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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 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
ON CENTRAL EXTENSIONS OF GYROCOMMUTATIVE
"... Central extensions of gyrocommutative gyrogroups (Kloops) are studied in order to clarify the status of a cocycle equation introduced by Smith and Ungar. A sufficient and necessary conditions under which a central invariant extension is a gyrocommutative gyrogroup are formulated in terms of a 2coc ..."
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Central extensions of gyrocommutative gyrogroups (Kloops) are studied in order to clarify the status of a cocycle equation introduced by Smith and Ungar. A sufficient and necessary conditions under which a central invariant extension is a gyrocommutative gyrogroup are formulated in terms of a 2cochain f(x, y). In particular, it is shown that for central invariant extensions of gyrocommutative gyrogroups defined by Cartan decompositions of simple Lie algebras, the corresponding f(x, y) satisfies the cocycle equation, provided an extension is a gyrocommutative gyrogroup. 1. Introduction. There has been a renewal of an interest in loop theory in recent years, concerning a special nonassociative loop structure called a gyrocommutative gyrogroup, known also under the name of a Kloop. It began with a paper by A. Ungar [15], who pointed it out that the addition law of relativistic velocities
(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 ..."
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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.
and its Applications In Honor of Stephen Smale’s 80th Birthday 123 Editors
"... the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or ..."
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the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for