. Then the fol lowing is tru e: D is comm utative (henc e a field) if and only if the p oints x 1 , x 2 , x 3 and x 123 ar e c oplanar as wel l (s e e Figur e 5). Pr o of of L emma 4. Co nsider a hyper cube with planar faces as in F o ur Dimensional Consistency of the Geometric Integrability Scheme. It cons ists with four “initial hexahedra ” shearing vertex x 0 , and the four “final hexahedra ” shear ing vertex x 1234 . T o demonstra te the first implication co nsider three “final hexahedra” containing the vertex x 123 . Because b y the B-re duction condition x 1 ∈ h x 123 , x 134 , x 124 i , x 2 ∈ h x 123 , x 124 , x 234 i , x 3 ∈ h x 123 , x 134 , x 234 i , then the three planes ab ov e (and therefor e the points x 1 , x 2 and x 3 ) are contained in the three dimensional subspace h x 123 , x 124 , x 134 , x 234 i . Notice that t his subspace contains also x 4 (as a p oin t of the plane h x 124 , x 134 , x 234 i ). Another three dimensiona l subspace h x 123 , x 12 , x 13 , x 23 i is the s ubspace of the initia l hexahedron containing x 123 . By constr uction (according to the Geo metric In teg rabilit y Scheme) it contains also the p oin ts x 1 , x 2 and x 3 . B oth subspaces are differen t (one co n tains x 4 and the other do es not), and b elong to the f our dimensional subs pace h x 0 , x 1 , x 2 , x 3 , x 4 i of the h ype rcube. Therefore their intersect in a plane. Therefor the p oints x 1 , x 2 , x 3 and x 123 are co planar, i.e., the implication holds for o ne of the “initial hexahedra ”; the s tatemen t for three others can b e shown analogo usly . T o show the backward implication we apply similar ar gumen ts, but for a “fina l hexahedron” of the h yp ercube — this time let us concen trate on t hat containing x 1 . No tice that, b y the assumption, the thr ee planes h x 1 , x 2 , x 3 i , h x 1 , x 2 , x 4 i , h x 1 , x 3 , x 4 i . contain, resp ectiv ely , x 123 , x 124 and x 134 . They b elong therefor e to the subspace h x 1 , x 2 , x 3 , x 4 i of dimension three, which contains a lso the p oint x 234 (as a p oint of the plane h x 2 , x 3 , x 4 i ). Another three dimensional subspace h x 1 , x 12 , x 13 , x 14 i , o f the final hexahedro n we ar e considering, by c on- struction (a ccording to the Geo metric Int egrability Sch eme) also contains the p oin ts x 123 , x 124 and x 134 . Notice that this subspa ce cannot contain x 234 , b ecause it would contain then all the vertices of the hyper cube). Both subspaces are different, a nd belong to the four dimens ional subspace h x 0 , x 1 , x 2 , x 3 , x 4 i = h x 1234 , x 123 , x 124 , x 134 , x 234 i , then they b oth int ersect in a plane. This plane co n tains the po in ts x 1 , x 123 , x 124 and x 134 , which shows t hat the he xahedron under in v estigation satisfies the B-r eduction condition; the sta temen t for three o thers can b e shown analogous ly .  The geometric pro of of Le mma 5 can b e obtained by a pplication: (i) its equiv a lence with certain theorem co ncerning the so called quadrang ular set of p oints [21], an (ii) equiv alence of that theor em with v a lidit y of the Pappus’ c onfiguration [12]. Belo w w e give a direct algebr aic pro of. 10 ADAM DOLIW A Alge br aic pr o of of L emma 5. In what follows, b y x ∈ D M +1 ∗ we denote the homogeneous coor dinates of a p oin t x ∈ P M ( D ); r ecall that we deal with right vector spaces. The coplanar it y of the four p oints x 0 , x 1 , x 2 and x 12 can b e algebra ically expressed as the line ar relation x 0 α + x 1 β + x 2 γ + x 12 δ = 0 , where, by the gener alit y assumption (no three of the p oin ts ar e collinear ), all the co efficien ts do no t v anish. Suitably rescaling the homogeneous coor dinates o f the points w e can transfer ab o ve equation to the form (4.3) x 12 = x 0 + x 1 + x 2 . In the equa tion ex pressing coplanarity of the p oint s x 0 , x 1 , x 3 and x 13 we can a gain res cale the homogeneous co ordinates of x 3 and x 13 to get (4.4) x 13 = x 0 + ( x 1 + x 3 ) a. How ever, the coplanarity o f x 0 , x 2 , x 3 and x 23 can b e ex pressed, by plaing with the gauge of x 23 , at most as (4.5) x 23 = x 0 + x 2 b + x 3 c. Then the a dditional condition of coplana rit y of x 0 , x 12 , x 13 and x 23 , which is equiv alent to existence of λ, µ, ν ∈ D such that the expr ession x 12 λ + x 13 µ + x 23 ν is pro portiona l to x 0 , gives c = − b . The homog eneous co ordinates of the point x 123 are given b y x 123 = x 1 A + x 12 B + x 13 C = x 2 ˜ A + x 23 ˜ B + x 12 ˜ C = x 3 A ′ + x 13 B ′ + x 23 C ′ , where the nine co efficien ts A ,. . . , C ′ (to b e determined) are given up to a c ommon factor. Using equations (4 .3 ), (4.4) and (4 .5) with c = − b we obtain decomp osition of x 123 in terms of the basis vectors x 0 , x 1 , x 2 , x 3 x 123 = x 0 ( B + C ) + x 1 ( A + B + aC ) + x 2 B + x 3 aC, = x 0 ( ˜ B + ˜ C ) + x 1 ˜ C + x 2 ( ˜ A + b ˜ B + ˜ C ) − x 3 b ˜ B , = x 0 ( B ′ + C ′ ) + x 1 aB ′ + x 2 bC ′ + x 3 ( A ′ + aB ′ − bC ′ ) . In consequence we obta in eight eq uations B + C = ˜ B + ˜ C = B ′ + C ′ , A + B + aC = ˜ C = aB ′ , B = ˜ A + b ˜ B + ˜ C = bC ′ , aC = − b ˜ B = A ′ + aB ′ − bC ′ , which allow to find the co efficien ts A ,. . . , C ′ . The additional r equiremen t x 123 ∈ h x 1 , x 2 , x 3 i alge braically means that the co efficients in front of x 0 in the a bov e deco mpositions v a nish. Neglecting three of the ab ov e equa tions which simply express A , ˜ A a nd A ′ in terms of six other co efficien ts B ,. . . , C ′ , w e obtain the system B + C = ˜ B + ˜ C = B ′ + C ′ = 0 , ˜ C = aB ′ , B = bC ′ , aC = − b ˜ B , which allows for nontrivial solutio n if and only if ab = ba .  GEOMETRIC ALGEBRA AND QUADRILA TERAL LA TTICES 11 5. Conclusions and discussion Motiv ated by v a lidit y of the Geo metric Integrability Scheme in pro jective spaces ov er divisio n rings w e in vestigated basic properties of the quadrilateral lattices in s uc h spaces and the corre- sp onding version o f the dis crete Darb oux equations. In particular, we show ed that basic ingredients of the v ectorial f undament al transformation of quadrilateral lattices transfer to suc h a setting almost without changes (one has to take care of corre ct ordering o nly). W e would like to mention that in the incidence ge ometry one considers also more gener al spaces over rings [48], whic h should provide geometric interpretation for the matrix Darb oux equatio ns. W e also in vestigated p ossibility o f obtaining the noncomm utativ e version of the B-(Mouta rd) quadrilatera l lattices. It turns out that the a dditional incidence geometry assumptions which imply int egrability of such lattices hold if and only if the division ring under consideration is comm utative (hence a field). The questio n remains o pen for mo re general geometries over r ings. Ackno wledgements The author acknowledges n umero us discuss ions with Jaros law Kosior ek, Andrzej Matra ´ s a nd Mark P anko v on foundations of geo metry . The paper was supp orted b y the P olish Ministry o f Science and Higher Educatio n research g ran t 1 P 03B 017 2 8. References 1. A . A. Akhmet shin, I. M. Kr ic hever a nd Y. S. V olvo vski, Discr ete analo g ues of the Darb oux–Egor off metrics , Proc. Steklo v Inst. Math. 225 (19 99) 16–39. 2. E . Artin, Ge ometric algebr a , Wiley and Sons, New Y ork, 1957. 3. R . Baer, Line ar algebr a and pr oje ctive ge ometry , Academic Press Inc., New Y ork, 195 2. 4. F. Beukenhou t and P . Cameron, Pr oje ctive and affine ge ometry over division rings , [in:] Handb o ok of incidenc e ge ometry , F. Beukenhou t (ed.), pp. 27 –62, Elsevier, Amsterdam, 199 5. 5. A . Beutelspac her and U. Rosen baum, Pr oje ctive ge ometry: fr om foundations to applic ations , U niv ersity Press, Camb ridge, 199 8. 6. M . Bia lec ki, A. Doliw a, Algebr o-geo metric solution of t he discr ete KP e quation over a finite field out of a hyp er- el liptic curve , Comm. Math. Phys. 253 (200 5) 157–17 0. 7. A . I. Bob enk o and Y u. B. Suri s, Inte gr able nonc ommutative equa tions on quad-gr aphs. The c onsistency appr o ach , Lett. Math . Ph ys. 61 (2002) 241–254. 8. E . D. Belokolos, A. I. Bob enk o, V. Z. Enol’ skii, A. R. Its and V. B. Matveev , Algebr o-ge ometric appr o ach to nonline ar integr able e quations , Springer, Berlin, 1994. 9. L. V. Bogdano v and B. G. Konopelchenk o, Lattic e and q -differ enc e Darb oux–Zakhar ov–Manakov systems via ¯ ∂ metho d , J. Phys. A: Math. Gen. 28 L173–L17 8. 10. L. V. Bogdano v and B. G. Konop elc henko, Ana lytic-biline ar appr o ach to inte gr able hier ar chies II. Multic omp onent KP and 2D T o da hier ar chies , J. Math. Phys. 39 (1998) 4701–4728. 11. P . M. Cohn, Sk ew fields. Th e ory of genera l division rings , Cambridge Universit y Press, Cam br idge, 1995. 12. H. S. M . Coxete r, Intr o duction t o ge ometry , Wiley and Sons, New Y ork, 1961. 13. G. Darb oux , L e¸ cons sur les syst´ emes ortho gonaux et les c o or donn´ ees curvilig nes , Gauthier-Villars, Paris, 1910. 14. E. Date, M. Kashiwara, M. Jimbo and T. M iw a, T r ansformation gr oups for soliton e quations , [in:] Pr oceedings of RIMS Symposi um on Non-Linear Int egrable Systems — Classical Theo ry and Quan tum Theory (M. Jim bo and T. Miw a, eds.) W orld Scien ce Publishing Co., Singap ore, 1983, pp. 39–119. 15. E. Date, M. Ji m b o, M. Kashiw ara and T. Miw a, KP hier ar chies of ortho gonal and symple ct ic typ e. T r ansformation gr oups for soliton e quations VI , J. Ph ys. Soc. Japan 50 (1981 ) 381 3–3818. 16. E. Date, M. Jimbo and T. Miwa, Metho d for ge ner ating discr ete soliton e quations. III , J. Ph ys. So c. Japan 52 (1983) 38 8–393. 17. E. Date, M. Jimbo and T. Miwa, Metho d for ge ne r ating discr ete soliton equa tions. V , J. Phys. Soc. Japan 52 (1983) 76 6–771. 18. A. Dimakis and F. M ¨ uller-Hoissen, A new appr o ach to deformation e quations of no nc ommutative KP hier ar chies , arXiv:ma th-ph/07 03067 . 19. A. Doliwa, Geo metric discr eti sa tion of the T o da system , Ph ys. Let t. A 234 (19 97) 187–192. 20. A. Doliwa, Q ua dr atic r e ductions of quadrilater al lattic es , J. Geom. P h ys. 30 (1999) 169 –186. 21. A. Doliwa, The B-quadrilater al lattice, its tr ansformations and t he algebr o-ge ometric c onstruction , J. Geom. Ph ys. 57 (2007) 1171–1192. 22. A. Doliwa, The C-(symmetric) quadrilater al lattic e, its tr ansformations and the algebr o-ge ometric c onstruction , arXiv:07 10.5820 [nlin .SI] . 12 ADAM DOLIW A 23. A. D oliw a and P . M. Santini, Multidimensional quadrilater al lattic es ar e inte gr able , Phys. Lett. A 233 (1997), 365–372. 24. A. Doli w a and P . M. Sa n tini, The symmetric, D-invariant and Egor ov r ed uctions of the quadrilater al lattic e , J. Geom. Phys. 3 6 (2000) 60–102. 25. A. Doliwa and P . M. Sant ini, Inte gr able systems and discr et e ge ometry , [ in:] Encyc lopedia of Mathematical Ph ysics, J. P . F ran¸ cois, G. N aber an d T. S. Tsun (ed s.), Elsevier, 2006, V ol. III, pp. 78-87. 26. A. Doliwa, M . Bia lecki, P . Klim cz ewski, The Hir ota e quation over finite fields: algebr o-ge ometric appr o ach and multisoliton solutions J. Ph ys. A 36 (20 03) 4827– 4839. 27. A. Doliwa, P . M. San tini and M. Ma ˜ nas, T r ansformations of quadrilater al lattices , J. Math. Phys. 41 (20 00) 944–990. 28. A. Doliwa, M . Ma˜ nas, L. Mart ´ ınez Alonso, E. M ed ina, and P . M. Santini, M ul tic omp onent KP hier ar chy and classic al tr ansformations of c onjugate net s , J. Phys. A 32 (1999), 1197–1216. 29. L. P . Eisenhart,, T r ansformations of surfac es , Princeto n Univ er sit y Press, Princeton, 1923. 30. I. Gelfand, S. Gelfand, V. Retakh and R. L. Wilson, Quasideterminants , Adv. Math. 193 (200 5), 56–1 41. 31. D. Hilb ert, Grund lagen der Ge ometrie , B. G. T eubner, Leipzig 1899. 32. V. G. Kac and J. v an de Leur, The n - c omp onent KP hier ar chy and r epr esentation t heo ry , [in:] Importan t dev elopmen ts in soliton theory , (A. S. F ok as and V. E. Zakharo v, eds.) Spri ng er, Berl in, 1993, pp. 302 –343. 33. B. G. Konop elc henko and W. K. Schief, Thr ee-dimensiona l inte gr able lattic es in Euclide an sp ac es: Co njugacy and ortho gonality , Pro c. Roy . Soc. London A 454 (1998), 3075–3104. 34. C. X. Li and J. J. C. Nimmo, Quasideterminant solutions of a non-A b elian T o da lattic e and kink solutions of a matrix sine-Gor don e quation , ar Xiv:0711 .2594 . 35. M . Ma ˜ nas, F undamental t r ansformation for quadrilater al lattic es: first p otentials and τ -functions, symmetric and pseudo-Egor ov re ductions , J. Ph ys. A 34 (20 01) 10413 –10421. 36. M . M a˜ nas, A. Doliwa and P .M . Sant ini, Darb oux tr ansformations for multidimensional quadrilater al lattic es. I , Ph ys. Let t. A 232 (1997) 99–105. 37. T. Miwa, O n Hir ota’s differ ence e quations , Pr oc. Japan Acad. 58 (1982 ) 9–12 . 38. J. J. C. Nimmo, On a non-Ab elian Hir ota-Miwa e quation , J. Ph ys. A: Math. Ge n. 39 (2006) 5053–5065. 39. J. J. C . Nimmo and W. K. Sc hief, Sup erp osition principles asso ciate d with t he Moutar d tr ansformation. An inte gr able discr et isation of a (2+1)-dimensional sine -Gor don system , Proc. R. Soc. London A 4 53 (1997), 255– 279. 40. A. N. Parshin, On a ring of formal pseudo-differ ential op er ators , Proc. Ste klo v Math. Inst. 22 4 (1999) 266–280. 41. C. Rogers and W. K. Sch ief, B¨ acklund and Darb oux tr ansformations. Geo metry and mo dern applic ations in soliton the ory , Cam br idge Universit y Press, Cambridge, 2002. 42. M . Sato and Y. Sato, Soliton equations as dynamic al system on infinit e dimensiona l Gr assmann Manifold , [in:] Nonline ar Partial Diffe r ential Equations in Applie d Sci e nc e , H. F ujita, P . D. Lax and G. Strang (eds.), Lectu re Notes in Num. Appl. Anal., vol. 5, pp. 259–271, North-Hol land, A msterdam, 1982. 43. R. Sauer, Diff er enzenge ometrie , Springer, Berlin, 1970. 44. S. M. Sergeev, Quantization of thr e e-wave equa tions , J. Ph ys. A: Math. Theor. 40 (2007) 12 709-12724. 45. W. K. Schief, La ttic e ge ometry of the discr et e Darb oux, KP, BKP and CKP e q ua tions. Menelaus’ and Carnot’s the or ems , J. Nonl. Math. Phys. 10 Supplemen t 2 (2003) 194–208 . 46. W. K. Sc hief, Discr ete Chebyshev nets and a universal p ermutability the or em , J. Phys. A: Math. T he or. 40 4775–480 1. 47. A. Sym, Soliton surfac es and their applic ations , [in:] Geometric asp ects of the Einstein equations and inte grable systems, Lec ture Note s i n Physics 2 39 , (R. Martini, ed.), Springer, 1985, pp. 154–231. 48. F. D. V eldk amp, Ge ometry over rings , [ in:] Handb o ok of incidence ge ometry , F. Beuke nhout (ed.), pp. 1033–1084, Elsevier, A msterdam, 1995. 49. V. E. Zakharo v and S. V. Manak o v, Construction of multidimensional nonline ar inte gr able systems and their solutions , F unct. A na l. Appl. 19 (1985) 89–101. Adam Doliw a, Wydzia l M a tema tyki i Informa tyki, Uniwersytet W armi ´ nsko-Mazurski w Olsztynie, ul. ˙ Zo lnierska 14, 10-561 Olszt yn, Poland E-mail addr ess : doliwa@matma n.uwm.ed u.pl