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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 10 (4 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 ..."
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Cited by 6 (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.
(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 ..."
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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.
(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.
The Structure Group of Certain KLoops
, 1995
"... this paper we'll compute the structure group of a class of Kloops described in [1], which includes the classical example of the relativistic velocity addition (cf. [6]). 1. Preliminaries We review the construction of the Kloops under consideration. Proofs can be found in [1]. Let R be an orde ..."
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this paper we'll compute the structure group of a class of Kloops described in [1], which includes the classical example of the relativistic velocity addition (cf. [6]). 1. Preliminaries We review the construction of the Kloops under consideration. Proofs can be found in [1]. Let R be an ordered field, let K := R(i), where i
Left Distributive Quasigroups and Gyrogroups
"... Abstract. The connection between gyrogroups and some types of left distributive quasigroups is established by means of isotopy considerations. Any quasigroup of reflection is isotopic to some gyrocommutative gyrogroup and any left distributive quasigroup satisfying some specific condition is isot ..."
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Abstract. The connection between gyrogroups and some types of left distributive quasigroups is established by means of isotopy considerations. Any quasigroup of reflection is isotopic to some gyrocommutative gyrogroup and any left distributive quasigroup satisfying some specific condition is isotopic to some nongyrocommutative gyrogroup. The geometry of reductive homogeneous spaces and the semidirect product for homogeneous loops are used to produce local exact decompositions of groups and local gyrocommutative gyrogroups. 1.