DEMONSTRATE MATIIEMATJCA Vol. XXVII No 1 1994

Michal Muzalewski

ON ORTHOGONALITY RELATION DETERMINED BY A GENERALIZED SEMI-INNER PRODUCT

In the paper we arc dealing with structures QJD = (V, W; IL), where V and W are vector spaccs over a skew-field K. We assume that the relation

IL C V X W satisfies some very weak axioms which are usually satisfied by the orthog- onality relations and wc prove that then the relation IL is induced by an appriopiate form f: V x W A', i.e IL = iL^, where xlL^y :<£> £(x,y) = 0, for x € V, y

1. Introductory definitions and facts

DEFINITION 1. Let V, W be left vector spaccs over skew-fields K and L, respectively. Let us denote L(V) := {L: L is a subspace of V A dim L = 1} , H(V) := {H: H is a subspace of V A codim H = 1} and S(V) := {U: U is a subspace of V} . In the case of finite dimensional projective spaccs by a correlation in ^3(V) := (L(V), H(V); C) wc mean any isomorphism K:

A map K: L(V) H(W)[7 : H(V) L(W)], is said to be correlation [dual correlation], if it satisfies the following condition:

Corrn : L0 C Lt + ... + in kL0 D kLj n... n «Ln,

[DCorrn : H0 D H[ n ...nHn » 7H0 C 7H1+ ... + tH„] for all n > 1.

A correlation K is strong if it satisfies SlrCorrn: for all Hq G H(W),

Li, ..., Ln G L(V) such that

Hq 2 Li H ... n L£ there is L0 C Li + ... + Ln with = L£

THEOREM 1. Let V, W be left vector spaces over a skew-field K and dimW > 3. A transformation K: L(W) H(V), where dimW > 3, is a correlation [strong correlation] iff there exist anti-monomorphism [anti-auto- morphism] J : K K and a nondegcnerate J-sesquilinear form £ : V x W —> K such that KL = AL for L e L(W), where 1L = 11^.

DEFINITION 2. A groupoid (5;©) is said to be a midpoint algebra if it satisfies the following axioms: Ml a © a = a (idemptotency), M 2 a © 6 = ¿> © a (commutativity), MS (a © b) © (c © d) = (a © c) © (b © d) (bi-commutativity), M 4 Vab3x x®a = b. Let 5 be a set and V be a group. A map u: S2 —V is said to be an atlas if it satisfies the following conditions: (1) •) : 5 —s- V is a bijcction for a £ S , (2) b) + ui(b, c) = u>(«, c) for a, 6, c € S . A midpoint algebra 9Jt = (5;©) and a group V are said LJ-associated iff u> : S2 —• V is an atlas and the following condition holds (©) c = a®b<&u(a, c) = w(c,6). A commutative group (V; 6, -f) is said to be a group with the operator \ if it satisfies the following conditions: (3) Va3x x + x = a ,

(4) a + a = 0=i>a = 0. THEOREM 2. A structure iXTi = (5;©), rvhere ©: S2 —s- S, is a midpoint algebra iff there exist a group V with operator | and an atlas OJ: S2 —> V, such that 231 and V are UJ-associated. Orthogonality relation 55

In the monograph [5] W. Szmiclcw introduced the parallclity planes (5; ||), where || C S'1. In the paper wc will omit upper bound axiom, which must be satisfied by the parallclity planes. Obtained in that manner struc- tures are called the parallclity spaces.

DEFINITION 3. A pair IL = (5; ||) is said to be a parallelity space if S is a set, || C 54 and il satisfies the following axioms P 0 ab\\ba, P1 ab\\cc, P 2 a ^ b Aab\\pq A ab\\rs pq\\rs, P 3 ab\\ac ba\\bc, PA 3abc->ab\\ac, P 5 3q (p / q A ab\\pq). A structure il = (5; ||. ©) is referred to as a midpoint space if (1) (5; ||) is a parallclity space, (2) (5; ©) is a midpoint algebra, and 11 satisfies the following axioms: T o/flA oa||o& => 3d [oc||or/ A ac||&cZ] (trapezoid axiom), F a, 6||a © c, b © c (Fa.no axiom). A midpoint space (.S';||,©) and a vector space = (V, A'; •) over a skew-field K are said to be u>-associated if >15 0 u>: S2 —* V is an atlas, AS 1 the midpoint algebra (5; ©) and V are w-associated, -45 2 ab\\cd vectors u(ab), uj(cd) are linearly dependent in for all a, b, c, d € S.

THEOREM 3. A midpoint space il = (S, ||, ©) is a Desarguean midpoint space iff there exist a vector space OT = (V, F\ •) over a skew-field K with characteristic different than 2u>-associated with space il.

2. Left orthogonality In Section 2 wc shall give an axiomatic description of the class of all structures 9JD = (V,W; IL^), where

V x W —K is an arbitrary left-linear form. Wc begin this section with some basic defini- tions concerning alFine and metric afiine geometry. One can regard them as a motivation for adopting a axiom system (see Definition 1), characterizing a class of corresponding "metric-vector" spaces. 56 M. Muzalcwski

DEFINITION 4. Let il = (S; ||,ffi) be a Desarguean midpoint space (see Section 1), V be a vector spacc over a skew-field 5" with characteristic ^ 2, and u: S2 V be an atlas such that: (1) ab\\cd vectors u(a, b),u:{c, d) are linearly dependent, (2) a@b = v(a, c) = w(c, b); consider arbitrary relations 1C 5"1 and IL C V2. The structures 1I'D := (il, 1) = (5; -L) and 235) = (V, JL) are said to be u>-associated, if (3) ab I cd& u;(«, b) IL w(c, d). Let ii = (5"; ||,©) be a Desarguean midpoint spacc and ICS4. The structure Hi) is said to be a Desarguean midpoint space with nondegenerate left orthogonality if Hi) satisfies the following axioms: ALO 1 oo ± op, ARO 1 op JL oo, ALO 2 oa -L op A ob A. op => o, a ® b ± o, p, ALO 3 oa 1 op A oa\\ob A o ^ a =>• ob I op, ALO 4 oa / op =>• 3b(xb ± op A ob\\oa), and ARO 5 o / a => 3p {op JL oa). DEFINITION 5. Let us consider a structure 235) = (V,W; 1L), where V, W are left vector spaces over a skew-field K and IL C V X W is an arbitrary binary relation. The structure 23® is said to be a vector space with left orthogonality, if it satisfies the following axioms: VLO 1 6 IL b, VLO 2 ai, a2 IL b aj + a2 IL b, and VLO 3 alLb =*> AalLb. Let us note that if HD = (ii; 1} and 935) = (V, V; IL ) are a»-a.ssociatcd, then ii5) is a Desarguean midpoint space satisfying ALO 1, ALO 2, and ALO 3 iff 235) is a vector spacc with left orthogonality. We will use the following symbols: ^B := {xe V:xJLLB}, := {y e W : AlLy}, and ^b := ^-{b}, a* := {a}^ . PROPOSITION 1. Let 235) = (V,W; IL), where V, W are left vector spaces over a skew-field K and J! C V x W be an arbitrary binary relation. (1) The structure 235) is a vector space with left-orthogonality iff -^b C V is a subspace for any b 6 W. (2) //2J5) is a vector space with left-orthogonality and B C W is a subset, then ^B C V is a subspace. Orthogonality relation 57

Proof. (1) follows directly by VLO 1, VLO 2 and VLO 3 • (2) By (1) and the evident equality: -^B = n{1b:beB}, we get the thesis • •

Definition 6. A map V x W K is said to be a lefl-linear form if

(1) £(xi + x2,y) = £(xi,y) + £(x2,y), (2) f(ax,y) = ai(x,y). Remark 1. Let C be the complex field and V be a vector space over C. A left-linear form £ : V x V —> C is called a semi-inner product if £ satisfies the following conditions: (3) £(x, x) > 0 when x ^ 0, (4) |£(x,y)|2<£(x,xK(y,y). The investigations on semi-inner products originated in the papers [3] and [2], and were motivated by Quantum Mechanics. Let us note that if £ is a semi-inner product, then the orthogonality relation need not be sym- metric. It seems that the notion of left-linear form may be considered as a natural and convenient basis for studying semi-inner products.

Proposition 2. // V x W — K is a left-linear form, over left K- spaces V and W, then the structure (V, W; _LL^) is a vector space with left orthogonality m

Definition 7. Let Q3D = (V, W; JL ) be a vector space with left orthog- onality; 03*1) is said to be a vector space with nondegenerate left orthogonality if it satisfies VLO 4 a-ji b =>• 3a(x — aa 1L b) (about projection, see fig. 1.5), VRO 5 b ± 6 =» 3a (a.j£b), VRO 1 aJLfl.

Proposition 3. //HD, 935) are ui-associated, then: (1) HD 1= ALOl iff WD (= VLOl, (2) 115) 1= ARO1 iff WD 1= VRO 1, (3) 11® t= ALOl A AL02 A ALOZ iff WD 1= VLO 1 A VL02 A VLO?,, (4) WD N ALOA iffWB t= VLOA, and (5) UV N AR05 iff WD 1= VROh m Let V and W be arbitrary vector spaces; let !L C V x W. In the structure (V,W; 1L) wc define: Lkcr( iL ) := ^W, Rker( JL ) := V^ . 58 M. Muzalewski

Fig. 1.5

If Lker( IL ) C V is a subspace, then we set

V0 := V/ Lkcr( il ). and if Rker( IL ) C W is a subspace, then we set

W0 := W/ Rkcr( iL ).

Proposition 4. 7/93D - (V, W; IL) is a vector space with left-orthogo- nality, then the following conditions are satisfies: (1) 2JX> 1= VL04 iff WD f= Va,b[a.j£ b => V = (a) © ^b], (2) t= VR05 iff 2JD t= Rkerf IL ) C O, (3) 23Î) t= VLOA A VROb A V RO1 iff WD N Vbcodim^b = dim(b), and (4) t= VR05 A VROl iff WD t= Rker( iL ) = O.

Proof. The conditions (1) and (2) are evident A (3) (=>) If b ^ 6, then, by VR05, there is a such that a_jl b. By (1) we get (a) © -"-b = V and thus codim -"-b = dim(a) = 1 A (•$=) Let b ^ 6. By the assumption, codim -^b = 1, so there is a with (a) © = V. Then a4l b, which proves VI10 5 A Let a. JL b. Then b 7i 6 and, by the assumption, codim -^b = 1. Hence (a) © ub = V, so by (1), we get VLO 4 A Orthogonality relation 59

(4) follows directly by (2) and VRO 1 A •

DEFINITION 8. A left-linear form V x W —• A" is said to be nonde- generate left-linear if Rkcr( 1L ^) = O. It is known (cf. [1]) that any two noiulcgenerate sesquilinear forms

£2 with lL^j = lI-£2 must be proportional. The fact strongly depends on the assumption that both £1 and £2 arc sesquilinear forms. Indeed, let K be an arbitrary skew-field, let V be a vector space over A', and let £1, £a be left-linear forms on V such that £1 is nondcgcncratc and is anisotroph

(i.e. fa(x,x) ^ 0 for all x ^ 0). Consider the map V X V —> K defined by the condition: 6(x,y) :=io(y,y)-6(x.y)- Clearly £1 and £> arc nondegenerate left-linear forms. It is easy to see, that iL^j = U_£2, but the forms £1, £> arc not proportional.

PROPOSITION 5. //£:V x W —>• K is a nondegenerate left-linear form over left K-spaces V and W, then the structure (V,W; -LL^), is a vector space with nondegenerate left-orthogonality.

Proof. The condition VLO 4 is an immediate consequence of Defini- tion 6, and VROb A VRO I is a consequence of Proposition 4 • As a direct consequence of Propositions 1 and 4 we get

PROPOSITION 6. Let V, W be vector spaces over the same skew-field. With every map k : W —2V, where 2V is the power set ofV, we correlate a relation iLK defined by

xiLKy :«xS fey, and with every relation JLL we correlate a map KJJ. : W —2V defined by b • "^"b. The following conditions are equivalent: (i) a structure (V,W; 1L) is a vector space with nondegenerate left- orthogonality;

(ii) there is a map k: W —S(V) such that iL = _1LK and

codim Kb = dim(b) for all b 6 W .

Moreover iL = _LLK in the case (i), and k = kjik in the case (ii) •

THEOREM 4. A structure (V,W; iL) is a vector space with nonde- generate left-orthogonality iff there is a nondegenerate left-linear form £ :

V X W -»• IC such that iL = ALi.

Proof. The implication (<=) follows by Proposition 5 A (=>) Put

(1) *0(x) := 0 . 60 M. Muzalcwski

Let y / tf. By Proposition 4, there is a such that V = (a) © -"-y. Define (2) *y(x) = a x = aa + z k ziLy & y ^ 0. Dy (1) and (2), (3) *y € V' and (4) kcr y = -"-y for all y € W. Let us dcfiiie £(x,y) := V(x); by (3), the map £ is a left-linear form. By (4), £(x,y) = 0o*y(x) = 0^xiLy, so IL = IL^. This yields that V-11-£ = V-11- = O, so £ is nondegenerate • As an immediate consequence of Definition 7 and the above we obtain

COROLLARY 1. Let ilD, 9T0 be LJ-associated. The structure it® is a Desarguean midpoint space with nondegenerate left orthogonality iff there is a nondegenerate left-linear form (,: V2 —¡- K such that ah J_ cd O i»),o;(c,c/)) = 0 •

PROPOSITION 7. If (V,W; iL) is a vector space with left-orthogonality and 0 € Ai, A2 C V , then AL (Ax + A2) = Aj"- HAf . Proof. The inclusion "C" is obvious, and the inclusion "C" follows directly from VLO'2 m

PROPOSITION 8. Let (V, W; iL) be a vector space with left-orthogonality and let A C Lkcr( IL) be a subspace ofV. In the quotient space V/A we define the relation A-LL by the following condition:

x + AAiLy :<£> xlLy . The structure (V/A, W; A-U- ) is also a vector space with left-orthogonality.

If A = Lker(iL), then we denote • IL = A-U- • The structure (V0, W; • _IL ) satisfies (1) Lkcr(- iL ) = O and (2) Rker(-iL) = Rker(iL) •

PROPOSITION 9. VL03 A VLOA => VLO1. Proof. Assume aiLb; by VLOZ, aiLb implies that 0 • a!Lb, which proves VLOl A If aJ^L b, then, by VLOl we consider a with 0 - a • aiL b; by VLOZ we get 0 • (—a • a) IL b, which proves the thesis • Orlìiogonaìiltj relation 61

3. Sesquilinear orthogonality In Section 3 wc will give an axiomatic description of the class of the structures iOT> = (V,W; ILi), where is an arbitrary sesquilinear form.

DEFINITION 9. The structure il® is a Desarguccin midpoint space with sesquilinear orthogonality [nondegenerate sesquilinear orthogonality] if il© satisfies AL01,..., ALOA, AR01,..., AR0A[AL0l,..., AL05,AR01,..., AR05].

DEFINITION 10. Let us consider a structure Q3D = (V,W; 1L), where V and W arc left vector spaccs over a skew-field K and IL C V x W is an arbitrary binary relation. The structure OUT) is said to be vector space with orthogonality, if it satisfies the following axioms (cf. Definitions 5 and 7): VLO 1 0iL b, VRO 1 a 1L 6,

VLO 2 al5a2 ILb => at + a2 Jib,

VR02 a iL bi, b-2 a IL bi + b2, VLO 3 a IL b => aa iL b, and VRO 3 a iL b => a iL /3h. Proposition 8 makes it obvious that the following holds.

PROPOSITION 10. Let. Q3Î) = (V,W; iL) be a vector space with orthog- onality, we define the relation • iL • C V0 X W0 by the condition x+ -^W- il - y + V11 roxily.

The quotient structure = (V0, W0; • iL •) is a vector space with orthog- onality, Lker(- iL •) = O and Rkcr(- iL •) = O m

DEFINITION 11. A vector space with orthogonality QJD = (V,W; iL) is said to be a vector space with sesquilinear orthogonality, if it satisfies the following conditions: VLO 4 a 4L b =>• 3a (x - oalLb), VRO 4 a^L b =» 3/3 (aiLy - /3b). A vector space with sesquilinear orthogonality Q3Î) = (V, W; iL ) is said to be vector space with nondegenerate sesquilinear orthogonality, if it satisfies the following conditions: VLO 5 a^0=>3b(a.jlb), VRO 5 b # 0 3a (a^b). 62 M. Muzalcwski

As an immediate consequence of Propositions 10, 2 and 11 we conclude with

PROPOSITION 11. Let ilD, QJD be u-associated. The structure Î1D is a Desarguean midpoint space with [nondcgencrale] sesquilinear orthogonality iff the structure 23 X) is a. vector space with [nondegenerate] sesquilinear or- thogonality m

EXAMPLE 1. Let £ : V x W —A' be a [nondegenerate] sesquilinear form over left A'-spaces V and W. The structure (V, W; IL^) is a vector space with [nondegenerate] sesquilinear orthogonality.

PROPOSITION 14. If (V,W; IL) is a vector space with nondegenerate sesquilinear orthogonality, then (1) (iLb)Ji=(b) and il(aiI) = (a), (2) if dim U = 2, then codim ^U > 2.

Proof. (1) Let b 6. By VRO'J and VLOA, there is a such that (a)+ ^b = V. Then, by Proposition 7, a-11 D ( -""b)-11- = V-11- ; thus, by Proposition 4, a-"- n ( -"-b)-"- = O. From this we get dim( -"-b)-"- < codim a11 . By Propo- sition 4 (applied to a right-orthogonality!), codim a-11 = 1. From this we have dim( ^b)-"- < 1. We have also (b) Ç ( ^b)11-. Thus (b)•"• = (b) A

(2) Let U = Pj + P2, where Pi ^ P2. Then by (1), ^Pi ^ P2 and codim """Pi = codim "U"P2 = 1. It may be concluded that

JJ codim ^U = codim "^(Pi + P2) > codim( ^Pi + "P2) > 2 A •

THEOREM 5. Let 2ÎD = (V,W; 1L), where V, W are left vector spaces over a skew-field K, and IL C V x W. A structure 23Î) is a vector space with [nondegenerate] sesquilinear orthogonality and dim Wo, dim Vo >3 iff there are an anti-isomorphism J : K — A* and a [nondegenerate] J-sesquilinear form (:VxW->A' such that 1L = iL^. Proof. Special case: 53Î) is a nondegenerate sesquilinear vector space. The implication (<=) directly follows by Example 1 A (=>) By Propositions 12, and 7 a transformation

PW 9 (b) A -"-b € HV Orthogonality relation 63 is a correlation. Wc next, prove that the map K is a strong correlation. Let

H D bi n -"-b^, where bi ^ b2 and b1( b2 £ 0. Then by Proposition 12,

11 Ji JL H C ( """b]) +(\) =(b1) + (b2), and dim H-0- < 2. Wc need only consider three cases: 1° dim Hu = 0. Set U = bi n -"-b^. Let us observe that codim U = 2. Thus there is b ^ 6 such that H = U + (b). By Proposition 7,

JL H-"- =V Db* and U-"- = (bi) + (b2). Then dimU-11 = 2, codm^b)^ = 1 and dim H-"- = din^U-"- nbu)>l, contrary to dimH-11 = 0. 2°dimHJL = 1. Then codim ^(H ^ ) = dim H11 = 1 = codim H . Since ^(H^ ) D H, wc conclude with ^(H^- ) = H. 3° dimH-11 = 2. Then by Proposition 12, codim ^(H11 ) > 2 > 1 = codim H, contrary to -"-(H-11- ) D H. Then the thesis follows from Theorem 1 A

General case: Q3Î) is an arbitrary scsquilincar vector space. By Pro- position 10, QJDo = (Vo, W(); • 1L •) is a vcctor space with nondegenerate sesquilinear orthogonality. Then, by the spécial case, there is a (nondegen- erate !) J-scsquilincar form

£0:V0 X Wo - A' such that • IL • = We define a J-scsquilincar form £: V X W —• K, by the following formula: i(x,y):=io(x+iLW,y + VJ1). To close the proof it suffices to observe that IL = IL^ •

Following the above theorem, by Example 1, wc conclude with the rep- resentation theorem for the class of all Desargucan midpoint spaces with sesquilinear (and with nondegenerate sesquilinear) orthogonality. From the point of view of foundations of metric affine geometry, the last theorem may be regarded as one of the most important.

THEOREM 6. Let ilî», be u-associated. The structure UV is a De- sarguean midpoint space with [nondegenerate] sesquilinear orthogonality iff there is a [nondegenerate] sesquilinear form Ç: V2 —• K such that ab J_ cd £(u;(a, b),u(c, d)) = 0 • 64 M. Muzalewski

Rcfcrcncos

[1] E. Artin: Geometric algebra, New York, London, 1957. [2] J. R. Giles: Classes of semi-inner-product spaccs, Trans. Amer. Math. Soc. 129 (1967), 436-441. [3] G. Lumer: Scmi-inncr-product-spaces, Trans. Amcr. Math. Soc. 100 (1961), 29-43. [4] M. Muzalewski, K. Prazmowski: Axiomatic investigations on symplectic geom- etry, Demonstratio Math. 2 (1993) 295-306. [5] W. Szmielew: From ajjinc to Euclidean geometry, An axiomatic approach, Warsaw, Dordrecht, 1983.

INSTITUTE OF MATHEMATICS WARSAW UNIVERSITY, BIAI-YSTOK BRANCH ul. Akademicka 2 12-267, BIALYSTOK, POLAND

Received July 8, 1991.