## (1.1) (2000)

### BibTeX

@MISC{Kinyon00(1.1),

author = {Michael K. Kinyon},

title = {(1.1)},

year = {2000}

}

### 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 a nonremovable 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

1212 |
Differential geometry, Lie groups and symmetric spaces, Pure and
- Helgason
- 1978
(Show Context)
Citation Context ...This paper is in final form and no version of it will be submitted for publication elsewhere. 12 MICHAEL K. KINYON 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... |

48 |
Symmetric Spaces I
- 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 through the origin, 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, thinkin... |

41 |
Quasigroups and Loops
- Pflugfelder
- 1990
(Show Context)
Citation Context ... \ agrees 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... |

32 |
On loops of odd order
- Glauberman
- 1968
(Show Context)
Citation Context ...HERES 5 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... |

23 |
Theory of K-loops
- Kiechle
- 2002
(Show Context)
Citation Context ...omorphism. 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 ... |

17 |
precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic
- Ungar, Thomas
- 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. Near-domains were introduced by Karzel [8], and the additive loop structure was later axiomatized and name... |

14 |
Geometry of homogeneous Lie loops
- Kikkawa
- 1975
(Show Context)
Citation Context ...ls only on the set S ∩ V of “measure zero”. 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... |

14 |
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.6 MICHA... |

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. Near-domains were introduced by Karzel [8], and the additive l... |

8 | Loops and semidirect products
- Kinyon, Jones
(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... |

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)... |

7 |
On some quasigroups of algebraic models of symmetric spaces
- Kikkawa
- 1974
(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... |

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 |
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 ... |

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 |
Methods of non-associative algebra in differential geometry
- 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... |

6 |
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.10 MICHAEL K. KINYON Finally, recall that the 2-sphere S2 may be identified with the Riemann sphere Ĉ = C ∪ {∞} by stereo... |

5 |
A construction of Bruck loops
- Kepka
- 1984
(Show Context)
Citation Context ...alent to what we call a B-loop.GLOBAL LEFT LOOP STRUCTURES ON SPHERES 5 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 Glauberman [6] to describe a finite, odd order Bol loop with the ... |

3 |
Point-reflection geometries, geometric K-loops and unitary geometries
- Gabriele, 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; (i... |

2 |
A Survey of Binary Systems” (3rd printing
- Bruck
- 1971
(Show Context)
Citation Context ...sion \ agrees 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 |
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) ∀... |

2 |
Zusammenhänge zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit
- Karzel
- 1968
(Show Context)
Citation Context ...ubbed “gyrogroups” [29] and later “gyrocommutative gyrogroups” [31]. Bruck loops are also equivalent to “K-loops”, which are the additive loops of near-domains. Near-domains 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 ... to what we call a B-loop.GLOBAL LEFT LOOP STRUCTURES ON SPHERES 5 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 ...ex conjugate of x, and we are using the usual conventions of complex arithmetic: 1/0 = ∞ and 1/∞ = 0. For x, y ̸= ∞, (3.12) agrees with the local geodesic operation on Ĉ found by Nesterov and Sabinin =-=[18]-=-. For the most part, results about ( Ĉ, ⊙) are just special cases of results about (S, ⊙). 4. Algebraic Structure It is clear from (3.8) that the left translation mappings Lx : S → S defined by Lxy = ... |

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... |