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Abraham A Ungar - One of the best experts on this subject based on the ideXlab platform.
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the intrinsic beauty harmony and interdisciplinarity in einstein velocity addition law Gyrogroups and gyrovector spaces
Mathematics Interdisciplinary Research, 2016Co-Authors: Abraham A UngarAbstract:The only justification for the Einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in Einstein addition remained for a long time a mystery to be conquered. Accordingly, the aim of this expository article is to present (i) the Einstein relativistic vector addition, (ii) the resulting Einstein scalar multiplication, (iii) the Einstein relativistic mass, and (iv) the Einstein relativistic kinetic energy, along with remarkable analogies with classical results in groups and vector spaces that these Einstein concepts capture in Gyrogroups and gyrovector spaces. Making the unfamiliar familiar, these analogies uncover the intrinsic beauty and harmony in the underlying Einstein velocity addition law of relativistically admissible velocities, as well as its interdisciplinarity.
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bi Gyrogroup the group like structure induced by bi decomposition of groups
Mathematics Interdisciplinary Research, 2016Co-Authors: Teerapong Suksumran, Abraham A UngarAbstract:The decomposition $Gamma=BH$ of a group $Gamma$ into a subset B and a subgroup $H$ of $Gamma$ induces, under general conditions, a group-like structure for B, known as a Gyrogroup. The famous concrete realization of a Gyrogroup, which motivated the emergence of Gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary mbox{operation} given by the Einstein velocity addition law of special relativity theory. The latter leads to the Lorentz transformation group $so{1,n}$, $ninN$, in pseudo-Euclidean spaces of signature $(1, n)$. The study in this article is motivated by generalized Lorentz groups $so{m, n}$, $m, ninN$, in pseudo-Euclidean spaces of signature $(m, n)$. Accordingly, this article explores the bi-decomposition $Gamma = H_LBH_R$ of a group $Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of $Gamma$, along with the novel bi-Gyrogroup structure of $B$ induced by the bi-decomposition of $Gamma$. As an example, we show by methods of Clifford mbox{algebras} that the quotient group of the spin group $spin{m, n}$ possesses the bi-decomposition structure.
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parametric realization of the lorentz transformation group in pseudo euclidean spaces
arXiv: Mathematical Physics, 2015Co-Authors: Abraham A UngarAbstract:The Lorentz transformation group $SO(m,n)$ is a group of Lorentz transformations of order $(m,n)$, that is, a group of special linear transformations in a pseudo-Euclidean space of signature $(m,n)$ that leave the pseudo-Euclidean inner product invariant. A parametrization of $SO(m,n)$ is presented, giving rise to the composition law of Lorentz transformations of order $(m,n)$ in terms of parameter composition. The parameter composition, in turn, gives rise to a novel group-like structure called a bi-Gyrogroup. Bi-Gyrogroups form a natural generalization of Gyrogroups where the latter form a natural generalization of groups. Like the abstract Gyrogroup, the abstract bi-Gyrogroup can play a universal computational role which extends far beyond the domain of pseudo-Euclidean spaces.
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parametric realization of the lorentz transformation group in pseudo euclidean spaces
Journal of Geometry and Symmetry in Physics, 2015Co-Authors: Abraham A UngarAbstract:Appears in: Journal of Geometry and Symmetry in Physics 38(2015), 39-108. Abstract. The Lorentz transformation group SO(m,n), m,n ∈ N, is a group of Lorentz transformations of order (m,n), that is, a group of special linear trans- formations in a pseudo-Euclidean space R m,n of signature (m,n) that leave the pseudo-Euclidean inner product invariant. A parametrization of SO(m,n) is pre- sented, givingriseto thecompositionlaw ofLorentztransformationsoforder(m,n) in terms of parameter composition. The parameter composition, in turn, gives rise to a novel group-like structure that R m,n possesses, called a bi-Gyrogroup. Bi-Gyrogroups form a natural generalization of Gyrogroups where the latter form a natural generalization of groups. Like the abstract Gyrogroup, the abstract bi- Gyrogroup can play a universal computational role which extends far beyond the domain of pseudo-Euclidean spaces.
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Gyrogroups and the decomposition of groups into twisted subgroups and subgroups
2015Co-Authors: Tuval Foguel, Abraham A UngarAbstract:Gyrogroups are generalized groups modelled on the Ein-stein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Ein-stein’s Gyrogroup fails to form a group since it is nonassocia-tive. 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 vec-tor space. The existence of striking analogies shared by gy
Teerapong Suksumran - One of the best experts on this subject based on the ideXlab platform.
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complete reducibility of Gyrogroup representations
Communications in Algebra, 2020Co-Authors: Teerapong SuksumranAbstract:AbstractIn this article, we show that any finite Gyrogroup can be represented on a space of complex-valued functions. In particular, we prove that any linear representation of a finite Gyrogroup on...
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the isometry group of n dimensional einstein Gyrogroup
arXiv: Metric Geometry, 2020Co-Authors: Teerapong SuksumranAbstract:The space of n-dimensional relativistic velocities normalized to c = 1, $$\displaystyle \mathbb {B} = \{\mathbf {v}\in \mathbb {R}^n\colon \|\mathbf {v}\| < 1\}, $$ is naturally associated with Einstein velocity addition ⊕E, which induces the rapidity metric dE on \(\mathbb {B}\) given by \(d_E(\mathbf {u}, \mathbf {v}) = \tanh ^{-1}\|-\mathbf {u}\oplus _E\mathbf {v}\|\). This metric is also known as the Cayley–Klein metric. We give a complete description of the isometry group of \((\mathbb {B}, d_E)\), along with its composition law.
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left regular representation of Gyrogroups
Mathematics, 2019Co-Authors: Teerapong SuksumranAbstract:In this article, we examine a subspace L gyr ( G ) of the complex vector space, L ( G ) = { f : f is a function from G to C } , where G is a nonassociative group-like structure called a Gyrogroup. The space L gyr ( G ) arises as a representation space for G associated with the left regular representation, consisting of complex-valued functions invariant under certain permutations of G. In the case when G is finite, we prove that dim ( L gyr ( G ) ) = 1 | γ ( G ) | ∑ ρ ∈ γ ( G ) | Fix ( ρ ) | , where γ ( G ) is the subgroup of Sym ( G ) generated by a class of permutations of G and Fix ( ρ ) = { a ∈ G : ρ ( a ) = a } .
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extension of maschke s theorem
Communications in Algebra, 2019Co-Authors: Teerapong SuksumranAbstract:In the present article, we examine linear representations of finite Gyrogroups, following their group-counterparts. In particular, we prove Maschke’s theorem for Gyrogroups, along with its converse...
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extension of maschke s theorem
arXiv: Representation Theory, 2018Co-Authors: Teerapong SuksumranAbstract:In the present article, we examine linear representations of finite Gyrogroups, following their group-counterparts. In particular, we prove the celebrated theorem of Maschke for Gyrogroups, along with its converse. This suggests studying the left regular action of a Gyrogroup $(G, \oplus)$ on the function space $$ L^{\mathrm{gyr}}(G) = \{f\in L(G)\colon \forall a, x, y, z\in G, f(a\oplus\mathrm{gyr}[x, y]z) = f(a\oplus z)\} $$ in a natural way, where $L(G)$ is the space of all functions from $G$ into a field.
Ferreira Milton - One of the best experts on this subject based on the ideXlab platform.
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Orthogonal gyrodecompositions of real inner product Gyrogroups
'MDPI AG', 2020Co-Authors: Ferreira Milton, Suksumran TeerapongAbstract:In this article, we prove an orthogonal decomposition theorem for real inner product Gyrogroups, which unify some well-known Gyrogroups in the literature: Einstein, Möbius, Proper Velocity, and Chen’s Gyrogroups. This leads to the study of left (right) coset partition of a real inner product Gyrogroup induced from a subGyrogroup that is a finite dimensional subspace. As a result, we obtain gyroprojectors onto the subGyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subGyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned Gyrogroups.publishe
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Harmonic analysis on the proper velocity Gyrogroup
'Duke University Press', 2017Co-Authors: Ferreira MiltonAbstract:In this paper we study harmonic analysis on the Proper Velocity (PV) Gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator, and its eigenfunctions, the generalized Poisson transform, and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.info:eu-repo/semantics/publishedVersio
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Harmonic Analysis on the Möbius Gyrogroup
'Springer Science and Business Media LLC', 2015Co-Authors: Ferreira MiltonAbstract:In this paper, we propose to develop harmonic analysis on the Poincaré ball B, a model of then-dimensional real hyperbolic space. The Poincaré ball B is the open ball of the Euclidean n-space $\bkR^n$ with radius t >0, centered at the origin of $\bkR^n$ and equipped with Möbius addition, thus forming a Möbius Gyrogroup where Möbius addition in the ball plays the role of vector addition in $\bkR^n.$ For any t>0 and an arbitrary parameter $\sigma \in \bkR$ we study the $(\sigma,t)$-translation, the $(\sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on B tends to the standard Euclidean harmonic analysis on $\bkR^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. As an application, we construct diffusive wavelets on B.info:eu-repo/semantics/publishedVersio
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Harmonic Analysis on the Einstein Gyrogroup
'Prof. Marin Drinov Publishing House of BAS (Bulgarian Academy of Sciences)', 2014Co-Authors: Ferreira MiltonAbstract:In this paper we study harmonic analysis on the Einstein Gyrogroup of the open ball of ${\mathbb R}^n, n \in \mathbb{N},$ centered at the origin and with arbitrary radius $t \in \mathbb{R}^+,$ associated to the generalised Laplace-Beltrami operator $$ L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right)$$where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$-convo\-lution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis.info:eu-repo/semantics/publishedVersio
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Möbius Gyrogroups: a Clifford algebra approach
'Elsevier BV', 2011Co-Authors: Ferreira Milton, Ren G.Abstract:Using the Clifford algebra formalism we study the Möbius Gyrogroup of the ball of radius t of the paravector space, 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 gyro-subgroup 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.info:eu-repo/semantics/publishedVersio
Lawson Jimmie - One of the best experts on this subject based on the ideXlab platform.
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Smooth Bruck loops, symmetric spaces, and nonassociative vector spaces
LSU Digital Commons, 2011Co-Authors: Kim Sejong, Lawson JimmieAbstract:Our purposes in this work include the following: (1) Extend and expand earlier work on symmetric spaces, particularly that done from a nonassociative algebra point of view, from the finite-dimensional setting to the Banach space setting. (2) Take a careful look at the equivalence of the categories of smooth pointed reflection quasigroups (a special class of symmetric spaces) and uniquely 2-divisible Bruck loops (= K-loops = gyrocommutative Gyrogroups). (3) Propose a loop-theoretic analog of topological vector spaces. (4) Derive algebraic consequences and equivalences of smoothness notions, particularly the notion of parallel transport. (5) Illustrate the effective interaction of the algebraic operations of reflection, Bruck addition, and coaddition in the test case of parallelograms in symmetric spaces
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Clifford algebras, Möbius transformations, Vahlen matrices, and B-loops
LSU Digital Commons, 2010Co-Authors: Lawson JimmieAbstract:In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or Gyrogroup on the open unit ball B in n-dimensional euclidean space ℝn. One notable achievement is a compact, convenient formula for the Möbius loop operation a * b = (a + b)(1 - ab)-1, where the operations on the right are those arising from the Clifford algebra (a formula comparable to (w+z)(1+wz)-1 for the Möbius loop multiplication in the unit complex disk)
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Clifford algebras, Möbius transformations, Vahlen matrices, and $B$-loops
'Walter de Gruyter GmbH', 2010Co-Authors: Lawson JimmieAbstract:summary:In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or Gyrogroup on the open unit ball $B$ in $n$-dimensional euclidean space $\mathbb R^n$. One notable achievement is a compact, convenient formula for the Möbius loop operation $a\ast b=(a+b)(1-ab)^{-1}$, where the operations on the right are those arising from the Clifford algebra (a formula comparable to $(w+z)(1+\overline wz)^{-1}$ for the Möbius loop multiplication in the unit complex disk)
Ungar, Abraham A. - One of the best experts on this subject based on the ideXlab platform.
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Bi-Gyrogroup: The group-like structure induced by bi-decomposition of groups
2016Co-Authors: Suksumran Teerapong, Ungar, Abraham A.Abstract:The decomposition $\Gamma=BH$ of a group $\Gamma$ into a subset $B$ and a subgroup $H$ of $\Gamma$ induces, under general conditions, a group-like structure for $B$, known as a Gyrogroup. The famous concrete realization of a Gyrogroup, which motivated the emergence of Gyrogroups into the mainstream, is the space of all relativistically admissible velocities along with a binary operation given by the Einstein velocity addition law of special relativity theory. The latter leads to the Lorentz transformation group $\mathrm{SO}(1,n)$, $n\in\mathbb{N}$, in pseudo-Euclidean spaces of signature $(1, n)$. The study in this article is motivated by generalized Lorentz groups $\mathrm{SO}(m, n)$, $m, n\in\mathbb{N}$, in pseudo-Euclidean spaces of signature $(m, n)$. Accordingly, this article explores the bi-decomposition $\Gamma = H_LBH_R$ of a group $\Gamma$ into a subset $B$ and subgroups $H_L$ and $H_R$ of $\Gamma$, along with the novel bi-Gyrogroup structure of $B$ induced by the bi-decomposition of $\Gamma$. As an example, we show by methods of Clifford algebras that the quotient group of the spin group $\mathrm{spin}(m, n)$ possesses the bi-decomposition structure.Comment: The published version of the article is accessible via http://mir.kashanu.ac.ir/article_13911_2153.htm
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On the Study of Hyperbolic Triangles and Circles by Hyperbolic Barycentric Coordinates in Relativistic Hyperbolic Geometry
2013Co-Authors: Ungar, Abraham A.Abstract:Barycentric coordinates are commonly used in Euclidean geometry. Following the adaptation of barycentric coordinates for use in hyperbolic geometry in recently published books on analytic hyperbolic geometry, known and novel results concerning triangles and circles in the hyperbolic geometry of Lobachevsky and Bolyai are discovered. Among the novel results are the hyperbolic counterparts of important theorems in Euclidean geometry. These are: (1) the Inscribed Gyroangle Theorem, (ii) the Gyrotangent-Gyrosecant Theorem, (iii) the Intersecting Gyrosecants Theorem, and (iv) the Intersecting Gyrochord Theorem. Here in gyrolanguage, the language of analytic hyperbolic geometry, we prefix a gyro to any term that describes a concept in Euclidean geometry and in associative algebra to mean the analogous concept in hyperbolic geometry and nonassociative algebra. Outstanding examples are {\it Gyrogroups} and {\it gyrovector spaces}, and Einstein addition being both {\it gyrocommutative} and {\it gyroassociative}. The prefix "gyro" stems from "gyration", which is the mathematical abstraction of the special relativistic effect known as "Thomas precession".Comment: 78 pages, 26 figure
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The hyperbolic triangle centroid
Charles University in Prague Faculty of Mathematics and Physics, 2004Co-Authors: Ungar, Abraham A.Abstract:summary:Some gyrocommutative Gyrogroups, also known as Bruck loops or K-loops, admit scalar multiplication, turning themselves into gyrovector spaces. The latter, in turn, form the setting for hyperbolic geometry just as vector spaces form the setting for Euclidean geometry. In classical mechanics the centroid of a triangle in velocity space is the velocity of the center of momentum of three massive objects with equal masses located at the triangle vertices. Employing gyrovector space techniques we find in this article that, in full analogy, the centroid of a hyperbolic triangle in relativity velocity space is the velocity of the center of momentum of three massive objects with equal rest masses located at the triangle vertices. Being guided by the relativistic mass correction of moving massive objects in special relativity theory, we express the hyperbolic triangle centroid in terms of the triangle vertices, resulting in a novel hyperbolic triangle centroid identity that captures remarkable analogies with its Euclidean counterpart