DOCSLIB.ORG

Quaternionic Kleinian Modular Groups and Arithmetic Hyperbolic Orbifolds Over the Quaternions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quaternionic Kleinian Modular Groups and Arithmetic Hyperbolic Orbifolds Over the Quaternions Università degli Studi di Trieste ar Ts Archivio della ricerca – postprint Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions 1 1 Juan Pablo Díaz · Alberto Verjovsky · 2 Fabio Vlacci Abstract Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quater- nion division algebra H, we define several Kleinian discrete subgroups of PSL(2, H).We define first a Kleinian subgroup PSL(2, L) of PSL(2, H(Z)). This group is a general- ization of the modular group PSL(2, Z). Next we define a discrete subgroup PSL(2, H) of PSL(2, H) which is obtained by using Hurwitz integers. It contains as a subgroup PSL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and 1 1 describe the orb-ifold quotients HH/PSL(2, L) and HH/PSL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space. Keywords Modular groups · Arithmetic hyperbolic 4-manifolds · 4-Orbifolds · Quaternionic hyperbolic geometry Mathematics Subject Classification (2000) Primary 20H10 · 57S30 · 11F06; Secondary 30G35 · 30F45 Dedicated to Bill Goldman on occasion of his 60th birthday. B Alberto Verjovsky [email protected] Juan Pablo Díaz [email protected] Fabio Vlacci [email protected]fi.it 1 Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Av. Universidad s/n. Col. Lomas de Chamilpa, Código Postal 62210 Cuernavaca, Morelos, Mexico 2 Dipartimento di Matematica e Informatica DiMaI “U. Dini”, Università di Firenze, Viale Morgagni 67/A, 50134 Florence, Italy 1 1 Introduction Since the time of Carl Friedrich Gauss and the foundational papers by Richard Dedekind and Felix Klein the classical modular group PSL(2, Z) and its action on the hyperbolic (complex) 1 upper half plane HC ={z ∈ C :(z)>0} have played a central role in different branches of mathematics and physics like number theory, Riemann surfaces, elliptic curves, hyperbolic geometry, theory of modular forms and automorphic forms, crystallography, string theory and others. Similarly, discrete subgroups of PSL(2, C) are very important in the construction of lattices to study of arithmetic hyperbolic 3-orbifolds (see, for instance, [20]) and many other fields of mathematics. We consider the action defined by isometries of the modular group PSL(2, Z) on the 1 hyperbolic plane HC. A fundamental domain is a triangle with one ideal point an two other vertices were the sides have an angle of π/3. This is the triangle with Coxeter notation (3, 3, ∞). See the Fig. 1. The modular group PSL(2, Z) is a subgroup of the group of 1 symmetries of the regular tessellation of HC whose tiles are congruent copies of the triangle (3, 3, ∞). We can describe the Cayley graph and a presentation of the group PSL(2, Z) in terms of 2 generators and 2 relations to obtain: PSL(2, Z) =a, b|a2 = (ab)3 = 1=Z/2Z ∗ Z/3Z. 2 1 2 The quotient O := HC/PSL(2, Z) has underlying topological space the plane R (or C)andO2 consists of two distinguished conical points. The local groups of the singular points are Z/2Z and Z/3Z modeled on groups of two and three elements which consist of hyperbolic rotations of angles π and 2π/3, respectively. The Euler characteristic of the orbifold O2 is −1/6. Thus a minimal surface Selberg cover is of order 6. Now we consider the action defined by isometries of the modular group PSL(2, Z[i]) 3 which is the Picard group related to the Gauss integers on the hyperbolic real 3-space HR. 3 3 3 The quotient O := HR/PSL(2, Z[i]) has underlying space the 3-space R . Its singular locus O3 is the 1-skeleton of a squared pyramid without the apex. The Euler characteristic of the orbifold O3 is 0. In this paper, we introduce two generalizations of the modular group in the four- dimensional setting of the quaternions and the rings of Lipschitz and Hurwitz integers (see [14,15],[19]) and then focus our attention to their actions on hyperbolic (quaternionic) half | |2 space H1 := {q ∈ H : (q)>0} with metric d q . We then explore new results which H ( q)2 give a very detailed description of the orbifolds related to the corresponding quaternionic modular groups. Secondly we define the group generated by translations by the imaginary 1 Fig. 1 A fundamental domain for the action of the modular group PSL(2, Z) on the hyperbolic plane HC 2 parts of Lipschitz integers, the inversion T and a finite group related to the Hurwitz units. We give a thorough description of its properties and the corresponding orbifolds. Furthermore, we also give a geometric description of the fundamental domains for the actions of the quaternionic modular groups and a detailed analysis of the topology and of the isotropy groups of the singularities of the orbifolds introduced. We study the corresponding modular Lipschitz and Hurwitz groups in the Lorentz–Minkowski model. 1 2 The quaternionic hyperbolic 4-space HH and its isometries 1 2.1 Isometries in the half-space model of the hyperbolic 4-space HH Consider the quaternions 2 2 2 H := {x0 + x1i + x2j + x3k : xn ∈ R, n = 0, 1, 2, 3, i = j = k =−1, ij =−ji = k}. If q = x0 + x1i + x2j + x3k ∈ H then (q) := x0 ∈ R, q := x0 − x1i − x2j − x3k ∈ H and |q|2 := qq ∈ R+. 1 Let HH := {q ∈ H : (q)>0} be the half-space model of the one-dimensional quaternionic hyperbolic space. This set is isometric to the hyperbolic real space in four 1 ∼ 4 4 dimensions, namely HH = HR ={(x0, x1, x2, x3) ∈ R : x0 > 0} with the element ( )2+( )2+( )2+( )2 ( )2 = dx0 dx1 dx2 dx3 of hyperbolic metric given by ds 2 where s measures length x0 along a parametrized curve. Even though the (natural) algebraic structures carried by the two sets are deeply different. Let GL(2, H) denote the general linear group of 2 × 2 invertible1 matrices with entries in the quaternions H. The next definitions can be found in [3,10,16]and[26]. ab Definition 2.1 For any A = ∈ GL(2, H), the associated real analytic function cd −1 FA : H ∪{∞}→H ∪{∞}defined by FA(q) = (aq + b) · (cq + d) is called the Möbius transformation associated with A. −1 −1 We set FA(∞) =∞if c = 0, FA(∞) = ac if c = 0andFA(−c d) =∞. Let F := {FA : A ∈ GL(2, H)} the group of Möbius transformations. Definition 2.2 Let SL(2, H) be the special linear group which consists of matrices in GL(2, H) with Dieudonnè determinant 1. Then : GL(2, H) → F defined as (A) = FA is a surjective group antihomomorphism with ker() ={tI : t ∈ R \{0}}. Furthermore, the restriction of to the special linear group SL(2, H) is still surjective and whose kernel is {±I}. 1 Let M 1 be the set of Möbius transformations that leave invariant H . Any transformation HH H 1 F ∈ M 1 is conformal and preserves orientation, moreover is an isometry of H .We A HH H 1 1 conclude that M 1 is isomorphic to the groups Conf+(H ) and Isom+(H ) of conformal HH H H 1 diffeomorphisms and isometries orientation–preserving of the half-space model HH (see [2] and [1]). Moreover, the group M 1 acts by orientation-preserving conformal transformations HH 1 on the sphere at infinity of the hyperbolic 4-space defined as ∂HH := {q ∈ H : (q) = ∼ 3 ∼ 1 ∼ 1 0}∪{∞}. In other words M 1 = Conf+(S ) = Conf+(H ) = Isom+(H ). HH H H 1 By this we mean that a 2 × 2 quaternionic matrix A has a right and left inverse; in [4] it is shown that this is equivalent for A to have non zero Dieudonné determinant (see [3]). 3 Finally we recall the conditions found by Ahlfors (see [2])andthenappliedbyBisiand Gentili in the next form (see [4]): Proposition 2.3 [Ahlfors conditions] The subgroup M 1 of P SL(2, H) can be charac- HH terized as the group induced by matrices which satisfy one of the following (equivalent) conditions: ⎧ ⎪ ⎪ = ab , , , ∈ H : t = = 01 , ⎪ A a b c d A KA K with K ⎪ cd 10 ⎪ ⎨⎪ ab A = a, b, c, d ∈ H : (ac) = 0, (bd) = 0, bc + da = 1 , ⎪ cd ⎪ ⎪ ⎪ ⎪ ab ⎩ A = a, b, c, d ∈ H : (cd) = 0, (ab) = 0, ad + bc = 1 . cd 1 2.2 The affine subgroup A(H) of the isometries of HH Consider now the affine subgroup A(H) of M 1 consisting of transformations which are HH λab induced by matrices of the form with |a|=1,λ>0and (ba) = 0. The group 0 λ−1a A(H) is the maximal subgroup of M 1 which fixes the point at infinity and its a Lie group HH of real dimension 7. Each matrix in A(H) acts as a conformal transformation on the hyperplane at infinity 1 ∂HH. Moreover A(H) is the group of conformal and orientation preserving transformations acting on the space of pure imaginary quaternions at infinity which can be identified with R3 so that A(H) is isomorphic to the conformal group Conf+(R3). 1 2.3 Isotropy subgroup of the isometries of HH which fixed one point αβ Let K := ∈ M 1 be the subgroup of symmetric matrices in M 1 .Forthematrix βα HH HH αβ |α|2 +|β|2 = (αβ) = βα the conditions 1and 0 are equivalent to Ahlfors conditions in 1 Proposition 2.3.
Load more

Recommended publications
  • Matrix Groups for Undergraduates Second Edition Kristopher Tapp
    STUDENT MATHEMATICAL LIBRARY Volume 79 Matrix Groups for Undergraduates Second Edition Kristopher Tapp https://doi.org/10.1090//stml/079 Matrix Groups for Undergraduates Second Edition STUDENT MATHEMATICAL LIBRARY Volume 79 Matrix Groups for Undergraduates Second Edition Kristopher Tapp Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 20-02, 20G20; Secondary 20C05, 22E15. For additional information and updates on this book, visit www.ams.org/bookpages/stml-79 Library of Congress Cataloging-in-Publication Data Names: Tapp, Kristopher, 1971– Title: Matrix groups for undergraduates / Kristopher Tapp. Description: Second edition. — Providence, Rhode Island : American Mathe- matical Society, [2016] — Series: Student mathematical library ; volume 79 — Includes bibliographical references and index. Identifiers: LCCN 2015038141 — ISBN 9781470427221 (alk. paper) Subjects: LCSH: Matrix groups. — Linear algebraic groups. — Compact groups. — Lie groups. — AMS: Group theory and generalizations – Research exposition (monographs, survey articles). msc — Group theory and generalizations – Linear algebraic groups and related topics – Linear algebraic groups over the reals, the complexes, the quaternions. msc — Group theory and generalizations – Repre- sentation theory of groups – Group rings of finite groups and their modules. msc — Topological groups, Lie groups – Lie groups – General properties and structure of real Lie groups. msc Classification: LCC QA184.2 .T37 2016 — DDC 512/.2–dc23 LC record available at http://lccn.loc.gov/2015038141 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.
    [Show full text]
  • The Classification and the Conjugacy Classesof the Finite Subgroups of The
    Algebraic & Geometric Topology 8 (2008) 757–785 757 The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LGONÇALVES JOHN GUASCHI Let n 3. We classify the finite groups which are realised as subgroups of the sphere 2 braid group Bn.S /. Such groups must be of cohomological period 2 or 4. Depend- ing on the value of n, we show that the following are the maximal finite subgroups of 2 Bn.S /: Z2.n 1/ ; the dicyclic groups of order 4n and 4.n 2/; the binary tetrahedral group T ; the binary octahedral group O ; and the binary icosahedral group I . We give geometric as well as some explicit algebraic constructions of these groups in 2 Bn.S / and determine the number of conjugacy classes of such finite subgroups. We 2 also reprove Murasugi’s classification of the torsion elements of Bn.S / and explain 2 how the finite subgroups of Bn.S / are related to this classification, as well as to the 2 lower central and derived series of Bn.S /. 20F36; 20F50, 20E45, 57M99 1 Introduction The braid groups Bn of the plane were introduced by E Artin in 1925[2;3]. Braid groups of surfaces were studied by Zariski[41]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition[16]. Let M be a compact, connected surface, and let n N . We denote the set of all ordered 2 n–tuples of distinct points of M , known as the n–th configuration space of M , by: Fn.M / .p1;:::; pn/ pi M and pi pj if i j : D f j 2 ¤ ¤ g Configuration spaces play an important roleˆ in several branches of mathematics and have been extensively studied; see Cohen and Gitler[9] and Fadell and Husseini[14], for example.
    [Show full text]
  • Uniform Polychora
    BRIDGES Mathematical Connections in Art, Music, and Science Uniform Polychora Jonathan Bowers 11448 Lori Ln Tyler, TX 75709 E-mail: [email protected] Abstract Like polyhedra, polychora are beautiful aesthetic structures - with one difference - polychora are four dimensional. Although they are beyond human comprehension to visualize, one can look at various projections or cross sections which are three dimensional and usually very intricate, these make outstanding pieces of art both in model form or in computer graphics. Polygons and polyhedra have been known since ancient times, but little study has gone into the next dimension - until recently. Definitions A polychoron is basically a four dimensional "polyhedron" in the same since that a polyhedron is a three dimensional "polygon". To be more precise - a polychoron is a 4-dimensional "solid" bounded by cells with the following criteria: 1) each cell is adjacent to only one other cell for each face, 2) no subset of cells fits criteria 1, 3) no two adjacent cells are corealmic. If criteria 1 fails, then the figure is degenerate. The word "polychoron" was invented by George Olshevsky with the following construction: poly = many and choron = rooms or cells. A polytope (polyhedron, polychoron, etc.) is uniform if it is vertex transitive and it's facets are uniform (a uniform polygon is a regular polygon). Degenerate figures can also be uniform under the same conditions. A vertex figure is the figure representing the shape and "solid" angle of the vertices, ex: the vertex figure of a cube is a triangle with edge length of the square root of 2.
    [Show full text]
  • A Symplectic Resolution for the Binary Tetrahedral Group
    A SYMPLECTIC RESOLUTION FOR THE BINARY TETRAHEDRAL GROUP MANFRED LEHN & CHRISTOPH SORGER Abstract. We describe an explicit symplectic resolution for the quo- tient singularity arising from the four-dimensional symplectic represen- ation of the binary tetrahedral group. Let G be a nite group with a complex symplectic representation V . The symplectic form σ on V descends to a symplectic form σ¯ on the open regular part of V=G. A proper morphism f : Y ! V=G is a symplectic resolution if Y is smooth and if f ∗σ¯ extends to a symplectic form on Y . It turns out that symplectic resolutions of quotient singularities are a rare phenomenon. By a theorem of Verbitsky [9], a necessary condition for the existence of a symplectic resolution is that G be generated by symplectic reections, i.e. by elements whose x locus on V is a linear subspace of codimension 2. Given an arbitrary complex representation V0 of a nite group G, we obtain a symplectic representation on ∗, where ∗ denotes the contragradient V0 ⊕V0 V0 representation of V0. In this case, Verbitsky's theorem specialises to an earlier theorem of Kaledin [7]: For ∗ to admit a symplectic resolution, the V0 ⊕ V0 =G action of G on V0 should be generated by complex reections, in other words, V0=G should be smooth. The complex reection groups have been classied by Shephard and Todd [8], the symplectic reection groups by Cohen [2]. The list of Shephard and Todd contains as a sublist the nite Coxeter groups. The question which of these groups G ⊂ Sp(V ) admits a symplectic res- olution for V=G has been solved for the Coxeter groups by Ginzburg and Kaledin [3] and for arbitrary complex reection groups most recently by Bellamy [1].
    [Show full text]
  • On the Solutions of Linear Matrix Quaternionic Equations and Their Systems
    Mathematica Aeterna, Vol. 6, 2016, no. 6, 907 - 921 On The Solutions of Linear Matrix Quaternionic Equations and Their Systems Ahmet Ipek_ Department of Mathematics Faculty of Kamil Ozda˘gScience¨ Karamano˘gluMehmetbey University Karaman, Turkey [email protected] Cennet Bolat C¸imen Ankara Chamber of Industry 1st Organized Industrial Zone Vocational School Hacettepe University Ankara, Turkey [email protected] Abstract In this paper, our main aim is to investigate the solvability, exis- tence of unique solution, closed-form solutions of some linear matrix quaternion equations with one unknown and of their systems with two unknowns. By means of the arithmetic operations on matrix quater- nions, the linear matrix quaternion equations that is considered herein could be converted into four classical real linear equations, the solu- tions of the linear matrix quaternion equations are derived by solving four classical real linear equations based on the inverses and general- ized inverses of matrices. Also, efficiency and accuracy of the presented method are shown by several examples. Mathematics Subject Classification: 11R52; 15A30 Keywords: The systems of equations; Quaternions; Quaternionic Sys- tems. 1 Introduction The quaternion and quaternion matrices play a role in computer science, quan- tum physics, signal and color image processing, and so on (e.g., [1, 17, 22]. 908 Ahmet Ipek_ and Cennet Bolat C¸imen General properties of quaternion matrices can be found in [29]. Linear matrix equations are often encountered in many areas of computa- tional mathematics, control and system theory. In the past several decades, solving matrix equations has been a hot topic in the linear algebraic field (see, for example, [2, 3, 12, 18, 19, 21 and 30]).
    [Show full text]
  • Maximal Subgroups of the Coxeter Group W(H4) and Quaternions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 412 (2006) 441–452 www.elsevier.com/locate/laa Maximal subgroups of the Coxeter group W(H4) and quaternions Mehmet Koca a, Ramazan Koç b,∗, Muataz Al-Barwani a, Shadia Al-Farsi a aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Muscat, Oman bDepartment of Physics, Faculty of Engineering, Gaziantep University, 27310 Gaziantep, Turkey Received 9 October 2004; accepted 17 July 2005 Available online 12 September 2005 Submitted by H. Schneider Abstract The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H2) × W(H2)]Z4 and W(H3) × Z2 possess non- crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems. © 2005 Elsevier Inc. All rights reserved. AMS classification: 20G20; 20F05; 20E34 Keywords: Structure of groups; Quaternions; Coxeter groups; Subgroup structure ∗ Corresponding author. Tel.: +90 342 360 1200; fax: +90 342 360 1013.
    [Show full text]
  • On Regular Polytopes Is [7]
    ON REGULAR POLYTOPES Luis J. Boya Departamento de F´ısica Te´orica Universidad de Zaragoza, E-50009 Zaragoza, SPAIN [email protected] Cristian Rivera Departamento de F´ısica Te´orica Universidad de Zaragoza, E-50009 Zaragoza, SPAIN cristian [email protected] MSC 05B45, 11R52, 51M20, 52B11, 52B15, 57S25 Keywords: Polytopes, Higher Dimensions, Orthogonal groups Abstract Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension n 1; now in dim. 2, 3 and 4 there are extra polytopes, while in general≥− dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron ex- ist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the SO(2) = U(1) group being (abelian and) divisible, is related to the existence of arbitrarily- sided plane regular polygons, and the splitting of the Lie algebra of the O(4) group will be seen responsible for the Schl¨afli special polytopes in 4-dim., two of which percolate down to three. In spite of dim. 8 being also special (Cartan’s triality), we argue why there are no extra polytopes, while it has other consequences: in particular the existence of the three division algebras over the reals R: complex C, quater- nions H and octonions O is seen also as another feature of the special arXiv:1210.0601v1 [math-ph] 1 Oct 2012 properties of corresponding orthogonal groups, and of the spheres of dimension 0,1,3 and 7. 1 Introduction Regular Polytopes are the higher dimensional generalization of the (regu- lar) polygons in the plane and the (five) Platonic solids in space.
    [Show full text]
  • Platonic Solids Generate Their Four-Dimensional Analogues.', Acta Crystallographica Section A., 69 (6)
    Durham Research Online Deposited in DRO: 26 January 2014 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Dechant, Pierre-Philippe (2013) 'Platonic solids generate their four-dimensional analogues.', Acta crystallographica section A., 69 (6). pp. 592-602. Further information on publisher's website: http://dx.doi.org/10.1107/S0108767313021442 Publisher's copyright statement: Additional information: Published on behalf of the International Union of Crystallography. Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk electronic reprint Acta Crystallographica Section A Foundations of Crystallography ISSN 0108-7673 Platonic solids generate their four-dimensional analogues Pierre-Philippe Dechant Acta Cryst. (2013). A69, 592–602 Copyright c International Union of Crystallography Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr.
    [Show full text]
  • Perturbation Analysis of Matrices Over a Quaternion Division Algebra∗
    Electronic Transactions on Numerical Analysis. Volume 54, pp. 128–149, 2021. ETNA Kent State University and Copyright © 2021, Kent State University. Johann Radon Institute (RICAM) ISSN 1068–9613. DOI: 10.1553/etna_vol54s128 PERTURBATION ANALYSIS OF MATRICES OVER A QUATERNION DIVISION ALGEBRA∗ SK. SAFIQUE AHMADy, ISTKHAR ALIz, AND IVAN SLAPNICARˇ x Abstract. In this paper, we present the concept of perturbation bounds for the right eigenvalues of a quaternionic matrix. In particular, a Bauer-Fike-type theorem for the right eigenvalues of a diagonalizable quaternionic matrix is derived. In addition, perturbations of a quaternionic matrix are discussed via a block-diagonal decomposition and the Jordan canonical form of a quaternionic matrix. The location of the standard right eigenvalues of a quaternionic matrix and a sufficient condition for the stability of a perturbed quaternionic matrix are given. As an application, perturbation bounds for the zeros of quaternionic polynomials are derived. Finally, we give numerical examples to illustrate our results. Key words. quaternionic matrices, left eigenvalues, right eigenvalues, quaternionic polynomials, Bauer-Fike theorem, quaternionic companion matrices, quaternionic matrix norms AMS subject classifications. 15A18, 15A66 1. Introduction. The goal of this paper is the derivation of a Bauer-Fike-type theorem for the right eigenvalues and a perturbation analysis for quaternionic matrices, as well as a specification of the location of the right eigenvalues of a perturbed quaternionic matrix and perturbation bounds for the zeros of quaternionic polynomials. The Bauer-Fike theorem is a standard result in the perturbation theory for diagonalizable matrices over the complex −1 field. The theorem states that if A 2 Mn(C) is a diagonalizable matrix, with A = XDX , and A + E is a perturbed matrix, then an upper bound for the distance between a point µ 2 Λ(A + E) and the spectrum Λ(A) is given by [4] (1.1) min jµ − λj ≤ κ(X)kEk: λ2Λ(A) Here, κ(X) = kXkkX−1k is the condition number of the matrix X.
    [Show full text]
  • Platonic Solids Generate Their Four-Dimensional Analogues
    This is a repository copy of Platonic solids generate their four-dimensional analogues. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/85590/ Version: Accepted Version Article: Dechant, Pierre-Philippe orcid.org/0000-0002-4694-4010 (2013) Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. pp. 592-602. ISSN 1600-5724 https://doi.org/10.1107/S0108767313021442 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a,b,c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone.
    [Show full text]
  • Quaternions and Quantum Theory
    Quaternions and Quantum Theory by Matthew A. Graydon A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Science in Physics Waterloo, Ontario, Canada, 2011 c Matthew A. Graydon 2011 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract The orthodox formulation of quantum theory invokes the mathematical apparatus of complex Hilbert space. In this thesis, we consider a quaternionic quantum formalism for the description of quantum states, quantum channels, and quantum measurements. We prove that probabilities for outcomes of quaternionic quantum measurements arise from canonical inner products of the corresponding quaternionic quantum effects and a unique quaternionic quantum state. We embed quaternionic quantum theory into the framework of usual complex quantum information theory. We prove that quaternionic quantum measurements can be simulated by usual complex positive operator valued measures. Furthermore, we prove that quaternionic quantum channels can be simulated by completely positive trace preserving maps on complex quantum states. We also derive a lower bound on an orthonormality measure for sets of positive semi-definite quaternionic linear operators. We prove that sets of operators saturating the aforementioned lower bound facilitate a reconciliation of quaternionic quantum theory with a generalized Quantum Bayesian framework for reconstructing quantum state spaces. This thesis is an extension of work found in [42]. iii Acknowledgements First and foremost, I would like to deeply thank my supervisor, Dr.
    [Show full text]
  • The Fundamental Group of SO(N) Via Quotients of Braid Groups Arxiv
    The Fundamental Group of SO(n) Via Quotients of Braid Groups Ina Hajdini∗ and Orlin Stoytchevy July 21, 2016 Abstract ∼ We describe an algebraic proof of the well-known topological fact that π1(SO(n)) = Z=2Z. The fundamental group of SO(n) appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group Bn, obtained by imposing one additional relation and turns out to be a nontrivial central extension by Z=2Z of the corresponding group of rotational symmetries of the hyperoctahedron in dimension n. 1 Introduction. n The set of all rotations in R forms a group denoted by SO(n). We may think of it as the group of n × n orthogonal matrices with unit determinant. As a topological space it has the structure of a n2 smooth (n(n − 1)=2)-dimensional submanifold of R . The group structure is compatible with the smooth one in the sense that the group operations are smooth maps, so it is a Lie group. The space SO(n) when n ≥ 3 has a fascinating topological property—there exist closed paths in it (starting and ending at the identity) that cannot be continuously deformed to the trivial (constant) path, but going twice along such a path gives another path, which is deformable to the trivial one. For example, if 3 you rotate an object in R by 2π along some axis, you get a motion that is not deformable to the trivial motion (i.e., no motion at all), but a rotation by 4π is deformable to the trivial motion.
    [Show full text]