Möbius gyrogroups: A Clifford algebra approach

AbstractUsing 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 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

Received: 6 May 2020; Accepted: 30 May 2020; Published: date

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, M\"{o}bius, 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.info:eu-repo/semantics/publishedVersio

Factorizations of Möbius gyrogroups

In this paper, we consider a Möbius gyrogroup on a real Hilbert space (of finite or infinite dimension) and we obtain its factorization by gyrosubgroups and subgroups. It is shown that there is a duality relation between the quotient spaces and the orbits obtained. As an example, we present the factorization of the Möbius gyrogroup of the unit ball in R^n linked to the proper Lorentz group Spin+(1, n).info:eu-repo/semantics/publishedVersio

Gyroharmonic analysis on relativistic gyrogroups

Einstein, Möbius, and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time \bkR^{n,1}. Using the gyrolanguage we study their gyroharmonic analysis. Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them. Our study focus on the translation and convolution operators, eigenfunctions of the Laplace-Beltrami operator, Poisson transform, Fourier-Helgason transform, its inverse, and Plancherel's Theorem. We show that in the limit of large $t,$ $t \rightarrow +\infty,$ the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis

Harmonic analysis on the Möbius gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ 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 $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$

Möbius gyrovector spaces and functional analysis (Research on preserver problems on Banach algebras and related topics)

This is a survey and résumés of previously published articles, including some announcement of new results. We discuss some aspects of the Einstein and Möbius gyrovector spaces from a viewpoint of elementary functional analysis