Results 1  10
of
15
Loops and semidirect products
, 2000
"... A left loop (B, ·) is a set B together with a binary operation · such that (i) for each a ∈ B, the left translation mapping La: B → B defined by La(x) = a · x is a bijection, and (ii) there exists a twosided identity 1 ∈ B satisfying 1 · x = x · 1 = x for every x ∈ B. A right loop is similarly def ..."
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Cited by 7 (4 self)
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A left loop (B, ·) is a set B together with a binary operation · such that (i) for each a ∈ B, the left translation mapping La: B → B defined by La(x) = a · x is a bijection, and (ii) there exists a twosided identity 1 ∈ B satisfying 1 · x = x · 1 = x for every x ∈ B. A right loop is similarly defined, and a loop is both a right loop
Involutory decomposition of groups into twisted subgroups and subgroups
 J. Group Theory
, 2000
"... Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addit ..."
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Cited by 7 (2 self)
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Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups
From Pythagoras To Einstein: The Hyperbolic Pythagorean Theorem
, 1998
"... A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as the "Einstein sum" of the squares of ..."
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Cited by 5 (5 self)
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A new form of the Hyperbolic Pythagorean Theorem, which has a striking intuitive appeal and offers a strong contrast to its standard form, is presented. It expresses the square of the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as the "Einstein sum" of the squares of the hyperbolic lengths of the other two sides, Fig. 1, thus completing the long path from Pythagoras to Einstein.
Involutory Decomposition of Groups Into Twisted Subgroups and Subgroups
"... . An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are gro ..."
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Cited by 5 (3 self)
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. An involutory decomposition is a decomposition, due to an involution, of a group into a twisted subgroup and a subgroup. We study unexpected links between twisted subgroups and gyrogroups. Twisted subgroups arise in the study of problems in computational complexity. In contrast, gyrogroups are grouplike structures which first arose in the study of Einstein's velocity addition in the special theory of relativity. Particularly, we show that every gyrogroup is a twisted subgroup and that, under general specified conditions, twisted subgroups are gyrocommutative gyrogroups. Moreover, we show that gyrogroups abound in group theory and that they possess rich structure. x1. Introduction Under general conditions, twisted subgroups are near subgroups [1]. Feder and Vardi [4] introduced the concept of a near subgroup of a finite group as a tool to study problems in computational complexity involving the class NP . Aschbacher provided a conceptual base for studying near subgroups demonstrating...
On twisted subgroups and Bol loops of odd order
 Rocky Mountain J. of Math
"... Abstract. In the spirit of Glauberman’s fundamental work in Bloops and Moufang loops [17] [18], we prove Cauchy and strong Lagrange theorems for Bol loops of odd order. We also establish necessary conditions for the existence of a simple Bol loop of odd order, conditions which should be useful in t ..."
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Cited by 3 (2 self)
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Abstract. In the spirit of Glauberman’s fundamental work in Bloops and Moufang loops [17] [18], we prove Cauchy and strong Lagrange theorems for Bol loops of odd order. We also establish necessary conditions for the existence of a simple Bol loop of odd order, conditions which should be useful in the development of a FeitThompson theorem for Bol loops. Bol loops are closely related to Aschbacher’s twisted subgroups [1], and we survey the latter in some detail, especially with regard to the socalled Aschbacher radical. 1.
Hyperbolic Trigonometry and its Application in the Poincaré Ball Model of Hyperbolic Geometry
 in Proceedings of the Fourth European SGI/Cray MPP Workshop (Sep 1011
, 2001
"... Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincard ball model of **dimensional hyperbolic geometry, as well as their application. The Poincard bal ..."
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Cited by 2 (1 self)
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Hyperbolic trigonometry is developed and illustrated in this article along lines parallel to Euclidean trigonometry by exposing the hyperbolic trigonometric law of cosines and of sines in the Poincard ball model of **dimensional hyperbolic geometry, as well as their application. The Poincard ball model of 3dimensional hyperbolic geometry is becoming increasingly important in the construction of hyperbolic browsers in computer graphics. These allow in computer graphics the exploitation of hyperbolic geometry in the development of visualization techniques. It is therefore clear that hyperbolic trigonometry in the Poincard ball model of hyperbolic geometry, as presented here, will prove useful in the development of efficient hyperbolic browsers in computer graphics.
The Relativistic CompositeVelocity Reciprocity Principle
, 2000
"... Gyrogroup theory [A.A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), pp. 881951] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector add ..."
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Cited by 1 (1 self)
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Gyrogroup theory [A.A. Ungar, Thomas precession: its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics, Found. Phys. 27 (1997), pp. 881951] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the Relativistic CompositeVelocity Reciprocity Principle. The breakdown of commutativity in the Einstein velocity addition # of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic compositevelocity reciprocity principle, since seemingly (i) on one hand the velocity reciprocal to the composite velocity u#v is (u#v) and (ii) on the other hand it is (v)#(u). But, (iii) (u#v) #= (v)#(u). We remove the confusion in (i), (ii) and (iii) by employing the gyrocommutative gyrogroup structure of Einstein's addition and, subsequ...
From the Group SL(2, C) to Gyrogroups and Gyrovector Spaces and Huperbolic Geometry
"... this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and gyrovector spaces. This natural way convincingly demonstrates that the theory of gyrogroups and gyrovector spaces provides a most powerful formalism for dealing with the Lorentz group and hyper ..."
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this paper is to present a natural way in which the algebra of the SL(2; C) group leads to gyrogroups and gyrovector spaces. This natural way convincingly demonstrates that the theory of gyrogroups and gyrovector spaces provides a most powerful formalism for dealing with the Lorentz group and hyperbolic geometry, the geometry that governs the special theory of relativity as well as other areas of physics (see, for instance, [9] and [10]). It is therefore hoped that, following this article, gyrogroup and gyrovector space theoretic techniques will provide standard tools in the study of relativity physics and, as such, will become part of the lore learned by all explorers who are interested in relativity physics. Links between gyrogroups and other mathematical objects are presented in [11] [12] [13] and [14]. Furthermore, our approach to gyrogroups and scalar multiplication in a gyrogroup of gyrovectors can be used as a preparation for the study of a related, but more abstract study of Sabinin's odules in [15]. A related study of quasigroups in differential geometry is presented by Sabinin and Miheev on pp. 357  430 of [16]. 6 2 THE ALGEBRA OF THE SL(2; C) GROUP Let R 3 c be the set of all relativistically admissible velocities, R 3 c = fv 2 R 3 : kvk < cg It is the ball of radius c, c > 0, of the Euclidean 3space R 3 , c being the vacuum speed of light. A boost L(v) is a pure Lorentz transformation, that is, a Lorentz transformation without rotation, parametrized by a velocity parameter v 2 R 3 c . The boost L(v) is a linear transformation of spacetime coordinates which has the matrix representation Lm (v), v = (v 1 ; v 2 ; v 3 ) t , Lm (v) = 0 B B B B B B @ v c 2 v v 1 c 2 v v 2 c 2 v v 3 v v 1 1 + c 2 2 v v +1 v 2 1 c ...
(1.1)
, 2000
"... Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space ..."
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Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided 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.
(1.1)
, 1999
"... Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space ..."
Abstract
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Abstract. On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S, ⊙) is a powerassociative Kikkawa left loop with twosided 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.