Einstein Gyrogroup as a B-loop

Suksumran, Teerapong; Wiboonton, Keng

doi:10.1016/s0034-4877(15)30019-7

Cited by 8 publications

(8 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we extend the results of Suksumran and Wiboonton [62] to an arbitrary real inner product space. Let V be an n-dimensional real inner product space with a positive definite inner product h ; i and the corresponding norm k k.…”

Section: Möbius and Einstein Gyrogroupsmentioning

confidence: 73%

“…This expository paper grew from our three research papers [60][61][62]. Let c be a positive constant representing the speed of light in vacuum, and let R where u is the Lorentz factor or gamma factor given by u D 1 q 1…”

Section: Introductionmentioning

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 3 more Smart Citations

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

Suksumran

2016

Essays in Mathematics and Its Applications

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Möbius and Einstein Gyrogroupsmentioning

confidence: 73%

Section: Introductionmentioning

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…It became known as an Einstein gyrogroup. Einstein gyrogroups are employed, for instance, in References [5,[16][17][18][19][20][21]. Möbius gyrogroups are employed as well, for instance, in References [4,[22][23][24].…”

mentioning

confidence: 99%

Binary Operations in the Unit Ball: A Differential Geometry Approach

Barabanov¹,

Ungar²

2020

Symmetry

View full text Add to dashboard Cite

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.

show abstract