## (1.1) (1999)

### BibTeX

@MISC{Kinyon99(1.1),

author = {Michael K. Kinyon},

title = {(1.1)},

year = {1999}

}

### OpenURL

### Abstract

Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a power-associative Kikkawa left loop with two-sided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space structure of S. (S, ⊙) is not a loop, and the right translations which fail to be injective are easily characterized. (S, ⊙) satisfies the left power alternative and left Bol identities “almost everywhere” but not everywhere. Left translations are everywhere analytic; right translations are analytic except at −e0 where they have an essential discontinuity. The orthogonal group O(H) is a semidirect product of (S, ⊙) with its automorphism group. The left loop structure of (S, ⊙) gives some insight into spherical geometry.

### Citations

1124 |
Differential Geometry, Lie Groups and Symmetric Spaces. Number 34
- Helgason
- 2001
(Show Context)
Citation Context ...e M allows. Thus if M is a smooth manifold, ∗ is assumed to be smooth, and not just continuous. In the smooth case, Definition 1.1 is due to Loos [16]. It is equivalent to the more standard ones (cf. =-=[7]-=-), but has the advantage of depending only on the topology. This motivates our trivial adaptation of his definition. The magma (S, ∗) defined by (1.2) is easily seen to satisfy (L1), (L2) and (L3). Al... |

39 |
Symmetric spaces
- Loos
- 1969
(Show Context)
Citation Context ...ght into spherical geometry. 1. Introduction Let H be a real Hilbert space with inner product 〈·, ·〉. Let S = {x ∈ H : 〈x,x〉 = 1} be the unit sphere in H. For x,y ∈ S, set (1.2) x ∗ y = 2〈x,y〉x − y. (=-=[16]-=-, p.66). If x and y span a plane Π, then x ∗ y is a point lying in S ∩ Π which is obtained by reflecting y in Π about the line passing through x and the origin. Equivalently, thinking of S∩Π as the gr... |

35 |
Quasigroups and Loops
- Pflugfelder
- 1990
(Show Context)
Citation Context ...N SPHERES 5 with the multiplication ∗.) Define (1.8) x · y = x 1/2 ∗ (e ∗ y) for x,y ∈ M. Then (M, ·) is a loop with identity element e, and in fact, (M, ·) is the principal e,e-isotope of (M, ∗) [1] =-=[19]-=-: x · y = (x/e) ∗ (e\y). Following some definitions, we will identify the class of loops to which (M, ·) belongs. In a left loop, denote the left translations by Lx : y ↦−→ x · y, and the left inner m... |

26 |
On loops of odd order
- Glauberman
- 1968
(Show Context)
Citation Context ...B-loop. Contemporary usage of the term “Bruck loop” in the literature (with some exceptions) tends to be as we have given it here, cf. [14] [15]. This usage seems to stem from a remark of Glauberman (=-=[6]-=-, p.376). Bruck loops are equivalent to the class of loops Ungar dubbed “gyrogroups” [29] and later “gyrocommutative gyrogroups” [31]. Bruck loops are also equivalent to “K-loops”, which are the addit... |

18 |
Theory of K-loops
- Kiechle
- 2002
(Show Context)
Citation Context ...tomorphism. A left loop with two-sided inverses is said to have the automorphic inverse property if (AIP) (x · y) −1 = x −1 · y −1 for all x,y. An Al, LIP, AIP left loop is called a Kikkawa left loop =-=[9]-=-. A loop is called a (left) Bol loop if it satisfies the (left) Bol identity: (Bol) LxLyLx = L x·(y·x) for all x,y,z. Bol loops necessarily satisfy LIP (see, e.g., [9], 6.4). A Bol loop with AIP is ca... |

17 |
Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic
- Ungar
- 1997
(Show Context)
Citation Context ...f. [14] [15]. This usage seems to stem from a remark of Glauberman ([6], p.376). Bruck loops are equivalent to the class of loops Ungar dubbed “gyrogroups” [29] and later “gyrocommutative gyrogroups” =-=[31]-=-. Bruck loops are also equivalent to “K-loops”, which are the additive loops of near-domains. Neardomains were introduced by Karzel [8], and the additive loop structure was later axiomatized and named... |

13 |
Thomas precession and its associated grouplike structure
- Ungar
- 1991
(Show Context)
Citation Context ...ons) tends to be as we have given it here, cf. [14] [15]. This usage seems to stem from a remark of Glauberman ([6], p.376). Bruck loops are equivalent to the class of loops Ungar dubbed “gyrogroups” =-=[29]-=- and later “gyrocommutative gyrogroups” [31]. Bruck loops are also equivalent to “K-loops”, which are the additive loops of near-domains. Neardomains were introduced by Karzel [8], and the additive lo... |

12 |
Smooth quasigroups and loops
- Sabinin
- 1999
(Show Context)
Citation Context ...f any distinguished point e, there exists a local B-loop structure with e as its identity element. This theory was worked out in detail primarily by Sabinin [23]; expositions can be found in [17] and =-=[24]-=-. For a smooth reflection quasigroup, the globally-defined B-loop operation agrees with the locally-defined operation guaranteed by the general theory wherever the latter operation is defined. Every B... |

11 |
Geometry of homogeneous Lie loops
- Kikkawa
- 1975
(Show Context)
Citation Context ...ails only on the set of “measure zero” S∩V. In the finite dimensional case, we may remove the quotation marks. A globally smooth Kikkawa left loop necessarily satisfies the left alternative property (=-=[11]-=-, Lemma 6.2. The result is stated for loops, but the proof clearly works for left loops.) It follows from Corollary 4.8 that the multiplication ⊙ on S is not globally smooth. We will examine the conti... |

8 |
The holomorphic automorphism group of the complex disk
- Ungar
- 1994
(Show Context)
Citation Context ... to the usual definition of the spherical distance function. Before establishing properties of the distance function, we require a lemma. The following result was established by Ungar for Bruck loops =-=[30]-=-. Here we extend it to Kikkawa left loops. Lemma 7.2. For all x,y,z in a Kikkawa left loop, (7.4) Lxy · (Lxz) −1 = L(x,y)(y · z −1 ). Proof. In any LIP left loop, we have the identity (7.5) L(x,y) −1 ... |

7 | Loops and semidirect products
- Kinyon, Jones
- 2000
(Show Context)
Citation Context ...eral by Sabinin [22] (see also [17], [24]), and was later rediscovered in the particular case of Al left loops with LIP by Kikkawa [11] and Ungar [28]. A survey with recent extensions can be found in =-=[12]-=-. Since (S, ⊙) is an Al left loop, we may form its standard semidirect product with Aut(S, ⊙). This is the group denoted by S ⋊ Aut(S, ⊙), consisting of the set S × Aut(S, ⊙) with multiplication defin... |

7 |
On the equivalence of categories of loops and homogeneous spaces
- Sabinin
- 1972
(Show Context)
Citation Context ...group O(H) relative to the subgroup O(V). This discussion is a particular case of the general theory of semidirect products of left loops with groups. This theory was worked out in general by Sabinin =-=[22]-=- (see also [17], [24]), and was later rediscovered in the particular case of Al left loops with LIP by Kikkawa [11] and Ungar [28]. A survey with recent extensions can be found in [12]. Since (S, ⊙) i... |

7 |
Weakly associative groups
- Ungar
- 1990
(Show Context)
Citation Context ...t loops with groups. This theory was worked out in general by Sabinin [22] (see also [17], [24]), and was later rediscovered in the particular case of Al left loops with LIP by Kikkawa [11] and Ungar =-=[28]-=-. A survey with recent extensions can be found in [12]. Since (S, ⊙) is an Al left loop, we may form its standard semidirect product with Aut(S, ⊙). This is the group denoted by S ⋊ Aut(S, ⊙), consist... |

6 |
On some quasigroups of algebraic models of symmetric spaces
- Kikkawa
- 1973
(Show Context)
Citation Context ...considerations lead us to the following definition. Definition 1.4. A reflection quasigroup (M, ∗) is a left keyesian, left distributive quasigroup. The term “reflection quasigroup” is due to Kikkawa =-=[10]-=-. Reflection quasigroups are also known as “left-sided quasigroups”, following a convention of Robinson [21]. Reflection quasigroups are also equivalent to the recently studied “point reflection struc... |

5 |
Methods of nonassociative algebra in differential geometry. Suppl. to the Russian transl. of S.Kobayashi and K.Nomizu, Foundations of differential geometry. V.1
- Sabinin
- 1981
(Show Context)
Citation Context ...y which guarantees that in a neighborhood of any distinguished point e, there exists a local B-loop structure with e as its identity element. This theory was worked out in detail primarily by Sabinin =-=[23]-=-; expositions can be found in [17] and [24]. For a smooth reflection quasigroup, the globally-defined B-loop operation agrees with the locally-defined operation guaranteed by the general theory wherev... |

5 |
A left loop on the 15-sphere
- Smith
- 1995
(Show Context)
Citation Context ...omorphic to) the Moufang loop of unit octonions. This is because (S, ⊙) turns out to be a left loop, but not a loop for dimH > 2. Also (S 15 , ⊙) is not (isomorphic to) the left loop studied by Smith =-=[26]-=- using sedenian multiplication; the latter left loop is not power-associative. Finally, recall that the 2-sphere S 2 may be identified with the Riemann sphere Ĉ = C ∪ {∞} by stereographic projection, ... |

5 |
The relativistic noncommutative nonassociative group of velocities and the Thomas rotation
- Ungar
- 1989
(Show Context)
Citation Context ... [8], and the additive loop structure was later axiomatized and named in unpublished work of Kerby and Wefelscheid (the first appearance of the term “K-loop” in the literature was in a paper of Ungar =-=[27]-=-). The aforementioned equivalences have been established independently by various authors. Kreuzer showed the equivalence of Bruck loops with K-loops [15], and Sabinin et al showed the equivalence of ... |

4 |
A construction of Bruck loops
- Kepka
- 1984
(Show Context)
Citation Context ...aring is a permutation, and is thus equivalent to what we call a B-loop. Contemporary usage of the term “Bruck loop” in the literature (with some exceptions) tends to be as we have given it here, cf. =-=[14]-=- [15]. This usage seems to stem from a remark of Glauberman ([6], p.376). Bruck loops are equivalent to the class of loops Ungar dubbed “gyrogroups” [29] and later “gyrocommutative gyrogroups” [31]. B... |

4 |
On the notion of gyrogroup, Aeq
- Sabinin, Sabinina, et al.
- 1998
(Show Context)
Citation Context ...stablished independently by various authors. Kreuzer showed the equivalence of Bruck loops with K-loops [15], and Sabinin et al showed the equivalence of Bruck loops with (gyrocommutative) gyrogroups =-=[25]-=-. (The direct equivalence of gyrocommutative gyrogroups with K-loops is a well-known folk result.) The term “B-loop” was introduced by Glaubermann [6] to describe a finite Bol loop with the automorphi... |

2 |
A Survey of Binary Systems” (3rd printing
- Bruck
- 1971
(Show Context)
Citation Context ...ES ON SPHERES 5 with the multiplication ∗.) Define (1.8) x · y = x 1/2 ∗ (e ∗ y) for x,y ∈ M. Then (M, ·) is a loop with identity element e, and in fact, (M, ·) is the principal e,e-isotope of (M, ∗) =-=[1]-=- [19]: x · y = (x/e) ∗ (e\y). Following some definitions, we will identify the class of loops to which (M, ·) belongs. In a left loop, denote the left translations by Lx : y ↦−→ x · y, and the left in... |

2 |
Reflection geometries over loops
- Gabrieli, Karzel
- 1997
(Show Context)
Citation Context ...lso known as “left-sided quasigroups”, following a convention of Robinson [21]. Reflection quasigroups are also equivalent to the recently studied “point reflection structures” of Gabrieli and Karzel =-=[3]-=- [4] [5]. Given a nonempty set P and a mapping ˜ : P → Sym(P) : x ↦−→ ˜x, Gabrieli and Karzel call the pair (P,˜) a “point-reflection structure” if the following hold: (i) ∀x,y ∈ P, (˜x ◦ ˜x)(y) = y; ... |

2 |
Point-reflection geometries, geometric K-loops and unitary geometries
- Gabrieli, Karzel
- 1997
(Show Context)
Citation Context ...known as “left-sided quasigroups”, following a convention of Robinson [21]. Reflection quasigroups are also equivalent to the recently studied “point reflection structures” of Gabrieli and Karzel [3] =-=[4]-=- [5]. Given a nonempty set P and a mapping ˜ : P → Sym(P) : x ↦−→ ˜x, Gabrieli and Karzel call the pair (P,˜) a “point-reflection structure” if the following hold: (i) ∀x,y ∈ P, (˜x ◦ ˜x)(y) = y; (ii)... |

2 |
The reflection structures of generalized co-Minkowski spaces leading to K-loops
- Gabrieli, Karzel
- 1997
(Show Context)
Citation Context ...n as “left-sided quasigroups”, following a convention of Robinson [21]. Reflection quasigroups are also equivalent to the recently studied “point reflection structures” of Gabrieli and Karzel [3] [4] =-=[5]-=-. Given a nonempty set P and a mapping ˜ : P → Sym(P) : x ↦−→ ˜x, Gabrieli and Karzel call the pair (P,˜) a “point-reflection structure” if the following hold: (i) ∀x,y ∈ P, (˜x ◦ ˜x)(y) = y; (ii) ∀x,... |

2 |
Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit
- Karzel
- 1968
(Show Context)
Citation Context ...dubbed “gyrogroups” [29] and later “gyrocommutative gyrogroups” [31]. Bruck loops are also equivalent to “K-loops”, which are the additive loops of near-domains. Neardomains were introduced by Karzel =-=[8]-=-, and the additive loop structure was later axiomatized and named in unpublished work of Kerby and Wefelscheid (the first appearance of the term “K-loop” in the literature was in a paper of Ungar [27]... |

2 |
Riemannian Geometry”, 2nd Ed
- Klingenberg
- 1995
(Show Context)
Citation Context ...desic such that x 1/2 ∗ e = s x 1/2(e) = x. However, in compact symmetric spaces, the distinguished point e (or any point for that matter) has a nonempty cut locus, i.e., set of points conjugate to e =-=[13]-=-. For each such conjugate point x, there is a nontrivial family of distinct geodesics connecting e and x, and thus there is a nontrivial family of distinct points z such that z ∗ e = x. Since there is... |

2 |
Inner mappings of Bol loops
- Kreuzer
- 1998
(Show Context)
Citation Context ... is a permutation, and is thus equivalent to what we call a B-loop. Contemporary usage of the term “Bruck loop” in the literature (with some exceptions) tends to be as we have given it here, cf. [14] =-=[15]-=-. This usage seems to stem from a remark of Glauberman ([6], p.376). Bruck loops are equivalent to the class of loops Ungar dubbed “gyrogroups” [29] and later “gyrocommutative gyrogroups” [31]. Bruck ... |

2 | Smooth loops, generalized coherent states and geometric phases
- Nesterov, Sabinin
- 1997
(Show Context)
Citation Context ...lex conjugate of x, and we are using the usual conventions of complex arithmetic: 1/0 = ∞ and 1/∞ = 0. For x,y ̸= ∞, (3.14) agrees with the local geodesic operation on Ĉ found by Nesterov and Sabinin =-=[18]-=-. For the most part, the theory of the magma ( Ĉ, ⊙) is just a special case of the theory of the magma (S, ⊙). Those features special to ( Ĉ, ⊙) will be explored elsewhere. 4. Algebraic Structure It i... |

2 |
A loop-theoretic study of right-sided quasigroups
- Robinson
- 1979
(Show Context)
Citation Context ... keyesian, left distributive quasigroup. The term “reflection quasigroup” is due to Kikkawa [10]. Reflection quasigroups are also known as “left-sided quasigroups”, following a convention of Robinson =-=[21]-=-. Reflection quasigroups are also equivalent to the recently studied “point reflection structures” of Gabrieli and Karzel [3] [4] [5]. Given a nonempty set P and a mapping ˜ : P → Sym(P) : x ↦−→ ˜x, G... |