Modular Groups Over the Quaternions
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Subgroups of Division Rings in Characteristic Zero Are Characterized
Subgroups of Division Rings Mark Lewis Murray Schacher June 27, 2018 Abstract We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders. MSC(2010): Primary: 16A39, 12E15; Secondary: 16U60 1 Introduction Let F be a field, and fix a and b to be non-0 elements of F . The symbol algebra A =(a, b) is the 4-dimensional algebra over F generated by elements i and j that satisfy the relations: i2 = a, j2 = b, ij = ji. (1) − One usually sets k = ij, which leads to the additional circular relations ij = k = ji, jk = i = kj, ki = j = ki. (2) − − − arXiv:1806.09654v1 [math.RA] 25 Jun 2018 The set 1, i, j, k forms a basis for A over F . It is not difficult to see that A is a central{ simple} algebra over F , and using Wedderburn’s theorem, we see that A is either a 4-dimensional division ring or the ring M2(F ) of2 2 matrices over F . It is known that A is split (i.e. is the ring of 2 2 matrices× over F ) if and only if b is a norm from the field F (√a). Note that×b is a norm over F (√a) if and only if there exist elements x, y F so that b = x2 y2a. This condition is symmetric in a and b. ∈ − More generally, we have the following isomorphism of algebras: (a, b) ∼= (a, ub) (3) 1 where u = x2 y2a is a norm from F (√a); see [4] or [6]. -
Affine Reflection Group Codes
Affine Reflection Group Codes Terasan Niyomsataya1, Ali Miri1,2 and Monica Nevins2 School of Information Technology and Engineering (SITE)1 Department of Mathematics and Statistics2 University of Ottawa, Ottawa, Canada K1N 6N5 email: {tniyomsa,samiri}@site.uottawa.ca, [email protected] Abstract This paper presents a construction of Slepian group codes from affine reflection groups. The solution to the initial vector and nearest distance problem is presented for all irreducible affine reflection groups of rank n ≥ 2, for varying stabilizer subgroups. Moreover, we use a detailed analysis of the geometry of affine reflection groups to produce an efficient decoding algorithm which is equivalent to the maximum-likelihood decoder. Its complexity depends only on the dimension of the vector space containing the codewords, and not on the number of codewords. We give several examples of the decoding algorithm, both to demonstrate its correctness and to show how, in small rank cases, it may be further streamlined by exploiting additional symmetries of the group. 1 1 Introduction Slepian [11] introduced group codes whose codewords represent a finite set of signals combining coding and modulation, for the Gaussian channel. A thorough survey of group codes can be found in [8]. The codewords lie on a sphere in n−dimensional Euclidean space Rn with equal nearest-neighbour distances. This gives congruent maximum-likelihood (ML) decoding regions, and hence equal error probability, for all codewords. Given a group G with a representation (action) on Rn, that is, an 1Keywords: Group codes, initial vector problem, decoding schemes, affine reflection groups 1 orthogonal n × n matrix Og for each g ∈ G, a group code generated from G is given by the set of all cg = Ogx0 (1) n for all g ∈ G where x0 = (x1, . -
Generalized Quaternions
GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8. -
Feature Matching and Heat Flow in Centro-Affine Geometry
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 093, 22 pages Feature Matching and Heat Flow in Centro-Affine Geometry Peter J. OLVER y, Changzheng QU z and Yun YANG x y School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] URL: http://www.math.umn.edu/~olver/ z School of Mathematics and Statistics, Ningbo University, Ningbo 315211, P.R. China E-mail: [email protected] x Department of Mathematics, Northeastern University, Shenyang, 110819, P.R. China E-mail: [email protected] Received April 02, 2020, in final form September 14, 2020; Published online September 29, 2020 https://doi.org/10.3842/SIGMA.2020.093 Abstract. In this paper, we study the differential invariants and the invariant heat flow in centro-affine geometry, proving that the latter is equivalent to the inviscid Burgers' equa- tion. Furthermore, we apply the centro-affine invariants to develop an invariant algorithm to match features of objects appearing in images. We show that the resulting algorithm com- pares favorably with the widely applied scale-invariant feature transform (SIFT), speeded up robust features (SURF), and affine-SIFT (ASIFT) methods. Key words: centro-affine geometry; equivariant moving frames; heat flow; inviscid Burgers' equation; differential invariant; edge matching 2020 Mathematics Subject Classification: 53A15; 53A55 1 Introduction The main objective in this paper is to study differential invariants and invariant curve flows { in particular the heat flow { in centro-affine geometry. In addition, we will present some basic applications to feature matching in camera images of three-dimensional objects, comparing our method with other popular algorithms. -
The Affine Group of a Lie Group
THE AFFINE GROUP OF A LIE GROUP JOSEPH A. WOLF1 1. If G is a Lie group, then the group Aut(G) of all continuous auto- morphisms of G has a natural Lie group structure. This gives the semi- direct product A(G) = G-Aut(G) the structure of a Lie group. When G is a vector group R", A(G) is the ordinary affine group A(re). Follow- ing L. Auslander [l ] we will refer to A(G) as the affine group of G, and regard it as a transformation group on G by (g, a): h-^g-a(h) where g, hEG and aGAut(G) ; in the case of a vector group, this is the usual action on A(») on R". If B is a compact subgroup of A(n), then it is well known that B has a fixed point on R", i.e., that there is a point xGR" such that b(x)=x for every bEB. For A(ra) is contained in the general linear group GL(« + 1, R) in the usual fashion, and B (being compact) must be conjugate to a subgroup of the orthogonal group 0(w + l). This conjugation can be done leaving fixed the (« + 1, w + 1)-place matrix entries, and is thus possible by an element of k(n). This done, the translation-parts of elements of B must be zero, proving the assertion. L. Auslander [l] has extended this theorem to compact abelian subgroups of A(G) when G is connected, simply connected and nil- potent. We will give a further extension. -
Map Composition Generalized to Coherent Collections of Maps
Front. Math. China 2015, 10(3): 547–565 DOI 10.1007/s11464-015-0435-5 Map composition generalized to coherent collections of maps Herng Yi CHENG1, Kang Hao CHEONG1,2 1 National University of Singapore High School of Mathematics and Science, Singapore 129957, Singapore 2 Tanglin Secondary School, Singapore 127391, Singapore c Higher Education Press and Springer-Verlag Berlin Heidelberg 2015 Abstract Relation algebras give rise to partial algebras on maps, which are generalized to partial algebras on polymaps while preserving the properties of relation union and composition. A polymap is defined as a map with every point in the domain associated with a special set of maps. Polymaps can be represented as small subcategories of Set∗, the category of pointed sets. Map composition and the counterpart of relation union for maps are generalized to polymap composition and sum. Algebraic structures and categories of polymaps are investigated. Polymaps present the unique perspective of an algebra that can retain many of its properties when its elements (maps) are augmented with collections of other elements. Keywords Relation algebra, partial algebra, composition MSC 08A02 1 Introduction This paper considers partial algebras on maps (similar to those defined in [15]), constructed in analogy to relation algebras [5,12,16]. The sum of two maps f0 : X0 → Y0 and f1 : X1 → Y1 is defined if the union of their graphs is a functional binary relation R ⊆ (X0 ∪ X1) × (Y0 ∪ Y1), in which case the sum is the map (f0 + f1): X0 ∪ X1 → Y0 ∪ Y1 with graph R. This is defined similarly to the union operation + from [10]. -
LOW-DIMENSIONAL UNITARY REPRESENTATIONS of B3 1. Introduction Unitary Braid Representations Have Been Constructed in Several
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 9, Pages 2597{2606 S 0002-9939(01)05903-2 Article electronically published on March 15, 2001 LOW-DIMENSIONAL UNITARY REPRESENTATIONS OF B3 IMRE TUBA (Communicated by Stephen D. Smith) Abstract. We characterize all simple unitarizable representations of the braid group B3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ1 and σ2 denote the two generating twists of B3,thenasimple representation ρ : B3 ! GL(V )(fordimV ≤ 5) is unitarizable if and only if the eigenvalues λ1,λ2;::: ;λd of ρ(σ1) are distinct, satisfy jλij =1and (d) ≤ ≤ (d) µ1i > 0for2 i d,wheretheµ1i are functions of the eigenvalues, explicitly described in this paper. 1. Introduction Unitary braid representations have been constructed in several ways using the representation theory of Kac-Moody algebras and quantum groups (see e.g. [1], [2], and [4]), and specializations of the reduced Burau and Gassner representations in [5]. Such representations easily lead to representations of PSL(2; Z)=B3=Z ,where Z is the center of B3,andPSL(2; Z)=SL(2; Z)={1g,where{1g is the center of SL(2; Z). We give a complete classification of simple unitary representations of B3 of dimension d ≤ 5 in this paper. In particular, the unitarizability of a braid rep- resentation depends only on the the eigenvalues λ1,λ2;::: ,λd of the images of the two generating twists of B3. The condition for unitarizability is a set of linear in- equalities in the logarithms of these eigenvalues. In other words, the representation is unitarizable if and only if the (arg λ1; arg λ2;::: ;arg λd) is a point inside a poly- hedron in (R=2π)d, where we give the equations of the hyperplanes that bound this polyhedron. -
Construction of Free Subgroups in the Group of Units of Modular Group Algebras
CONSTRUCTION OF FREE SUBGROUPS IN THE GROUP OF UNITS OF MODULAR GROUP ALGEBRAS Jairo Z. Gon¸calves1 Donald S. Passman2 Department of Mathematics Department of Mathematics University of S~ao Paulo University of Wisconsin-Madison 66.281-Ag Cidade de S. Paulo Van Vleck Hall 05389-970 S. Paulo 480 Lincoln Drive S~ao Paulo, Brazil Madison, WI 53706, U.S.A [email protected] [email protected] Abstract. Let KG be the group algebra of a p0-group G over a field K of characteristic p > 0; and let U(KG) be its group of units. If KG contains a nontrivial bicyclic unit and if K is not algebraic over its prime field, then we prove that the free product Zp ∗ Zp ∗ Zp can be embedded in U(KG): 1. Introduction Let KG be the group algebra of the group G over the field K; and let U(KG) be its group of units. Motivated by the work of Pickel and Hartley [4], and Sehgal ([7, pg. 200]) on the existence of free subgroups in the inte- gral group ring ZG; analogous conditions for U(KG) have been intensively investigated in [1], [2] and [3]. Recently Marciniak and Sehgal [5] gave a constructive method for produc- ing free subgroups in U(ZG); provided ZG contains a nontrivial bicyclic unit. In this paper we prove an analogous result for the modular group algebra KG; whenever K is not algebraic over its prime field GF (p): Specifically, if Zp denotes the cyclic group of order p, then we prove: 1- Research partially supported by CNPq - Brazil. -
Math.AG] 17 Mar 2001 Ebro Uhafml Mle H Og Ojcuefrinfin field 5-Dimensional for Such with Identify Type Conjecture to Weil Hope Hodge We of the Algebra
QUATERNIONIC PRYMS AND HODGE CLASSES B. VAN GEEMEN AND A. VERRA Abstract. Abelian varieties of dimension 2n on which a definite quaternion algebra acts are parametrized by symmetrical domains of dimension n(n 1)/2. Such abelian varieties have primitive Hodge classes in the middle dimensional cohomolo−gy group. In general, it is not clear that these are cycle classes. In this paper we show that a particular 6-dimensional family of such 8-folds are Prym varieties and we use the method of C. Schoen to show that all Hodge classes on the general abelian variety in this family are algebraic. We also consider Hodge classes on certain 5-dimensional subfamilies and relate these to the Hodge conjecture for abelian 4-folds. In this paper we study abelian varieties of dimension 8 whose endomorphism ring is a definite quaternion algebra over Q, we refer to these as abelian 8-folds of quaternion type. Such abelian varieties are interesting since their Hodge rings are not generated by divisor classes [M] and the Hodge conjecture is still open for most of them. The moduli space of 8-folds of quaternion type, with fixed discrete data, is 6-dimensional. One such moduli space was investigated recently by Freitag and Hermann [FrHe]. In [KSTT] a 6-dimensional family of K3 surfaces is studied whose Kuga-Satake varieties have simple factors of dimension 8 which are of quaternion type. Our first result, in section 3, is that one of these moduli spaces parametrizes Prym varieties of unramified 2:1 covers C˜ Cˆ, actually the genus 17 curves C˜ are 8:1 unramified covers of genus 3 curves with Galois→ group Q generated by the quaternions i and j. -
A Note on the Quaternion Group As Galois Group
proceedings of the american mathematical society Volume 108, Number 3, March 1990 A NOTE ON THE QUATERNION GROUP AS GALOIS GROUP ROGER WARE (Communicated by Louis J. Ratliff, Jr.) Abstract. The occurrence of the quaternion group as a Galois group over cer- tain fields is investigated. A theorem of Witt on quaternionic Galois extensions plays a key role. In [9, §6] Witt proved a theorem characterizing quaternionic Galois exten- sions. Namely, he showed that if F is a field of characteristic not 2 then an extension L = F(y/ä,y/b), a,b e F, of degree 4 over F can be embedded in a Galois extension A" of 7 with Gal(K/F) = 77g (the quaternion group 7 7 7 of order 8) if and only if the quadratic form ax + by + abz is isomorphic to x 2 + y 2 + z 2 .In addition he showed how to explicitly construct the Galois extension from the isometry. An immediate and interesting consequence of this is the fact that 77g cannot be a Galois group over any Pythagorean field. In this note Witt's theorem is used to obtain additional results about the existence of 77g as a Galois group over certain fields. If 7 is a field (of char- acteristic not 2) with at most one (total) ordering such that 77g does not oc- cur as a Galois group over F then the structure of the pro-2-Galois groups GF(2) = Gal(7(2)/7), Gpy = Gal(7py/7) (where 7(2) and Fpy are the quadratic and Pythagorean closures of F) are completely determined. -
A Classification of Clifford Algebras As Images of Group Algebras of Salingaros Vee Groups
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT A CLASSIFICATION OF CLIFFORD ALGEBRAS AS IMAGES OF GROUP ALGEBRAS OF SALINGAROS VEE GROUPS R. Ablamowicz,M.Varahagiri,A.M.Walley November 2017 No. 2017-3 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookeville, TN 38505 A Classification of Clifford Algebras as Images of Group Algebras of Salingaros Vee Groups Rafa lAb lamowicz, Manisha Varahagiri and Anne Marie Walley Abstract. The main objective of this work is to prove that every Clifford algebra C`p;q is R-isomorphic to a quotient of a group algebra R[Gp;q] modulo an ideal J = (1 + τ) where τ is a central element of order 2. p+q+1 Here, Gp;q is a 2-group of order 2 belonging to one of Salingaros isomorphism classes N2k−1;N2k; Ω2k−1; Ω2k or Sk. Thus, Clifford al- gebras C`p;q can be classified by Salingaros classes. Since the group algebras R[Gp;q] are Z2-graded and the ideal J is homogeneous, the quotient algebras R[G]=J are Z2-graded. In some instances, the isomor- ∼ phism R[G]=J = C`p;q is also Z2-graded. By Salingaros Theorem, the groups Gp;q in the classes N2k−1 and N2k are iterative central products of the dihedral group D8 and the quaternion group Q8, and so they are extra-special. The groups in the classes Ω2k−1 and Ω2k are central products of N2k−1 and N2k with C2 × C2, respectively. The groups in the class Sk are central products of N2k or N2k with C4. Two algorithms to factor any Gp;q into an internal central product, depending on the class, are given. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group).