Geometry of kinematicK-loops

Abh. Math. Sem. Univ. Hamburg 65 (1995), 189-197

Geometry of Kinematic K-loops

BY E. KOLB and A. KREUZER

Abstract. A K-loop is called kinematic, if a further condition (K7) is valid. Such a loop (L,(9) can be provided in a natural way with a left and right structure ~ and t5 such that (L, ~) and (L, 15) become incidence (linear) spaces. For (L, ~) and t E L, each left translation t + : L ~ L; x ~ t (9 x is a collineation and (L, 15) can be turned in an incidence space with parallelism (L, 15, [[). Examples of kinematic K-loops are given for which the corresponding automorphisms 6~,b are either the identity or fixed point free.

In order to describe sharply 2-transitive groups, H. KARZEL introduced in [4] the notion of a neardomain (F, ~, .) (cf. [16], Kap. V). The properties of the additive structure (F, ~) lead W. KERBY and H. WEFELSCHEID to introduce the concept of a K-loop. In the last years the interest on K-loops is grown, since A.A. UNGAR showed in [13], [14] that the velocities ~.~ := {v E ~3 "lvl < c} with the relativistic velocity composition (9 form a non-commutative and non-associative K-loop, which UNGAR calls a gyrogroup.

Every K-loop satisfies the Bol identity (B) and the so called automorphic inverse property (K5) (cf. section 1), i.e., every K-loop is a Bruck loop. The Bol identity appears the first time 1937 in a note by G. BOL [1] in connection with the coordinatization of webs and was considered in the following years in many papers (cf. e.g. [2], [3], [12]). If in a Bruck loop (L, @), x ~ x ~ 0 for all x ~ 0, then (L, ~) is also a K-loop (cf. [6], Theorem 1).

An example (R, ~) of a non-associative kinematic K-loop with (K7) is given in section 3, by using a commutative ring R with Jacobson radical J ~ {0}. This example is a generalisation of the gyrogroup on the complex disc ~1 := {z E (I~ :lzl < 1} of [15]. (UNGAR remarks in [15] that the MObius transformations / / z) ) l + ~ z aEIE1

do not form a group.) Different from the examples of [6], [8] each auto-

morphism 8a,b ~ Aut(R, ~), which is not the identity, is here fixed point free on R\{0}.

1 Kinematic K-loops

Let L be a set with a binary operation ~. We call (L, ~) a loop if (K1), (K2) are valid, and a K-loop, if (K1) through (K6) are fulfilled for all a, b, c E L.

190 E. Kolb and A. Kreuzer

(K1) The equations a r x = b and y r a = b have unique solutions x, y E L.

(K2) There is a two-sided neutral element 0 6 L with a r 0 = a = 0 r a.

(K3) For a, b E L there exists an automorphism 6a,b E Aut(L, r with:

a q~ (b r x) = (a r b) r 6~,b(X) for all x E L.

(K4) If a ~ b = 0 then 6a,b = id.

For a, b E L with a ~ b = 0 we get a ~ ( b ~ a ) (K3} ( a C b ) r (~__4) O C a = a,

i.e., b @ a = 0 by (K1). Therefore b is the right and left inverse o f a and we write e a := b.

(K5) The automorphic inverse property is satisfied, i. e., ( e a ) ~ ( O b ) = e ( a O b ) .

(K6) (~a,b = 6a,bea.

We remark that (K4) is a consequence of (K1), (K2), (K3) and (K6) (cf. [8], (2.10)): Since a @ x = a ~ (0 @ x) = a @ 6a,O(X) for all x E L, 6a,0 = id.

Hence for a, b E L with a @ b = 0, id =fib,O = 6b,aeb (K6) 6b,a and b = b ~ (a ~ b) = (b r a) ~ 6b,a(b) = (b O) a) O) b. By (K1), we get b @ a = 0 and

(~a,b (K6) (~a,b~a = (~a,O = id.

A loop (L, r is called a Bol loop, if for all a, b, c E L the Bol identity

(B) a �9 (b �9 (a �9 c)) = (a �9 (b �9 a)) �9 c

is fulfilled. We call a Bol loop a Bruck loop (cf. ROBINSON [11]), if also (K5) is valid. Every K- loop fulfills the Bol identity, hence every K- loop is in part icular a Bruck loop (cf. [6], (1.2)). If a Bruck loop (L, r satisfies x r x :p 0 for x E L* := L\{0}, then (L, r is also a K- loop (cf. [6], Theorem i).

F rom now on let (L, ~) be a K- loop and for a 6 L* let

[a] : = { x E L :b,~,~,=id}.

Since 6,,a = 6a,o = id, 0, a E [a]. By [8], (2.2.c), (2.6), (2.9):

1.1. Let a, b, c c L, d E L* and c~ E Aut(L, r Then:

(a) ~ o 6a,b o o~ - 1 = (~ct(a),ct(b) (e) 6 - I = 6b,a , a,b ,

(b) (Ca) �9 (a �9 b) = b, (f) (~ea,eb = (~a,b,

(c) 6a,O = 6o,, = id, (g) (~a,bec o (~b,c = 6aeb,6,,b(c) o 6,,b,

(d) (~a,b(b ~ a) = a ~ b , (h) ~([d]) = [~t(d)].

P r o o f (a) to (g) by [81. (h): For x E

~)ct(d),o:(x) = ~ 0 (~d,x

L we get ~(x) E ct([d]) .,v--> x E [d] ~ ~d,x = id r o ~-1 = id by (a) -: :- ~(x) E [ct(d)], i.e., ~([d]) = [~(d)].

Geometry of Kinematic K-loops

Theorem 1.2. Let (L, 09) be a K-loop. Then the following statements

(K7) l f for a, x, y E L*, 6a,x = 6a,y = id, then 6~,y = id.

(K7') {[a]\{0} : a E L*} is a partition o f L*.

are equivalent and imply that for a E L*, ([a], 09) is a commutative subgroup.

191

Proof Assume (K7'). For x, y E [a] with x 4 : 0 we get y E [a] = [x], hence (~a,x = (~a,y ~" id implies 6~,y = id. Now we assume (K7). Then:

a) [a] is a subloop: Let x, y ~ [a], i.e. 6a,~ = 6~,y = id. By (K7) and (K6) (~yoc = f~yx~y = id. By (1.1.e) we get (~ya = ~ - - 1 = id, hence again by (K7), Oa,x~y = id. By (K4) 6a.ea = id, thus by (K7) 6ea,~ = id, and so 6~,ex = id by (1.1.f).

b) ([a],09) is associative and commutat ive: Let x, y, z E [a]. Since 6x,y = id, x ~ (y ~ z) = (x 09 y) 09 6~,y(z) = (x 09 y) 09 z, and by (1.1.d) x 09 y = 6x,y (y 09 x) = y 09 x.

c) {[a]\{0} : a 6 L*} is a partition of L*, i.e., for a, b ~ L* either [a] n [b] = {0} or [a] = [b]: Let 0 4: x E [a] n [b]. Then 6,,~, = 6x,a = id and 6b,~ = 6~,b = id (cf. (1.1.e)) implies by (K7) 6~,b = id. Let z E [b]. Then 6b,~ = id and 6a,b = 6b,a = id imply 6,,~ = id, i.e., z E [a] and [b] c [a].

We call a K- loop kinematic, if one of the properties (K7) or (K7') are fulfilled.

1.3. Let (L, 09) be a kinematic K- loop and let a, x, y E L with a 4: 0. If y E [a] 09 x, then [a] 09 x = [a] 09 y and 0b~ := { [a] 09 x : x E L} is a part i t ion of L.

Proof Let z ~ [a] w i t h y = z @ x c [a]09x. Let w c [a] ,hence w09(z09x) c [a]09y. Since z, w E [a], 6w,z = id. We obtain w 0 9 y = w @ ( z @ x ) = (w 09 z) 09 6w,z(x) = (w 09 z) 09 x E [a] 09 x, since ([a], 09) is a subloop of (L, 09) by (1.2). This shows [a] 09 y c [a] 09 x. On the other hand by (1.1.b), x = (Oz) 09 (z 09 x) = (@z) 09 y E [a] 09 y, hence [a] 09 x c [a] 09 y. By (1.1.b) for any p 6 L, p = 0 09p E [a] @p ~ ff~,.

1.4. If for a, b, x E L, 6a,b = id implies 6~,x = 6b,x or if L = [a] for an element a E L, then (L, 09) is a commutat ive group.

Proof Let a, b c L, a 4: 0. By (1.1.c) 6a,0 = id, hence (~a,b = (~O,b = id, i.e., L = [a].

2 Geometries of Kinematic K-loops

Let (L, 09) be a kinematic K- loop with ILl ~ 2. We define a right and left geometry (L, ff)) and (L, ~), respectively. We consider L as the set of points, f f i : = { [ a ] 0 9 x : a , x ~ L w i t h a : f i 0 ) and ~ := {x 09 [a] : a , x ~ L w i t h a ~ 0 } ,


respectively, as the sets o f lines. For lines G = [a] r x, H = [b] r y ~ 15 we define

G II n if and only if [a] = [b]

and call G, H parallel. For t E L let

t + : L ~ L , x~ ~ t ~ x .

Theorem 2,1. (L, 15, II) is an incidence (l inear) space with a parallelism, i.e., the following axioms are satisfied:

(I1) For every two distinct points x, y E L, there exists a unique line G E 15 with x, y E G.

(I2) For every line G ~ 15, IGI > 2.

(P) For every line G E 15 and every point x E L, there is a unique line H E 15

with G II n and x E H.

Proof. Let x, y E L be distinct points. By (K1) there exists an element a E L with a C x = y. Hence x, y E [ a ] @ x , since 0 E [a] (cf. (1.2)). Now let [b] r z ~ 15 with x, y E [b] r z. By (1.3), [b] r z = [b] r x. Hence there is a w E [b] with y = w r x, which implies a = w ~ 0 by (K1) and [b] = [w] = [a] by (1.2), i.e., [ a ] r x = [b] r w is the uniquely determined line of 15 joining x and y. (I2) is a consequence o f 0, a E [a] and (1.3) implies (P).

Theorem 2.2. (L, ~) is an incidence space with the additional property

(L) For s, t E L, t + is an automorphism o f (L, yd) with t + o s + = (t ~ s) + o ~t,s.

Proof. (L) is valid, since t+(x r [a]) = t r (x @ [a]) = (t ~ x) ~ 6t,x[a] = (t ~ x) ~ [6t,x (a)] E Yd. t + o s + (x) = t ~ (s ~ x) = (t ~ s) ~ 6t,s (x) = (t ~ s) + o 6t,s (x) by (K3). (I1): Let x, y ~ L with x ~ y. By (K1) there is an element a ~ L with x r a = y, hence x, y E L := x r [a]. By (L) we may assume x = 0 and so y ~ [a] with y :~ 0, i.e., [y] = [a] by (1.2). Now let 0, y ~ w r [b]. Since 0 E w r [b] it follows e w E [b] and by (1.2) w ~ [b], i. e., w r [b] = [b]. Hence y E [b] implies [y] = [b] and L = w �9 [b]. (I2) is a consequence of 0, a ~ [a].

2.3. For y E x @ [a] it follows x r [a] = y r [6ey,x(a)].

Proof. By (1.1.b), y @ [b] = x @ [a] .*~ [b] = ( ey ) @ (x @ [a]) = ( (ey) @ x) [~ey,x(a)] r ( e y ) �9 x E [b] -- [6ey,x(a)], since [b] is a group by (1.2). Hence

x @ [a] = y @ [6ey,x(a)].

For a E L ~ in general yda = {x ~ [a] : x E L} is not a par t i t ion of L. The following proper ty

(K8) For a, x ~ L, (~a,ba,x(a) = id.

Geometry of Kinematic K-loops 193

implies that ~ is a par t i t ion of L, but for the example (3.5), (K8) is not valid (set for example a = 1, x = i).

2.4. Let a, b E L* and x, y E L. Then ~a is a par t i t ion of L if and only if (K8) is fulfilled. Then for

x �9 [al II Y �9 [bl r [a] = [b]

(L, E, II) is an incidence space with a parallelism.

Proof . Clearly x E x @ [a]. We have to prove that for y r x @ [a] we have x @ [a] = y @ [a] if and only if (K8) is fulfilled: By (1.2) we m a y assume y = x ~ a. Then x @ [a] = y @ [a] r there is a bijection ' : [a] ~ [a], b ~-* b' such that x ~ b' = y ~ b = (x ~ a) @ b = x ~ (a ~ ~a,x(b)) (cf. (K3) and 6x_a 1 = 6a,~ by (1.1.e)) r b' = a ~ 6~,x(b) r [a] = ~a,x([a]) -- [~a,x(a)], since [a] is a group r ~a,x(a) E [a] r (K8).

3 Example

Let (R, +, .) be a commuta t ive ring with identity and denote by J the Jacobson radical o f R. Let - be an involutory au tomorph i sm o f (R, +, .) and fix some e E J with ~ = e. For an example see (3.5). Recall that J consists o f all a E R such that 1 + ar is a unit for any r E R ([9]). Thus one can define a new compos i t ion ~ on R as follows:

x + y @ : R x R ) R , ( x , y ) , ~ x @ y . - l + e N y

Theorem 3.1. (R, ~ ) is a K- loop, where 0 is the neutral element, @x = - x and the automorphism 6~,b is given by

1 + eab 5a,b(x)= l + e ~ b x .

(R, ~ ) is non-commutat ive and non-associative, i f there is an e lement b E R with

~ ~ eb.

Proof . We have to show the axioms (K1) to (K6). So let a, b E R.

a+x . . b ( l + e - a x ) = a + x c = ~ x - b-a (K1): b = a ~ x = ~ ~ l-~bn"

y + a (*) b = y ~ a -

1 + e~a r b(1 + eya) = y + a .

An appl icat ion of - yields b(1 + ay~) = y + ~ and by el imination o f y we

b-a+~ab(~-a) which is a solution of (*). obta in y = l_b-~a~e 2 ,


(K3):

a ~ (b ~ x) = a + ~+~ a ( l + ~ b x ) + b + x l +~bx

l + ~ a b+5 l + g b x + ~ a b + e a x l+ebx

a+b 1 +eab X

l+,~b + l+,~b = (a ~ b) ~ G,b(X) l + e ~+~_l+~a~ l+gabl-'~'~'~ ~

with 1 + ea-b

6a,b(X) := 1 + e-d--~-b x = 5a,b(1) " X.

TO show that (~a,b is an automorphism, observe that for r := Oa,b(1) we have Fr = 1 and consequently

r x + r y x + y = 6 a b ( X ~ y ) . 6 a ' b ( X ) ~ 6 a ' b ( Y ) - - l+eg-~ry - r l + e 2 y '

(K4) : a ~ b = 0 -: :. a = - b =~ (~a,b = id.

(K5): ( -a ) ~ ( -b) = ~'~-~--a-b --(a I~ b).

(K6):

3 ~ b ~ a = 3 a , b

ea b + a ~ = ( b + a ~ ( l + e a b ) (1 + e~b) 1 + 1 ~ , ] 1 + ~ 1 + ~ba ,]

1 + e~b + ga(b + ~) = 1 + ~ab + ~ (b + a).

Suppose that e b ~ eb for some b E R. Then 61,b-~id, l ~ b ~ b ~ l and 1 ~ ( b ~ 1) @ (1 ~ b ) ~ 1.

3.2. 6a,b = id ~ eab= g~b. Hence [a] = {x E R �9 ~(aX -- ~x) = 0}.

If eaX = eax and eay = g~y, then eaxy = eaXyy = e-d-~y. Therefore

3.3. If ~ is not a zero divisor and if 6a,x = Oa,y = id, then 5x,y = id.

3.4. If (R, +, .) is a domain, then (R, ~) is a kinematic K-loop and 6a,b is either fixed point free on R\{0} or the identity.

Example 3.5. Let R = Cl[t]] be the power series ring in t over the complex numbers. (J = {x = ~i~o ai ti E R :a0 = 0} is the maximal ideal and R \ J is the group of units.) Choose for - the natural extension of the conjugation in 112 and set e = t. From (3.4) and (3.1) it follows that (R, ~) is a non-commutative and non-associative kinematic K-loop.


Now consider the incidence space with parallelism (R, t5, II), ~ = {[a] ~ x : a, x e R, a :fi 0}, associated with (R, ~) (see section 2). Let S = {s E R :~ = s}. (For the example (3.5), S = Rl[t~.)

For the rest of this paper, suppose that R is a local domain with maximal ideal J satisfying the following condition

(C) For all b E R there is an a E R \ J such that b E Sa.

which is obviously valid in (3.5). Then it is possible to give an explicit description of the lines:

3.6. Any line G E ~ has the form G = Sa @ x with x E R and a 6 R \ J .

Proof. Let G = [b] ~ x. Then by (C), b = sa with a E R \ J and s e S. By (3.2), [b] = [sa] = {x E R : sa~ = s~x} = [a] --- {x E R : a - l x = a - i x } ~- {X E R : a - i x 6 S} = Sa.

Finally we ask, when two different lines [a] ~ x, [b] ~ y intersect. Clearly by (2.1), a # b. But this is not sufficient as example (3.5) shows, since [1] ~ 0 N [1 + it] ~ i = 0. (Note that also (K8) does not hold in (3.5): choose a = b = l a n d x = i . )

In order to give sufficient conditions, we assume that R is even a local principal domain with maximal ideal J such that R is complete with respect to the uniform topology generated by {a + J" : a e R, n e N} (the J-adic topology). For the terminology see [11], II 16. In other words, there exists a discrete non-archimedian valuation v on the quotient field K of R such that K is complete under v and R is the valuation ring of v. Note that these conditions are valid in example (3.5), if one takes the leading degree of a power series as valuation. Then we ccan prove the following:

Theorem 3.7. I f R satisfies (C) then for two different lines [a] ~ x, [b] (9 y (a, b ~ O) the following hold:

(i) ab - ~b E R \ J implies [a] @ x A [b] @ y ~ 0.

(ii) I f a-b - ~b E J and (x - y)b - (7 - -~)b E R \ J then [a] ~ x N [b] @ y = 0.

Proof. Because of (3.5) we may assume that [a] ~ x = Sa~gx, [b] ~ y = S b ~ y with a, b E R \ J and Sa --fi Sb, i.e., a-b ~ S. Then for 2, # E S

2 a + x # b + y

1 + e2-dx 1 + epby (*)

# [e(b-dx - bay)2 + b - e-byx] = (a - e-dxy)2 + x - y .

With the abbreviations A = a - eaxy, B = b - e-bxy and C = b-dx - -bay this is equivalent to: 3 2 e S such that ~ = # for # := ~ r 3 ~. e S such that eC2+B p(2) := e(A-C--AC)22 + ( e ( x - y ) U - - e ( ~ - y ) C + A B - A B ) 2 + ( x - - y ) - B - - ( ~ - - y ) B =


~ S is a solution o f (*), iff p(2) = 0 for some 0. Hence 2 and /~ = ~B~+c 2 G S .

(i) p(t) is a po lynomia l in R[t] and for 2o = (~-Y)b-(x-e)~ G S it follows that ab-~b p(20) -= (ab - ~b)),0 + (x - y)b - (~ - y)b = 0(mod J) and p'(2o) = a-b - -db ~ 0

(mod J), because of our assumption. By Hensel 's L e m m a [10], I I w Prop. 2, there is a 2 e R with p(2) = 0. To show 2 e S, we observe that p(s) = - p ( s ) and p'(s) = -p ' ( s ) for all s e S. Since 2 is the limes o f the sequence (2n),~N with 2,+i = An - z(h)_~ and 20 e S, we conclude 2n = An for all n G hi. But now p'(~)

= 2 by the continuity o f - . So 2 G S.

(ii) Suppose [a] ~ x n [b] ~ y :fi 0. Then there is a 2 ~ S with p(2) = 0 and the contradict ion p(2) -- (x - y)b - (~ - y)b - 0 (mod J) follows.

References

[1] G. BOL. Gewebe und Gruppen. Math. Ann. 114 (1937), 414-431.

[2] R.H. BRUCK. A Survey of Binary Systems. Springer-Verlag, Berlin 1958.

[3] G. GLAUBERMAN. On Loops of Odd Order. J. Algebra 1 (1966), 374-396.

[4] H. KARZEL. Zusammenh~inge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom. Abh. Math. Sere. Univ. Hamburg 32 (1968), 191-206.

[5] G. KIST. Theorie der verallgemeinerten kinematischen R~iume. Beitrtige zur Geo- metrie und Algebra 14, TUM-Bericht M8611, MiJnchen 1986.

[6] A. KREUZER. Beispiele endlicher und unendlicher K-Loops. Res. Math. 23 (1993), 355-362.

[7] A. KREtJZER. Algebraische Struktur der relativistischen Geschwindigkeitsaddition. Betrl~ge zur Geometrie und Algebra 23 (1993), 31-44.

[8] A. KREUZER and H. WEFELSCHEID. On K-loops of finite order. Res. Math. 25 (1994), 79-102.

[9] J. LAMBEK. Rings and Modules, second edition, New York 1976.

[10] S. LANG. Algebraic Numbers. Addison-Wesley 1964.

[11] M. NAGATA. Local Rings. Interscience 1962.

[12] D.A. ROBINSON. Bol-Loops. Trans. Amer. Math. Soc. 123 (1966), 341-354.

[13] A.A. UNGAR. Thomas rotation and the parametrization of the Lorentz trans- formation group. Found. Phys. Lett. 1 (1988), 57-89.

[14] A.A. UNGAR. Weakly associative groups. Res. Math. 17 (1990), 149-168.

[15] A.A. UNGAR. The holomorphic automorphism group of the complex disc. Aequat. Math. 47 (1994), 240-254.

[16] H. WXHLING. Theorie der Fastki~rper. Thales Verlag, Essen 1987.


Eingegangen am: 17.12.1993 in revidierter Fassung am: 07.09.1994

Authors' addresses: Emanuel Kolb, Fakult~it for Mathematik, Technische Hochschule Darmstadt, 64289 Darmstadt, Germany.

Alexander Kreuzer, Mathematisches Institut, Technische Universit~it Miinchen, 80290 Miinchen Germany.

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Geometry of kinematicK-loops