DEMONSTRATIO MATHEMATICA Vol. XXVI No 2 1993
Michal Muzalewski, Krzysztof Praimowski
AXIOMATIC INVESTIGATIONS ON SYMPLECTIC GEOMETRY
Introduction In this paper we investigate metric affine and metric projective geome- tries induced by skew-symmetric forms. Usually, the corresponding geome- tries are called symplectic. We present axiom systems for the class of sym- plectic metric affine spaces and for the class of symplectic metric projective spaces. By Artin's results (cf. [1]), which we generalize to spaces of arbitrary dimension (Thm. 0.6), if a "metric" space [metric affine of metric projec- tive] is determined by a polarity, then this polarity must be defined by a form. Either a polarity has a non selfconjugate point, or all points are self conjugate, and then, in accordance, either such a form must be sesquilinear (hermitean), or skew-symmetric. The geometry associated with sesquilinear (especially with bilinear symmetric forms) is very well known; both from an- alytical point of view, and from the standpoint of Foundations. At the same time foundations of geometries induced by skew-symmetric forms are rather postponed and very few intuitions concerning this geometry can be get from the mathematical folklor. The aim of our paper is to make an attempt to built foundations of symplectic geometry. The results we obtain generalize papers [3] and [4] to structures with orthogonality (polarity) determined by skew-symmetric forms. In particu- lar it follows from this that a symplectic orthogonality determines a form inducing this orthogonality up to proportionality.
Basic notions and algebraic preliminaries 1. Affine and projective spaces Let 3" = (F; +, •, 0,1) be a skew field (i.e. not necessarily commutative filed) and V = (V; -f, 6, •) be a left vector space over J. For every left vector 296 M. Muzalewski and K. Prazmowski
space V we denote by V* the dual vector space over 5°, where 5° is the skew field opposite to 3". For X C V let (X) be the linear subspace of V spanned by X. Let V be the set of all subspaces of V. Note that U\ + U2 = {U\ U U2) for all
UuU2e V. We set
TiiiV) = {(») : u G V, u ± 6} = {U G V : dim U = 1}
Hl(V) = {U G V : codim U = 1}, Vf = {U G V : dim U < 00},
V1 = {U G V : codim U < 00}. In the set V we define the relation of parallelity of vectors as follows:
u || v (v) C (u) V (u) C {v).
Let us denote n(v):= iy\ ||). The affine space over V is the structure
A(V)= (K; ||>, where ||C V4 and
v}u2 || u3u4 (ti1 - u2) |'| (u3 - u4).
It is known that for every u G V the structures II(V) and (V; ||,u) are definitionally equivalent. The class of all structures A(V) with dimV > 2 and arbitrary J is axiomatizable (cf. [5], [6]). Let us consider the following formulas: A1 : ab || 6a;
A2 : ab || cc;
A3 : pq || ab A pq || cd A p ^ q ab || cd\
A4 : ab || ac ^ ba || be;
A5 : (3d)[ab || cdAac || bd];
A6 : pa || pc A p ^ a => (3d)\jpb || pd A ab || cd];
A7 : (3p, q, r, s)\pq rs A (Va)[ap || aq => ar ft as]].
THEOREM 0.1 ([5]). A structure ® = (5; ||) satisfies the axioms A1-A7 iff 0 A(V) for some at least ^-dimensional vector space V over a skew field 3". • Axiomatic investigations on symplectic geometry 297
For UUU2,U3 £ Hi{V) we define
L(Ui, U2, U3) dim(i7i + U2 + U3) < 2. By the projective space with collinearity over V we understand the struc- ture P(V) = (Wi(V);L). If dim V < oo, then P(V) and the structure H(V) = (7ii(V),7i1(V);C) are definitional^ equivalent; therefore we shall also frequently refer to H(V) as a projective space.
2. Collineations between projective spaces Let /z : 3"! —• 3*2 be a field monomorphism and let V,- be a vector space over fo (i = 1,2); let A : V\ —• be a map. The map A is /x-semilinear iff A is additive and it satisfies X(au) = a"A(tt) for all a in and ti in Vi. Note that if A is /z-semilinear, where fi is a field isomorphism, as above, then A««» = (A(u)) for every vector u. If fi is not onto, then there is vector u with A((«)) ^ (\(u)). With every /i-semilinear transformation A : V\ —* V2 we correlate the map (A) : fti(Vi) Wi(V2) given by (A)««» := (A(tO). If n is onto (i.e. it is an isomporphism) and A is a 1-1 /x-semilinear map, then the following hold: (1) (A) is 1-1; (2) A-1 : A(Vi) —>• Vi is /z-1-semilinear; (3) (A)"» = (A"1). If/i is a field isomorphism, Ai, A2 are 1-1 /z-semilinear maps, and (Ai) = (A2), then there exists a in fo such that
A2(u) = aA^u) for every vector u. Given two projective spaces P(Vi) and P(V2), by a collineation ip : P(Vi) —• P(V2) we mean any map tp defined on Wi(Vi) with values in T^i (V2) which preserves linear dependence, that means which satisfies 298 M. Muzalewski and K. Prazmowski
Co]ln : U0 C U! + ... + Un o Uf C U? + ... + ,
for all Uo, Ui,.. .Un € Tii(Vi) for all n > 1. By (Colin =>) and (Colln •<=) we mean the formula obtained from Colln by substituting (-4= resp.) instead of . Note that every collineation if) is, by Coll!, a 1-1 map. Note also, that the condition Colln implies Coll, for all i < n. THEOREM 0.2 [Artin's Scheme for Collineations]. If a transformation tp : Hi(Vi) —> Wi(V2) is a collineation mapping P(Vi) into P(V2), then an a there exists a field monomorphism fi : —• 5"2 d fJ-semilinear map A : Vi -»• V2 such that i/> = {A). Proof. This is based completely on Artin's idea. We choose a basis : £ < rj) of Vi, notice that ip preserves linear dependence, and choose for £ < r) a vector A'^ with
(4) = W- We modify the system {A^} in such a way that
(A0+A^ = (A'0 + A'i). Then we set
W = ({A'r.( StrColln : for all £/£ € Wi(V2), Uu...,Une Wi(Vi) such that U^cUf + .-. + U* there is U0 C Ux + ... + Un with U'Q = Uf. A collineation t/j : P(Vi) —»• P(V2) is a strong collineation iff it satisfies StrColln. PROPOSITION 0.3. A transformation V> : fti(Vi) —»• fti(V2) is a strong collineation iff tj) is 1-1 and satisfies (Coll2=») and StrColl2. Axiomatic investigations on symplectic geometry 299 Proof. The only nonevident step of the proof is that the conjunction (Co112=Q & StrColl2 implies (Colln<$=) for every n > 2. To this aim we prove by the induction that /\ (CoU^) & StrColl,- implies (Colln<=) & StrCoUn. i UQ = {A'0 + f5A'I). Let U0 = {JQAQ + Let us observe that 70 / 0; indeed, 70 = 0 yields that f/0 = (A{) and then UQ = (A'^) in contrary to _1 so Ut = {A'0 + 0A'(). Thus U0 = {A0 + 7o 7€^>, P = Kl^ld- • 3. Correlations In the case of finite dimensional projective spaces by a correlation in H(V) we mean any isomorphism x : H(V) —»• (?i1(V),?ii(V); 3) a /s (which is usually identified with its natural extension x : V —• V). If dim V = 00, then there is no such function x, therefore we must change and generalize our definition. Let us begin with some preparatory remarks. If /x is a monomorphism of the field 5" into itself and Vi, V2 are vector spaces over 3", then a map £ : Vi x V2 -+ J is a ¡i-sesquilinear form iff it is additive (on each of its arguments) and it satisfies Note that given any /i-semilinear transformation A : Vi —• one can define f : Vi x V2 F as follows: £(«,») = A(u)(t>). Evidently £ is a /¿-sesquilinear form (cf. [1]). With any /x-sesquilinear form £ we correlate the relation JL=JL^C Vi x V2 defined by the condition u -1L v £(u, v) = 0. Such a relation is called the orthogonality relation determined by For every U G V we set 1(U) = {/€ V*:Ker/D U}. 300 M. Muzalewski and K. Prazmowski By a correlation between P(VI) and P(V2) we understand any transfor- mation x mapping 7ii(Vi) into Til(V2) and satisfying Corin = U0 C Ux + ... + Un o U? n ... D Uf C for all Uo, U\, ...,Un€ Hi(Vi) for all n > 1. Analogously by a dual correlation between P(VI) and P(V2) we mean 1 any transformation x : 7i (Vi) —• 7ii(V2) satisfying the condition DualCorr : U0 2 Ux D ... n Un & U? + ... + 2 for all U0,U1,...,Un G W^Vi) for all n > 1 (note, that between above need not to be symmetric). In analogy to the previous section, in the class of all correlations we distinguish subclasses of strong correlations. A correlation x is strong if it satisfies 1 StrCoiI„ : for all G W (V2), Uu...,Une Wi(Vx) such that u'0 2 V* n ... n UZ there is U0 C Ux + ... + Un with £/£ = for all n > 1. A dual correlation x is strong if it satisfies StrDualCorr„ : for all U'Q G Wi(V2), Ui,..., Un G W^Vi) such that x £/o C U? + ... + there is U0 2 n ... fl Un with U'Q = £/0 , for all n > 1. One can note (analogously to Proposition 0.2) that a map x : Tii(Vi) —• 1 H (W2) is a strong correlation iff it is 1-1 and satisfies (Corr2 =>) and 1 StrCorr2; a map x : W (Vi) —• Wi(V2) is a strong dual correlation if it is 1-1 and satisfies (PualCorr2 =>•) and StrPualCorr2. PROPOSITION 0.5. (i) 7 is a strong dual correlation between P(V) and P(V). l (ii) If x 1 : Wi(Vi) H^V2), : H {V2) Wi(Vi) are correlation and dual correlation resp., then X2X1 is a collineation. X 1 (iii) If x1 : Wi(Vi) W (V2), : W (V2) -»• Wi(Vi) are strong corre- lation and dual strong correlation resp., then xix\ is a strong collineation. P r o o f. A straightforward calculation and the observation that codim U = 1 implies dim~/(U) = 1. • Combining this proposition and Theorem 0.2 we immediately get the following general characterization of correlations: THEOREM 0.6 [Artin's Scheme for Correlations]. If a transformation x : Hi{Vi) • W1(V2) is a correlation, then there exists a monomorphism (i : Axiomatic investigations on symplectic geometry 301 —• and a fi-sesquilinear nondegenerate form f : Vi x V2 —• J such that for every U £ Tii(Vi) we have U* = {v G V2 : U JL t>} = where JL=iL^. • Theorem 0.7. A transformation x : Wi(Vx) —• W1(V2) is a strong correlation iff there exists a isomorphism /z : 5" -h• and a fi-sesquilinear nondegenerate form £ : Vi x V2 suc/i that for every U G Wi(Vi) we have U" = U\ where iL=JL^. • Proposition 0.8. Let £1,^2 • V\ X —• F be two nondegenerate sesqui- linear forms with JL^2 where \i is a field anti-automorphism. Then there is a G F such that £1 (u,v) = a£2(u,v) for all u G Vi,v G V2. Proof. With every form we can correlate a /¿-semilinear transforma- tion A,- : V —> V* defined by A,(u)(v) = v) and the strong correlation "cWiOO-iWHv), (tt)x< = {v G V : A,(ti)(u) = 0} = {u £ V : it) = 0}. The condition JL^1=JL^2 implies x\ = x2 and thus i\)X\ = 7x2. Every two forms inducing 7*1 must be proportional, thus Aj = aA2 for some a / 0 i.e. £1 = a£2 as required. • In the sequel we shall consider more special forms, namely we will con- cerned with geometric forms. A sesquilinear form £ : V X V —• F is geomet- ric iff £ satisfies £(u, v) = 0 O- £(u, u) = 0 for all u, v £ V. In terms of the orthogonality relation this condition is equivalent to the symmetry of the relation JL^. Metric affine spaces and symplectic geometry By a metric affine space we understand any structure 97T = (5;||,JL) such that (0) (5; ||) is a model of A1.-AZ and the following formulas are satisfied in DJl 01 abLcd A ab || ef =• a = bV cdLef, 02 abLcd A abLce => abLde, 03 abLcd =>• cdLab, 302 M. Muzalewski and K. Prazmowski 04 (Vo6)(3c)[6 ± c A ablbc), 05 (Va6)(3c)[a ^ 6 =>• abLbc A ->a& || 6c], 06 (Va6cd)(3e)[-ia6±ac A ab || ac =>• de || ac A ae±ab], 07 (Va6c)(3d)[-ia6 || ac ad Lab A -iad±ac]. 971 will be called a sympletic a Sine space iff moreover 971 satisfies OS (Va6)[a6J_a&]. Let V be a left vector space over 3" and let JLC V X V. The structure N = (V; +, 0, ||, JL) will be called a metric vector space iff the relation JL satisfies the following conditions (cf. [3]) VI a JL 6,cA A,/i G F =>• a JL 6A + c/z, V2 a JL 6 =>• 6 JL a, V3 (Va)(36)[6 ± 0 A a JL b), V4 (Va)(36)[-io JL b A -.a || 6], V5 (VtiVtu)(3j/)[-iu JL t; A || v =>• 3/ JL u A (x — y) || v], V6 (Vut>)(3ti7)[-iu || v =>• w JL u A ->w JL v]. In virtue of the definition of the relation || the condition VI can be replaced by two, more elementary, conditions: VI.1 a JL 6, c=^aJL6-c, V1.2 a[|&A6JLc=»6 = 0VaJLc. A metric vector space 9T is symplectic iff a JL a for every a € V. Given any metric vector space we define the quaternary relation JL PROPOSITION 1.1. If VI is a metric vector space, then M(Orl) is a metric affine space. is symplectic iff M(9T) is symplectic. m PROPOSITION 1.2. Let 97t = (5;||, J.) be a metric affine space. Let V be a vector space such that (5; f|) = A(V). We define for u, v € V til»^ OulOv. Then 9T = (V; +,0, ||, JL) is a metric vector space and M(9t) = M. • Theorem 0.1 and Propositions 1.1 and 1.2 enable us to investigate metric vector spaces as convenient and more algebraic counterparts of the metric affine spaces. Axiomatic investigations on symplectic geometry 303 Let be a fixed metric vector space; for every U € V we consider UL = {v 6 V : (Vw € U)[w JL u]}. In particular u1 = (u)x = {r € V : u JL t?} (by VI). LEMMA 1.3. (i) uL e H^V) for (ii) (trL)J- = (t»> for u^e, (iii) u G v1 # « 6 n1 for u, v ^ 9, (iv) u1- = v1 & u || v for u, v ^ 0. Proof, (i) By VI we infer that it1- 6 V. There exists v such that v u and ti | a. By V5 for arbitrary x € V there exists y with y € uL and x — y || v. Thus x = y + (x — y) where y € tix, and {x — y) G (v); so V = (v) © «•*• and codim(«-L) = 1. A (ii) Clearly (ti) C (ti1)-1-). If (u) (tix)x), then there exists v with u || v and v € (u1)-1). Then v1 C ((u1)1)"1" = Since u jf v, by V6, there exists w with w JL v, w JL v, which is impossible. Thus (u) = (u-L)-L.A (iii) follows from V2. A (iv) follows from (ii). A • By Lemma 1.3, the transformation ip± : (u) i-* u1 is a involutive corre- lation of P(V) (more precisely: in H(V)). LEMMA 1.4. Let u M.v>v:-&u = 0Vue PROPOSITION 1.5. Let V be a left vector space overfi and let * : Wi(V)-» ftx(V) be a null correlation. Then the field ff is commutative and x is induced by a bilinear alternating form f : V X V J. Proof. From Theorem 0.7 it follows that there is an anti-automorphism a : 3" —> J and a sesquilinear form £ such that for every v € V\{0} we have (u)* = {u £ V : v) = 0}. Because x is a null correlation the following conditions hold: (i) £(u,u) = 0 for u e V; 304 M. Muzalewski and K. Prazmowski (ii) £ is nondegenerate. By (i), £(u,v) = —£(v,u) for arbitrary u,v. This implies £(u, v) = — £(v, u) for all u, v € V. Indeed: 0 = f (u + v, u + v) = u) + f (u, v) + f (u, u) + f (v, v). Since £ is nondegenerate there are it, v 6 V with v) ^ 0. Let a € 3" be arbitrary. Then £(u,av) = aa£(u,v) and £(u,av) = —f(av,ti) = —a£(v,ti) = a£(u, v), so a™ = a for a G J. Therefore £ is bilinear alternating form as required. • As a direct consequence of Propositions 1, 2, 5 and Lemma 4 we ob- tain the following characterization of symplectic affine spaces of arbitrary dimension: THEOREM 1.6. For any structure 2Jt = {S; ||, JL) the following conditions are equivalent (i) 271 is a model of formulas AJL-A7, 01-06, OS. (ii) there is a vector space V over a commutative field ff and a bilinear alternating form £ : V X V —> J such that ir, J.') where (5'; ||') = A(V) and uvL'wz £(u—v, w—z) = 0 for all u, v,w,z £ V. • Notice that in view of proposition 0.8 a symplectic space determines an inducing form uniquely up to proportionality. In virtue of Theorem 1.6 the system of axioms A1-A7. 01-06, OS can be considered as an axiom system of dimension-free symplectic (affine) ge- ometry. Notice that the "metric" axiom OS implies that every symplectic space must be Pappian. It is an essential distinction between the symplectic ge- ometry and the geometry induced by symmetric forms. In the case of second geometry one must add an extra assumption to obtain Pappian spaces and usually to this aim the theorem on three perpendiculars is considered. Metric projective symplectic geometry In accordance with the usual (common) definition by a metric projective space we understand a projective space equipped with some fixed correla- tion. Such a space is called symplectic iff the associated correlation is a null correlation. To avoid some complications concerning infinite dimension, instead of a correlation x we shall use a conjugacy relation u> defined by the condition Axiomatic investigations on symplectic geometry 305 Let us consider the following axiom system: Pi (Va, 6)[L(aafc) A L(abb) A L(ata)], 3 P2 (Va,b,pup2,p3)[a / 6A A (Habpi) => L(i>ii>2l>3)], >=i P3 (3a, b, c, d){Vp)[-iL(abp) V ->L(cdp)], P4 (Va, 6, c,p, g)(3x)[->L(a6c) A L(abp) A L(bcq) => L(pqx) A L(acx)], P5 (Va, 6, wc, d,p, g, r)[L(aftg) A L(adg) A L(acp) A L(bdp) A L(a L(aM) V L(a6c) V L(acd) V L(6cd)]. The following theorem was proved in [4]: THEOREM 2.1 ([4]). A structure 4 over a skew field $ with char3" ^ 2. • Thus the formulas P1-P5 form an axiom system for dimension-free pro- jective geometry. Let us consider the following sentences: CI u(a,b) u(b, a), C2 w(a, &i) A w(a, i>2) A L(i>i£>263) => = b2 V w(a, 63), C3 (Va, 6, c)(3d)[L(oM) A w(c, d)], C4 (Va)(36, c)[u(a, 6) A -i«(a, c) A a / 6, c]. Let vMuu:^u-eyv = ey u>({u),{v)). Conversely, let = (X; +, 0, [|, JL) and let JL satisfy V1.2 . We define in Wi(V) the relation WJL by the following condition: WJL((U), (v)) -O- u JL v for u, v £ 6 and we set B(2l) := (W^VJJL.wjl). The following theorem is evident. THEOREM 2.2. (i) A structure (V'y +,0, ||, JL^,) is a metric vector space ¡J(P(V),w) is a model of C1-C4. (ii) A structure 21 is a metric vector space iff B(2l) is a model of C1-C4. In both cases JLWJL=JL and u= u resp. • Let us denote by CS the following formula: CS (Va)[w(o,o)]. 306 M. Muzalewski and K. Prazmowski As a corollary we get the following theorem. THEOREM 2.3. Let © = (P; L,U) be a model O/Pl-P5, C1-C4 and let (P; L) = P(V) for some vector space V. Then (V; +, 9, ||, JLW) is a symplectic metric vector space iff © satisfies CS. • Thus the axiom system consisting of the formulas P1-P5, C1-C4 and CS characterizes dimension-free symplectic projective geometry. Combining the two theorems Theorem 1.5 and Theorem 2.3 we obtain the following characterization of metric projective symplectic spaces: THEOREM 2.4. The following conditions are equivalent: (i) A structure © is a model of P1-P5, C1-C4, and CS; (ii) There exists a vector space V over a commutative field 3" and an alternating form £ in V such that © = where , (v)) iff £(u,v) = 0, for all u,v £V \ {(?}. • Bibliography [1] E. Artin, Geometric algebra, New York 1957. [2] W. Burau, Mehrdimensionale projektive und höhere Geometrie, Berlin 1961. [3] E. Kusak, A geometric construction of a norm function, submitted to J. Geom. [4] E. Kusak and K. Prazmowski, An axiom system for dimension-free elliptic and hyperbolic geometry, mimeographed. [5] E. Kusak, A new approach to dimension-free affine geometry, Bull PAS 27 (1979), 875-882. [6] W. Szmielew, From affine to Euclidean geometry, Warsaw, Dordrecht, 1983. [7] M. Muzalewski, Foundations of Metric-Afine Geometry (An axiomatic approach to affine geometry with orthogonality, Politechnika Warszawska, Preprint No 43, Oc- tober 1991. INSTITUTE OF MATHEMATICS WARSAW UNIVERSITY, BIALYSTOK DIVISION Akademicka 2 12-567 BIALYSTOK, POLAND Received November 6, 1990.