Reflections on S and Quaternionic Möbius Transformations (original) (raw)

Reversible maps in the group of quaternionic Möbius transformations

Mathematical Proceedings of the Cambridge Philosophical Society, 2007

The reversible elements of a group are those elements that are conjugate to their own inverse. A reversible element is said to be reversible by an involution if it is conjugate to its own inverse by an involution. In this paper, we classify the reversible elements and the elements reversible by involutions in the group of quaternionic Möbius transformations.

Isometric actions of quaternionic symplectic groups

arXiv: Differential Geometry, 2018

Denote by Sp(k,l)Sp(k,l)Sp(k,l) the quaternionic symplectic group of signature (k,l)(k,l)(k,l). We study the deformation rigidity of the embedding Sp(k,l)timesSp(1)hookrightarrowHSp(k,l) \times Sp(1) \hookrightarrow HSp(k,l)timesSp(1)hookrightarrowH, where HHH is either Sp(k+1,l)Sp(k+1,l)Sp(k+1,l) or Sp(k,l+1)Sp(k,l+1)Sp(k,l+1), this is done by studying a natural non-associative algebra mathfrakm\mathfrak{m}mathfrakm comming from the affine structure of Sp(1)backslashHSp(1) \backslash HSp(1)backslashH. We compute the automorphism group of mathfrakm\mathfrak{m}mathfrakm and as a consecuence of this, we are able to compute the isometry group of Sp(1)backslashHSp(1) \backslash HSp(1)backslashH at least up to connected components. Using these results, we obtain a uniqueness result on the structure of Sp(1)backslashHSp(1) \backslash HSp(1)backslashH together with an isometric left Sp(k,l)Sp(k,l)Sp(k,l)-action and classify its finite volume quotients up to finite coverings. Finally, we classify arbitrary isometric actions of Sp(k,l)Sp(k,l)Sp(k,l) into connected, complete, analytic, pseudo-Riemannian manifolds admitting a dense orbit of dimension bounded by textrmdim(Sp(1)backslashH)\textrm{dim}(Sp(1) \backslash H)textrmdim(Sp(1)backslashH).

Transvections as generators of the special linear group over the quaternions

Journal of Pure and Applied Algebra, 1989

Let V be a finite-dimensional right vector space over the quaternions IH. Each transformation M in the special linear group of V is a product of transvections T,, i.e. M= Tt . . Ti. The smallest t is called the length of M, t = I(M). We show that I(M) = dim B(M) if dim B(M) 2 3, Mf 3Jt(l) where /I E @\ IR, and MP is neither simple nor the identity for any fl E I?. In any case I(M) = dimB(M)+i, where Osis3.

A characterization of the quaternion group

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2013

The goal of this note is to give an elementary characterization of the well-known quaternion group Q8 by using its subgroup lattice.

Moebius transformations and the Poincare distance in the quaternionic setting

Indiana University Mathematics Journal, 2009

In the space H of quaternions, we investigate the natural, invariant geometry of the open, unit disc ∆ H and of the open half-space H + . These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of Möbius transformations of ∆ H and H + and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the Möbius transformations, and use it to define the analogous of the Poincaré distances on ∆ H and H + . We easily deduce that there exists no isometry between the quaternionic Poincaré distance of ∆ H and the Kobayashi distance inherited by ∆ H as a domain of C 2 , in accordance with a direct consequence of the classification of the non compact, rank 1, symmetric spaces.

The Quaternions and the Spaces S 3, SU(2), SO(3), and ℝ ℙ3

Texts in Applied Mathematics, 2011

One of the main goals of these notes is to explain how rotations in R n are induced by the action of a certain group, Spin(n), on R n , in a way that generalizes the action of the unit complex numbers, U(1), on R 2 , and the action of the unit quaternions, SU(2), on R 3

Regular vs. Classical Möbius Transformations of the Quaternionic Unit Ball

Advances in Hypercomplex Analysis, 2013

The regular fractional transformations of the extended quaternionic space have been recently introduced as variants of the classical linear fractional transformations. These variants have the advantage of being included in the class of slice regular functions, introduced by Gentili and Struppa in 2006, so that they can be studied with the useful tools available in this theory. We first consider their general properties, then focus on the regular Möbius transformations of the quaternionic unit ball B, comparing the latter with their classical analogs. In particular we study the relation between the regular Möbius transformations and the Poincaré metric of B, which is preserved by the classical Möbius transformations. Furthermore, we announce a result that is a quaternionic analog of the Schwarz-Pick lemma.

Some Properties of Representation of Quaternion Group

KnE Engineering

The quaternions are a number system in the form + + +. The quaternions ±1, ± , ± , ± form a non-abelian group of order eight called quaternion group. Quaternion group can be represented as a subgroup of the general linear group 2 (C). In this paper, we discuss some group properties of representation of quaternion group related to Hamiltonian group, solvable group, nilpotent group, and metacyclic group.

Spectral correspondences for Maass waveforms on quaternion groups

Journal of Number Theory, 2016

We prove that in most cases the Jacquet-Langlands correspondence between newforms for Hecke congruence groups and newforms for quaternion orders is a bijection. Our proof covers almost all cases where the Hecke congruence group is of cocompact type, i.e. when a bijection is possible. The proof uses the Selberg trace formula.