Factorizations of Möbius Gyrogroups

Ferreira, Milton

doi:10.1007/s00006-009-0154-7

Cited by 31 publications

(36 citation statements)

References 20 publications

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“…Henceforth, we make the identification v $ v C I and regard V as a subspace of C`.V; Q/. By compatibility (14), one has the following fundamental relations in C`.V; Q/:…”

Section: Theorem 5 ([31]mentioning

confidence: 99%

“…The following definition of a subgyrogroup first appeared in [14,Sect. 4] with the term "gyro-subgroup":…”

Section: Subgyrogroupsmentioning

confidence: 99%

“…Einstein and Möbius gyrogroups play a major role in gyrogroup theory as they provide concrete models for the abstract theory. See, for instance, [1,14,18,38,58,62,[69][70][71]73].…”

Section: Introductionmentioning

confidence: 99%

“…(43). A Clifford algebra approach to study Möbius and Einstein gyrogroups is very fruitful [14,16,18,19,44,59,62].…”

Section: Introductionmentioning

confidence: 99%

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The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Suksumran

2016

Essays in Mathematics and Its Applications

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Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible gyrocommutative gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Möbius addition. In the second part of this paper, we show that the symmetric group of a gyrogroup admits the gyrogroup structure, thus obtaining an analog of Cayley's theorem for gyrogroups. We examine subgyrogroups, gyrogroup homomorphisms, normal subgyrogroups, and quotient gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange's theorem for gyrogroups and use this result to prove that gyrogroups of particular order have the Cauchy property.

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“…Henceforth, we make the identification v $ v C I and regard V as a subspace of C`.V; Q/. By compatibility (14), one has the following fundamental relations in C`.V; Q/:…”

Section: Theorem 5 ([31]mentioning

confidence: 99%

“…The following definition of a subgyrogroup first appeared in [14,Sect. 4] with the term "gyro-subgroup":…”

Section: Subgyrogroupsmentioning

confidence: 99%

“…Einstein and Möbius gyrogroups play a major role in gyrogroup theory as they provide concrete models for the abstract theory. See, for instance, [1,14,18,38,58,62,[69][70][71]73].…”

Section: Introductionmentioning

confidence: 99%

“…(43). A Clifford algebra approach to study Möbius and Einstein gyrogroups is very fruitful [14,16,18,19,44,59,62].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Suksumran

2016

Essays in Mathematics and Its Applications

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“…Thus, any non-zero paravector is invertible and the inverse is given by x −1 = x (x) . To keep the same notations as in [6] we shall also denote x 2 := (x), x ∈ W . The extension of the geometric product (6) to the paravector case is given by…”

Section: Clifford Algebrasmentioning

confidence: 99%

Möbius gyrogroups: A Clifford algebra approach

Ferreira

Ren

2011

Journal of Algebra

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Using the Clifford algebra formalism we study the Möbius gyrogroup of the ball of radius t of the paravector space R ⊕ V , where V is a finite-dimensional real vector space. We characterize all the gyro-subgroups of the Möbius gyrogroup and we construct left and right factorizations with respect to an arbitrary gyrosubgroup for the paravector ball. The geometric and algebraic properties of the equivalence classes are investigated. We show that the equivalence classes locate in a k-dimensional sphere, where k is the dimension of the gyro-subgroup, and the resulting quotient spaces are again Möbius gyrogroups. With the algebraic structure of the factorizations we study the sections of Möbius fiber bundles inherited by the Möbius projectors.

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An Introduction to Hyperbolic Barycentric Coordinates and their Applications

Ungar

2014

Mathematics Without Boundaries

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Barycentric coordinates are commonly used in Euclidean geometry. The adaptation of barycentric coordinates for use in hyperbolic geometry gives rise to hyperbolic barycentric coordinates, known as gyrobarycentric coordinates. The aim of this article is to present the path from Einstein's velocity addition law of relativistically admissible velocities to hyperbolic barycentric coordinates, along with applications.

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Factorizations of Möbius Gyrogroups

Cited by 31 publications

References 20 publications

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Möbius gyrogroups: A Clifford algebra approach

An Introduction to Hyperbolic Barycentric Coordinates and their Applications

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