The pentagon relation and incidence geometry

THE PENT A GON RELA TION AND INCIDENC E GEOMETR Y ADAM DO LIW A AND SERGEY M . S ERGE EV Abstra ct. W e define a map S : D 2 × D 2 99K D 2 × D 2 , where D is a n arbitrary division ring (skew field), associated with the V eblen configuration, an d w e show that su c h a map provides solutions to the functional dynamical p entagon equation. W e explain that fact in elemen tary geometric terms using the symmetry of the V eblen and Desargues configurations. W e introduce also anoth er map of a geometric origin with the p entago n prop erty . W e sh o w eq uiv alence of these maps with recently in tro duced Desargues maps which provide geometric in terpretation to a non- comm utative version of H irota’s discrete Kad om tsev–Petvias hvili equation. Finally we demonstrate that in an appropriate gauge the (commutative v ersion of the) maps preserves a natural Poisson structure – t he q uasiclassi cal limit of t he W ey l commutation relations. The correspondin g quantum reduction is then studied. In particular, we discuss uniqueness of the W eyl relations for the ultra-local reduction of the map. W e giv e th en th e corresp onding solution of the quantum pentagon equation in terms of the non- compact quantum d ilogari thm function. 1. Introduction Let A b e an asso ciativ e unital algebra ov er a field k , an elemen t S ∈ A ⊗ A is said to satisfy the quan tum p en tagon equation if (1.1) S 23 S 13 S 12 = S 12 S 23 in A ⊗ A ⊗ A , where S ij acts as S on i -th and j -th factors in the tensor pr o duct and lea v es unc hanged element s in the r emainin g factor. Equation (1.1) looks lik e a d egenerate version of the quant um Y ang– Baxter equation, well k n o wn in the th eory of exactly solv able mo d els of statistical m echanics and quan tum fi eld th eory [6, 51, 40], and qu an tum group s [43]. Ther e are also some s im ilarities in constructing solutions of b oth equations, for example th e role of the Drinfeld double construction [30] o f solutions of the quan tu m Y ang–Baxt er equation is replace d b y th e Heisen b erg double [72, 76, 55, 45]. Ho wev er, in mo dern th eory of qu an tum groups [4, 78, 53] (see also [74] for a review, and [58] for discussion of the finite dimens ional case) the quantum p enta gon equation seems to p lay more profound role. Remark ably , given a solution of (1.1 ) satisfying some additional non-degeneracy conditions, it allo ws to construct all the remaining structur e maps of a quantum group and of its Pon trjagin d ual simultaneously . Let X b e a set, S : X × X → X × X b e a map from its square into itself. W e call S p entagon map if it satisfies the functional (or set-theoretical) p en tagon equation [80] (1.2) S 12 ◦ S 13 ◦ S 23 = S 23 ◦ S 12 , on X × X × X , regarded as an equalit y of comp osite maps; here again S ij acts as S in i -th and j -th f actors of the C artesian p r o duct. On e can consider a parameter (it m a y b e fun ctional) dep endent v ersion, then the parameters of the five maps in (1.2) ma y b e constrained by r elations inv olving also the dynamical v ariables. The corresp onding f unctional Y ang–Ba xter equation [73 , 31] has b een studied recen tly [2 , 77, 63] in connection to integrabilit y (und ersto o d as the multidimensional consistency [59, 12, 13, 2]) Key wor ds and phr ases. p entago n equation, integrable discrete g eometry; Desargues configuration; Hirota equa- tion; Po isson maps; W eyl comm utation relations; qu antum rational functions; non- compact q uantum dilogarithm MSC 2010: Primary 37K10; Secondary 37K60, 39A14, 51A20, 16T20 P AC S 2010: 02.10.Hh, 02.30.Ik, 02.40.Dr, 03.65.-w. 1 2 A. DOLIW A AN D S. M. SERGEEV of tw o dimensional lattice equations. A generalization of the quantum Y ang-Baxter equation for three dimensional mo dels is th e tetrahedron equation p r op osed b y Zamolod c hik o v [81 ]. It is related [8] to four dimensional consistency of the discrete Darb oux equations [14], or equiv alen tly , to f our dimensional compatibilit y of the geometric construction scheme of the quadr ilateral lattice [27]. It turns out [9] that all pr esen tly known solutions of the quan tum tetrahedron equation can b e obtained, by a canonical quanti zation, from classical solutions of the fu n ctional tetrahedr on equation deriv ed from the quadrilateral lattices. Recen tly it has b een observed [25 ] that the qu adrilateral lattice theory can b e consid ered as a part of the theory of the Desargues maps, wh ic h d escrib e in geo metric terms Hirota’s discrete Kadom tsev–P etviash vili (KP) equ ation [39] an d its integ rabilit y prop erties. It is kno wn that the Hirota equation enco des [57] the KP h ierarc h y of inte grable equations [20]. It p la y s also an imp ortant role in man y bran ches of mathematics and theoretica l ph ysics related to int egrabilit y; see [52] for a recent review of some of its application. There is a natural qu estion what should replace Zamolo dc hiko v’s tetrahedron equation in the trans ition from qu adrilateral lattices do Desargues maps. W e demonstrate that the answ er is pro vided b y the p en tagon equation. T o make p r esen tation more transparen t we start in Section 2 fr om the geometric essence of the construction b ased on elemen tary consid erations on the V eblen and Desargues configurations [10] and th eir symmetry group s. Giv en fi v e p oints of the V eblen configuration we are (almost) uniquely given the last one, what w e call the V eblen flip . Th e pr esence of five V eblen confi gu - rations within the Desargues configuration giv es the p en tagon prop erty of the V eb len flip. By parametrizing the V eblen flip us in g h omogeneous co ordinates from a division ring D w e obtain in Section 3 a b irational map of D 2 × D 2 in to itself, wh ic h con tains a fu nctional gauge parameter. W e chec k th at the V eblen map s atisfies the f unctional p en tagon equation, provided th e fu nctional parameters are r estricted by some add itional r elations. In Section 4 we discus s another solution, without parameters, of the p entag on map together with its geometric interpretatio n. Section 5 is dev oted to presen tation of the equiv alence to the ab o v e solutions of th e p en tagon equation with the Desargues maps and th e n on-comm utativ e Hirota (discrete KP) equation. The quant ization pr o cedure, which we present in Section 6, can b e u ndersto o d in tw o wa ys. First, under appropriate c hoice of the gauge in the comm utativ e case the V eblen map lea ves in v ariant a natural Po isson stru cture. Such a m ap p reserv es also the ultra-lo cal W eyl commu- tation relations which qu an tize that s tr ucture. F rom other p oint of view the p ro cedure can b e considered as integ rable reduction of the generic division ring solution of the functional p enta gon equation to a particular division algebra. Th is allo ws to rewrite the m ap as inner automorphism , and to obtain the corresp ondin g solution of the dyn amical quant um p enta gon equation, which can b e expressed using the quantum dilogarithm function. Finally , in addition to th e conclud in g remarks w e present sev eral op en problems and researc h directions. 2. The Veb len flip a nd its pent agon proper ty Belo w we present elementa ry considerations, which form ho w ev er a geometric core of the pap er with far r eac h ing consequences. 2.1. Geometry and com binatorics of the V eblen and Desargues configurations. C on- sider the V eblen configuration (6 2 , 4 3 ) of six p oints and four lines, eac h p oin t/line is in ciden t w ith exactly t w o/three lines/p oin ts; see Fig. 1. T o elucidate the S 4 symmetry group of the configura- tion is con v enien t to lab el its p oin ts by tw o-elemen t subs ets of the four-elemen t set { A, B , C , D } , and lines by three-elemen t su bsets; the in cidence relation is d efined by con tainment . This combi- natorial description of the V eblen configuration has a geo metric origin [16 ], whic h asso ciates (in a non-u nique wa y) with the configur ation four p oints A, B , C, D in general p osition in P 3 . Six lines of edges of the simplex and its four p lanes intersect ed b y a generic plane form six p oin ts and four lines th e V eblen configuration (on the intersect ion plane). THE PENT A GON RELA TION AND INCIDENCE GEOMETR Y 3 2 1 1’ 2’ BD A C C A D B B D 2’ 1’ BC AB AC CD AD w u 1 2 1 1 Figure 1. The V eblen flip f AB C D and its tetrahedron repr esen tation. F aces of the tetrahedron repr esen t lines of the confi guration, and the arr o ws den ote p osition of the w and u co efficien ts in the norm alizati on of the corresp onding linear relations discussed in Section 3. The Desargues configur ation is of the t yp e (10 3 ), i.e. it consists of ten p oin ts and ten lines, eac h line/p oin t is incident with exactl y three p oints/ lines. It is known that com bin atorially there are ten suc h distinct configurations [38]. Th e Desargues configuration is selected by the prop ert y that it con tains five V eblen configurations. Again, it is p ossible to lab el p oints of the configuration by t w o-elemen t subsets of the five- elemen t s et { A, B , C , D , E } , and lines by three-elemen t sub s ets; the inciden ce r elation is d efined by con tainmen t. Similarly , like for th e V eblen configuration, giv en fiv e p oin ts in general p osition in P 4 , consider lines joinin g pairs of p oint s, and planes defined by the triples. A s ection of su c h a system of ten lines and ten planes by a generic three-dimensional h yp erp lane giv es a Desargues configuration, see Fig. 2. Five four-elemen t subsets give rise to five V eblen configurations. 2.2. Geometry and com binatorics of the V eblen flip. Definition 2.1. Given tw o ordered p airs ( P 1 , P 2 ), ( P 3 , P 4 ) of distinct and non-collinear p oin ts of a pro jectiv e space such that the lines h P 1 , P 2 i and h P 3 , P 4 i are coplanar, and th us in tersect in the p oin t P 5 . W e denote su c h fiv e p oin ts b y { ( P 1 , P 2 ) , ( P 3 , P 4 ) , P 5 } and say th at they satisfy the V eblen configuration condition. W e can complete these five p oints (and tw o lines), by the intersecti on p oin t P 6 of the tw o lines h P 1 , P 3 i and h P 2 , P 4 i (and th e lines), to the V eblen configuration; notice that the ordering in pairs is imp ortant, b ecause P 6 6 = h P 1 , P 4 i ∩ h P 2 , P 3 i . I f w e remo v e the old in tersection p oin t P 5 , whic h do es not b elong to t w o new intersec ting lines we obtain new system { ( P 1 , P 3 ) , ( P 2 , P 4 ) , P 6 } satisfying the V eblen configuration condition. Suc h an inv olutory transition we call a V eblen flip . Giv en fiv e p oints satisfying the V eblen configuration condition w e can lab el them, in the w a y describ ed ab o v e in Section 2.1, b y fi v e edges of the three-simplex or, equiv alently (up to an action of S 4 ), b y fiv e t w o-elemen t sub sets of a four elemen t set. T he ed ge represen ting the intersectio n p oint P 5 con tains tw o v ertices of v alence three, the sixth edge will represent the p oin t P 6 ; see Fig. 1 whic h illustrates the V eblen flip { ( AB , B C ) , ( AD , C D ) , AC } 7→ { ( AB , AD ) , ( B C , C D ) , B D } . 2.3. Geometry of t he p en tagon rela t ion satisfied b y the V eblen flip. T o un derstand the geometric origin of the p en tagon relation prop ert y of the V eblen flip let us start from tw o sets of p oint s satisfying the V eblen confi guration pr op ert y with three p oin ts and one lin e in common, see Fig. 2. Without loss of generalit y (the symm etry group of Desargues configuration is th e p ermutati on group S 5 ) we consider p oin ts AC , AD , C D of the line AC D , t w o p oints AB and 4 A. DOLIW A AN D S. M. SERGEEV B A C D E BE DE BD BC CE AB AC AE AD CD Figure 2. Th e Desargues configuration and its four-simplex com binatorics. Th e (initial) sev en p oin ts and three lines, and their four-simp lex coun terparts, used to study the p en tagon prop erty of the V eblen flip are d istinguished b y solid lin es. B C on the line AB C , and t w o p oints AE and D E on the line AD E . By a sequen ce of V eblen flips w e can reco ver all the other p oin ts of the Desargues configur ation. R emark. T he Desa rgues confi gu r ation is considered usu ally in relation to the celebrated Desargues theorem v alid in p ro jectiv e spaces o v er division r ings [10]. It states that tw o triangles (e.g. △ AB ,AC, AE and △ B D ,C D ,D E on Fig. 2) are in p er s p ectiv e from a p oin t ( AD in our case) if and only if they are in p ersp ectiv e fr om a line (that passing thr ou gh the remaining thr ee p oints B C , C E , B E of the configuration, whic h are constr u cted as in tersections of the corresp onding sid es of the tr iangles). If we wan t to app ly the V eblen fl ip, starting from the initial confi gu r ation describ ed ab o v e, it can b e either the flip f AB C D (in the configur ation lab elled by the tetrahedr on with ve rtices A, B , C, D ) or the flip f AC D E . After the firs t flip we ha ve a similar c hoice of t w o flips, bu t we exclude that related to the V eblen configuration just used (in order not to go bac k imm ediately to the initial configuration). Th erefore the choi ce of the first flip d etermines the next ones. By insp ection of Fig. 3 we obtain that sup erp osition f B C D E ◦ f AB C E ◦ f AC D E of transformations when applied to the initial configur ation, giv es the same r esult as f AB D E ◦ f AB C D . R emark. In other w ords, starting f r om the data d escrib ed ab o v e and going around the d iagram w e obtain the sequence of V eblen flips of p erio d fi v e. 3. Algebraic descript ion o f the Ve b len flip and of its pent a gon proper ty The p resen t section is devote d to algebraizat ion of the geometric considerations presented ab o v e. In d oing that we in tro duce n umerical coefficient s (in the non-comm utativ e Hirota equation in terpretation they will serve as soliton fields , see Section 5) attac hed to vertic es of the V eblen configuration. These d escrib e p ositions of p oint s on the corresp ondin g lines, and the V eblen flip giv es a map fr om a set of old coefficient s to the corresp onding set of new ones. The p en tagon THE PENT A GON RELA TION AND INCIDENCE GEOMETR Y 5 B A C D E B A C D E B A C D E B A C D E B A C D E f ABCD f f f ABCE BCDE ABDE ACDE f 2 3 1 S 12 1 3 2 S 23 1 3 2 S 12 1 2 3 S 13 2 1 3 S 23 U V X Y Z Figure 3. The p en tagon relation for V eblen flips in the 4-simplex rep r esen tation prop erty of the V eblen fl ip implies that the map satisfies the fun ctional dynamical p entag on equation. 3.1. Algebraic description of the V eblen flip. Con s ider three d istin ct collinear p oint s AB , AC , B C of the (righ t) pro jectiv e space P M ( D ) o v er division ring D . W e lab el them in the sp irit of the com binatorics of the V eb len and Desargues configur ations, b y edges of a triangle AB C . Their collinearit y , expressed in terms of the homogeneous coord in ates φ AB , φ AC , φ B C ∈ D M +1 , tak es the form of a linear dep en d ence r elatio n (3.1) φ B C = φ AB w − φ AC u, with non-v anishing co efficien ts u, w ∈ D × (w e add th e min us sign for conv enience). T he ab ov e linear dep end ence relatio n (3.1 ) is normalized at φ B C b y pu tting corresp onding co efficien t equal to one. W e can incorp orate this new information to the graphic representa tion on the triangle AB C b y adding an arrow from the w -edge (i.e. the edge whic h represents the p oint w ith homogeneous co ordinates m ultiplied by the co efficien t w ) to the u -edge, as visualized on Fig. 1. Consider five distinct p oin ts of pro jectiv e space P M ( D ), whic h satisfy the V eblen confi guration condition and are lab eled by five edges of a three-simplex, as explained in Section 2. T w o trip lets of collinear p oints give rise to t wo linear dep en d ence r elatio ns. In order to write them d o wn w e ha v e to fix en umeration of the lines, and to normalize the corresp ondin g linear relations. It turns out that in discussin g the p en tagon prop erty of th e V eblen map it is con v enien t to c ho ose the normalization in relatio n to lab elling of straight lin es as follo ws: (1) Pick up a p oin t ( B C on Fig. 1) differen t from the initial intersectio n p oint ( AC on Fig. 1) and declare it to b elong to th e lines 1 and 1 ′ , moreo ver th e p oin t is the normalizatio n p oint on the b oth lines 1 and 1 ′ . (2) Th e second p oin t on line 1 ( AB on Fig. 1) differen t from the in tersectio n p oin t is attrib- uted the w co efficien ts on b oth lines 1 and 2 ′ . 6 A. DOLIW A AN D S. M. SERGEEV (3) Th e old in tersection p oin t is attributed the w co efficien t on line 2, and the new intersect ion p oint ( B D on Fig. 1) is the w -p oin t on line 1 ′ . (4) Th e p oint on the lines 2 and 2 ′ ( AD on Fig. 1) is attributed the u co efficien ts on b oth the lin es. In algebraic terms we ha v e, as the starting p oin t, h omogeneous coord inates of five initial p oin ts and four co efficient s w 1 , u 1 , w 2 , u 2 , in the linear relations φ B C = φ AB w 1 − φ AC u 1 , (3.2) φ C D = φ AC w 2 − φ AD u 2 . (3.3) In writing down sim ilar linear dep endence relations f or coordin ates of p oin ts on t w o new lines of the V eblen configuration φ B C = φ B D w ′ 1 − φ C D u ′ 1 , (3.4) φ B D = φ AB w ′ 2 − φ AD u ′ 2 , (3.5) w e are lo oking for co ordinates φ B D of the new int ersection p oin t, and f or the corresp onding co efficien ts w ′ 1 , u ′ 1 , w ′ 2 , u ′ 2 . Th e tr an s formation form ulas read w ′ 1 = Gu 1 , u ′ 1 = w − 1 2 u 1 , (3.6) w ′ 2 = w 1 u − 1 1 G − 1 , u ′ 2 = u 2 w − 1 2 G − 1 . (3.7) Here G is a free parameter whic h expr esses p ossibilit y of multiplying φ B D b y a non-zero factor. Indeed, equations (3.4), (3.5) give (3.8) φ B D =  φ B C u − 1 1 + φ C D w − 1 2  G − 1 =  φ AB w 1 u − 1 1 − φ AD u 2 w − 1 2  G − 1 . R emark. The in v erse transformation is giv en as follo w s w 1 = w ′ 2 w ′ 1 , u 1 = G − 1 w ′ 1 , (3.9) w 2 = G − 1 w ′ 1 u ′ − 1 1 , u 2 = u ′ 2 w ′ 1 u ′ − 1 1 . (3.10) Actually , w e could tak e an arbitrary non-zero p arameter G ′ , in stead of G , which w ould result in m ultiplying of the original h omogeneous co ordinates φ AC b y an ap p ropriate factor. Equations (3.4) and (3.5) ha v e in itial interpretatio n on the line ar algebr a lev el as transfor- mation (in v olving a n umerical parameter) b et ween n umerical coefficients describing p ositions of p oint s of the V eblen configuration. Ther e is another lev el of lo oking on the form ulas, whic h can b e called the algebr aic ge ometry lev el , i.e. we consider w 1 , u 1 , w 2 , and u 2 as non-comm uting in- determinates in the corresp onding univ ersal skew field of fractions [18]. Then also the gauge co efficien t G can v ary when changing the indeterminates, what allo ws for in terpretation of G = G ( w 1 , u 1 , w 2 , u 2 ) as an arbitrary rational fun ction of the four v ariables. In this wa y by c ho osing fun ction G we may interpret form ulas (3.4)-(3.5) as a defin ition of th e rational map S G : D 2 × D 2 99K D 2 × D 2 . Demanding inv ertibility of the map S G w e restrict from n ow on our atten tion to b ir ational maps, what imp oses certain conditions on the form of admissible gauge functions G . R emark. On the algebraic geometry leve l in the case of birational map S G , we are lo oking for the in v erse map in the form of equations (3.9)-(3.10). Therefore w e can in terpret the gauge parameter G ′ as a n ew r ational fu nction G ′ = G ′ ( w ′ 1 , u ′ 1 , w ′ 2 , u ′ 2 ) of four indeterminates. Equalit y of the gauge p arameters on the lin ear algebra lev el is then transf ered into the follo wing functional relation b et we en the gauge functions, whic h w e write do wn in its fu ll exp an d ed form G ′  G ( w 1 , u 1 , w 2 , u 2 ) u 1 , w − 1 2 u 1 , w 1 u − 1 1 G ( w 1 , u 1 , w 2 , u 2 ) − 1 , u 2 w − 1 2 G ( w 1 , u 1 , w 2 , u 2 ) − 1  = G ( w 1 , u 1 , w 2 , u 2 ) . (3.11) THE PENT A GON RELA TION AND INCIDENCE GEOMETR Y 7 In short notation, the fu nctional relatio n (3.11) reads G ′ ( w ′ 1 , u ′ 1 , w ′ 2 , u ′ 2 ) = G ( w 1 , u 1 , w 2 , u 2 ) , where w ′ 1 , u ′ 1 , w ′ 2 , u ′ 2 are giv en b y equations (3.6)-(3.7). Example 3.1. Fix the gauge function G b y requiring w ′ 2 = w 2 , whic h giv es G ( w 1 , u 1 , w 2 , u 2 ) = w − 1 2 w 1 u − 1 1 , and the tran s formation form ulas read w ′ 1 = w − 1 2 w 1 , u ′ 1 = w − 1 2 u 1 , w ′ 2 = w 2 , u ′ 2 = u 2 w − 1 2 u 1 w − 1 1 w 2 . The in v erse transformation is of the form w 1 = w ′ 2 w ′ 1 , u 1 = w ′ 2 u ′ 1 , w 2 = w ′ 2 , u 2 = u ′ 2 w ′ 1 u ′ − 1 1 , and defines th e function G ′ ( w ′ 1 , u ′ 1 , w ′ 2 , u ′ 2 ) = w ′ 1 u ′ − 1 1 w ′ − 1 2 whic h satisfies condition (3.11). 3.2. Algebraic description of the p entagon relation. The algebraic count erpart of the p enta gon r elation satisfied by th e V eblen fl ips f will b e an analogous relation on the algebraic geometry lev el of the map S G . Let us first discus s the implication of the p en tagon relation b et w een the V eblen flips on the linear algebra lev el. S tarting f rom the initial configuration of s ev en p oin ts on three lines (see Fig. 2) w e ha v e three linear relations φ B C = φ AB w 1 − φ AC u 1 , φ C D = φ AC w 2 − φ AD u 2 , φ D E = φ AD w 3 − φ AE u 3 , normalized according to Fig. 3. W e p erform the tr ansformation in the first t wo equations with the gauge parameter U and then we p erform the transform ation in the second and the third equation with the gauge parameter V . At the end we obtain the final linear relations φ B C = φ B D ˆ w 1 − φ C D ˆ u 1 , φ B D = φ B E ˆ w 2 − φ D E ˆ u 2 , φ B E = φ AB ˆ w 3 − φ AE ˆ u 3 , compare Fig. 3 . The resulting transformation ( w i , u i ) 3 i =1 → ( ˆ w i , ˆ u i ) 3 i =1 of the coefficients reads (abusin g the notation w e write S U 12 and S V 23 lik e on the algebraic geomet ry lev el)   w 1 u 1 w 2 u 2 w 3 u 3   S U 12 − − →   U u 1 w − 1 2 u 1 w 1 u − 1 1 U − 1 u 2 w − 1 2 U − 1 w 3 u 3   S V 23 − − →   U u 1 w − 1 2 u 1 V u 2 w − 1 2 U − 1 w − 1 3 u 2 w − 1 2 U − 1 w 1 u − 1 1 w 2 u − 1 2 V − 1 u 3 w − 1 3 V − 1   . According to our previous geometric considerations the final set of linear relations can b e also obtained by th e sequ ence of thr ee transformations: (i) in th e second and the third relation with 8 A. DOLIW A AN D S. M. SERGEEV a parameter which we call X , (ii) in the firs t and third (new) r elation with a parameter Y , (iii) in the first and the second relation with a parameter Z   w 1 u 1 w 2 u 2 w 3 u 3   S X 23 − − →   w 1 u 1 X u 2 w − 1 3 u 2 w 2 u − 1 2 X − 1 u 3 w − 1 3 X − 1   S Y 13 − − →   Y u 1 X u 2 w − 1 2 u 1 X u 2 w − 1 3 u 2 w 1 u − 1 1 Y − 1 u 3 w − 1 3 u 2 w − 1 2 Y − 1   S Z 12 − − → S Z 12 − − →   Z X u 2 w − 1 2 u 1 w − 1 2 u 1 Y w 2 u − 1 2 X − 1 Z − 1 w − 1 3 X − 1 Z − 1 w 1 u − 1 1 Y − 1 u 3 w − 1 3 u 2 w − 1 2 Y − 1   . Notice ho w ev er, that the actual equalit y h olds only on the geometric lev el of p oin ts in the pro jectiv e space, while on the (linear algebra) leve l of their homogeneous coord inates w e should tak e int o accoun t p ossibile rescaling. Ho mogeneous co ordinates of the t w o new p oin ts B D and B E (the second sequence of transformations giv es also the p oin t C E which is n ot pr o duced by the first sequence) h a v e then double expressions φ B D =  φ B C u − 1 1 + φ C D w − 1 2  U − 1 =  φ B C u − 1 1 w 2 + φ C D  u − 1 2 X − 1 Z − 1 , φ B E =  φ AB w 1 u − 1 1 w 2 u − 1 2 − φ AE u 3 w − 1 3  V − 1 =  φ AB w 1 u − 1 1 − φ AE u 3 w − 1 3 u 2 w − 1 2  Y − 1 , found with the help of equation (3.8). Therefore, to ha v e equalit y of th e homogenous co ord in ates of the new p oin ts and of the co efficien ts in the b oth expressions of the final linear relations the gauge parameters should b e adju sted according to th e follo wing equations (3.12) U = Z X u 2 w − 1 2 , V = Y w 2 u − 1 2 . On th e algebraic geometry lev el equations (3.12) ha v e more complicated interpretatio n. Con- sider fi v e rational f unctions U, V , X , Y , Z : D 2 × D 2 99K D of four v ariables. W e call th e fu nctions p enta gon-compatible if they s atisfy in D 2 × D 2 × D 2 equations of the form (3.12), where we write (the order of fu nctions in the sys tem is imp ortan t) U = U ( w 1 , u 1 , w 2 , u 2 ) , X = X ( w 2 , u 2 , w 3 , u 3 ) , V = V ( w 1 u − 1 1 U − 1 , u 2 w − 1 2 U − 1 , w 3 , u 3 ) , Y = Y ( w 1 , u 1 , w 2 u − 1 2 X − 1 , u 3 w − 1 3 X − 1 ) , (3.13) Z = Z ( Y u 1 , X u 2 w − 1 2 u 1 , X u 2 , w − 1 3 u 2 ) . Notice that the argumen ts of the functions ab o v e coincide with arguments of the corresp ond - ing V eblen maps considered in the t w o sequences of trans f ormations. In the expanded form the functional equations lo ok rather complicated, and one can eve n w onder if there exists an y p enta gon-compatible system of functions. Example 3.2. If the gauge f unctions U, V , X , Y , Z are of the same f orm as G in Example 3.1 then this choice giv es a solution to equations (3.12)-(3.13). The Th eorem b elo w can b e ve rified directly , but actually it follo ws from the considerations ab o v e. Theorem 3.1. Given five r ational functions of four variables U, V , X, Y , Z : D 2 × D 2 99K D which ar e p entagon-c omp atible then the V eblen maps S G : D 2 × D 2 99K D 2 × D 2 , wher e G is one of U, V , X, Y , Z , satisfy the fu nctional dynamic al p entagon r elation on D 2 × D 2 × D 2 (3.14) S Z 12 ◦ S Y 13 ◦ S X 23 = S V 23 ◦ S U 12 . 4. The nor maliza tion map an d its p ent agon proper ty In th is Section we pr esent another map w ith th e p enta gon pr op ert y . W e give also its simple geometric meaning r elated to four collinear p oin ts. This map will b e u sed in the next Section, THE PENT A GON RELA TION AND INCIDENCE GEOMETR Y 9 A C C A D B B D 2’ 1’ x 1 1 y 1 2 A B C D Figure 4. Graphic represen tation of the linear relations f or the normalization map where we discuss its r ole in establishing the r elatio n of th e V eblen map to the Hirota equation. First w e discuss another auxiliary map of p er io d three. Consider a map of ve rtices A, B , C of a triangle into a pro jectiv e space P M ( D ) su c h that all the v ertices are mapp ed in to collinear (bu t differen t) p oin ts. The corresp onding linear constraint in v olving the homogeneous co ordinates of the p oin ts φ A = φ C y − φ B x w e call normalized at φ A , while φ B and φ C are called co ord inates of the x and y -p oint , resp ec- tiv ely . Graphically , w e pu t an arrow at the v ertex of the triangle repr esen ting the normalization p oint . Th e arro w is directed from the ed ge joining the (v ertex representi ng the) normalization p oint and the x -p oint , see Fig. 4. Notice that th is is the different normalization ru le than that used in Section 3.1. R emark. Notice th at b y changing the normalization p oint ”by 2 π / 3 rotation”, where th e orien- tation is giv en by the arr o w, w e arriv e at an equiv alen t linear relation (4.1) φ B = φ A ˜ y − φ C ˜ x, where (4.2) ( ˜ y , ˜ x ) = N ( y , x ) =  − x − 1 , − y x − 1  , ( y , x ) = ( ˜ x ˜ y − 1 , − ˜ y − 1 ) , N 3 = id . The birational map N : D 2 99K D 2 , whic h can b e called the non-commuta tiv e Newton map [21], is of order th ree. Let us consider four collinear (but distinct) p oin ts of a pro jectiv e sp ace P M ( D ). W e lab el them by vertic es A, B , C , D of a three-simplex. F our triangular f aces of the simplex give fou r linear relations (up to p ossible normalizations). By analogy with the V eblen map presented in Section 3.1 w e w rite first t w o of them φ A = φ C y 1 − φ B x 1 , (4.3) φ A = φ D y 2 − φ C x 2 , (4.4) and transfer them in to th e r emaining equations φ D = φ C ¯ y 1 − φ B ¯ x 1 , (4.5) φ A = φ D ¯ y 2 − φ B ¯ x 2 . (4.6) The relation b et w een old and new coefficients reads ¯ y 1 = ( y 1 + x 2 ) y − 1 2 , ¯ x 1 = x 1 y − 1 2 (4.7) ¯ y 2 = y 2 ( x 2 + y 1 ) − 1 y 1 , ¯ x 2 = x 1 ( x 2 + y 1 ) − 1 x 2 , (4.8) 10 A. DOLIW A AN D S. M. SERGEEV and defines a b irational map ¯ S : D 2 × D 2 99K D 2 × D 2 . Th e inv erse transformation is giv en b y y 1 = ¯ y 1 ¯ y 2 , x 1 = ¯ x 2 + ¯ x 1 ¯ y 2 (4.9) y 2 = ¯ y 2 + ¯ x − 1 1 ¯ x 2 , x 2 = ¯ y 1 ¯ x − 1 1 ¯ x 2 . (4.10) The map ¯ S : D 2 × D 2 99K D 2 × D 2 , giv en by equations (4.7)-(4.8 ) is called the (change of the) normaliza tion map. R emark. This time homogeneous co ordinates of all p oin ts are fixed, which resu lts in the absence of gauge p arameters in the map. R emark. Equations (4.8) can b e rewritten in the form (4.11) ¯ y − 1 2 y 2 − y − 1 1 x 2 = 1 , ¯ x − 1 2 x 1 − x − 1 2 y 1 = 1 , whic h will b e relev an t in Section 5. Finally , let us consider fiv e collinear p oin ts and lab el them by v ertices of th e four-simplex. Prop osition 4.1. The normalization map ¯ S : D 2 × D 2 99K D 2 × D 2 , gi v en by e quations (4.7) - (4.8) satisfies the fu nctional p entagon r elation (1.2) . Pr o of. Th e result can b e verified by direct calculation. Notice h o w ev er that, ev en ha ving m uc h simpler geometric in terpretation then the V eblen map S G , th e new map ¯ S has the same graphic represent ation (compare Fig. 1 with Fig. 4). Therefore the s tatemen t follo ws from consistency of arro ws on Fig. 3 .  5. Rela tion t o Desargues maps an d Hirot a ’s d isc rete KP eq ua tion In [25] it w as s h o wn (see also an earlier related w ork [50]) that the non-comm utativ e v ersion of Hirota’s discrete KP equation [39, 60, 61] can b e derived from the V eblen confi guration. Moreo ver, its four dimen sional compatibilit y follo ws from the Desargues th eorem. The imp ortan t observ ation that the f ou r dimensional consistency of the d iscrete KP equation in the Sc h w arzian form has com b inatorics of the Desargues configuration is d ue to W olfgang S chief, see [11] and remarks in [25, 26]. In view of the previous Sections it is clear that th e incidence geometry structures are the same for b oth the non-comm utativ e Hirota sys tem and the V eblen map. The goal of this Section is to establish a d ictionary b etw een b oth sub jects. Th e Reader not int erested in the theory of in tegrable discrete equations ma y go directly to the next part where we d iscuss the quan tum p en tagon equation. 5.1. Desargues maps a nd the Hirota system. W e collec t fi rst some facts on incidence geo- metric interpretatio n of the Hirota equation. T he Desargues maps, as defi ned in [25], are maps φ : Z N → P M ( D ) of multidimensional in teger lattice in to p ro jectiv e space of dimension M ≥ 2 o v er a division ring D , such that for any pair of indices i 6 = j th e p oin ts φ ( n ), φ ( n + ε i ) and φ ( n + ε j ) are collinear; h ere ε i = (0 , . . . , i 1 , . . . , 0) is th e i -th elemen t of the canonical basis of R N . W e will write F ( i ) ( n ) instead of F ( n + ε i ) for an y function F on Z N . Moreo v er we w ill often skip the argum en t n . R emark. Th e Desargues maps can b e d efined starting f r om the ro ot lattice Q ( A N ), instead of the Z N lattice . Suc h an approac h, prop osed in [26], mak es transparent the A N affine W eyl group symmetry of Desargues m aps and the discrete KP system. Its ”lo cal” v ersion is the p en tagon prop erty of the V eblen map. In the homogeneous co ordinates φ : Z N → D M +1 the map can b e d escrib ed in terms of the linear system (5.1) φ + φ ( i ) A ij + φ ( j ) A j i = 0 , i 6 = j, THE PENT A GON RELA TION AND INCIDENCE GEOMETR Y 11 where A ij : Z N → D × are certain n on-v anishing fu n ctions. The compatibilit y of the linear system (5.1) turns out to b e equ iv alen t to equations A − 1 ij A ik + A − 1 k j A k i = 1 , (5.2) A ik ( j ) A j k = A j k ( i ) A ik , (5.3) where the indices i, j, k are distinct. Equ ations (5.2) and (5.3) are equiv alen t, in an appropriate gauge giv en in [25], to the non-commutati ve Hirota sy s tem p r op osed in [61]. The pr ecise relation with the Hirota equation is as follo ws (see [25] for details and other gauge forms). Equations (5.2) and (5.3) imply existence of a non-v anish in g function F : Z N → D × satisfying (5.4) F ( i ) A ij = − F ( j ) A j i , i 6 = j. After rescaling the homogeneous coordin ates as (5.5) ˜ φ = φ F − 1 , w e obtain that ˜ φ satisfies the linear pr oblem [19, 61] (5.6) ˜ φ ( i ) − ˜ φ ( j ) = ˜ φ U ij , i 6 = j ≤ N , with the coefficients (5.7) U ij = F A − 1 j i F − 1 ( j ) = − U j i . Moreo ver, equations (5.2)-(5.3) reduce to the follo wing systems for distinct triples i, j, k U ij + U j k + U k i = 0 , (5.8) U k j U k i ( j ) = U k i U k j ( i ) . ( 5.9) Equations (5.9) allo w to in tro duce p oten tials r i : Z N → D × suc h that (5.10) r i ( j ) = r i U ij , i 6 = j. When D is comm u tativ e then the f u nctions r i can b e p arametrized in terms of a single p oten tial τ (5.11) r i = ( − 1) P k