Algebraic Groups and Arithmetic Groups
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Quotient Groups §7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I
Math 371 Lecture #33 x7.7 (Ed.2), 8.3 (Ed.2): Quotient Groups x7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I Recall that to make the set of right cosets of a subgroup K in a group G into a group under the binary operation (or product) (Ka)(Kb) = K(ab) requires that K be a normal subgroup of G. For a normal subgroup N of a group G, we denote the set of right cosets of N in G by G=N (read G mod N). Theorem 8.12. For a normal subgroup N of a group G, the product (Na)(Nb) = N(ab) on G=N is well-defined. We saw the proof of this before. Theorem 8.13. For a normal subgroup N of a group G, we have (1) G=N is a group under the product (Na)(Nb) = N(ab), (2) when jGj < 1 that jG=Nj = jGj=jNj, and (3) when G is abelian, so is G=N. Proof. (1) We know that the product is well-defined. The identity element of G=N is Ne because for all a 2 G we have (Ne)(Na) = N(ae) = Na; (Na)(Ne) = N(ae) = Na: The inverse of Na is Na−1 because (Na)(Na−1) = N(aa−1) = Ne; (Na−1)(Na) = N(a−1a) = Ne: Associativity of the product in G=N follows from the associativity in G: [(Na)(Nb)](Nc) = (N(ab))(Nc) = N((ab)c) = N(a(bc)) = (N(a))(N(bc)) = (N(a))[(N(b))(N(c))]: This establishes G=N as a group. -
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, . -
Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education)
Field Theory 1. Introduction to Quantum Field Theory (A Primer for a Basic Education) Roberto Soldati February 23, 2019 Contents 0.1 Prologue ............................ 4 1 Basics in Group Theory 8 1.1 Groups and Group Representations . 8 1.1.1 Definitions ....................... 8 1.1.2 Theorems ........................ 12 1.1.3 Direct Products of Group Representations . 14 1.2 Continuous Groups and Lie Groups . 16 1.2.1 The Continuous Groups . 16 1.2.2 The Lie Groups .................... 18 1.2.3 An Example : the Rotations Group . 19 1.2.4 The Infinitesimal Operators . 21 1.2.5 Lie Algebra ....................... 23 1.2.6 The Exponential Representation . 24 1.2.7 The Special Unitary Groups . 27 1.3 The Non-Homogeneous Lorentz Group . 34 1.3.1 The Lorentz Group . 34 1.3.2 Semi-Simple Groups . 43 1.3.3 The Poincar´eGroup . 49 2 The Action Functional 57 2.1 The Classical Relativistic Wave Fields . 57 2.1.1 Field Variations .................... 58 2.1.2 The Scalar and Vector Fields . 62 2.1.3 The Spinor Fields ................... 65 2.2 The Action Principle ................... 76 2.3 The N¨otherTheorem ................... 79 2.3.1 Problems ........................ 87 1 3 The Scalar Field 94 3.1 General Features ...................... 94 3.2 Normal Modes Expansion . 101 3.3 Klein-Gordon Quantum Field . 106 3.4 The Fock Space .......................112 3.4.1 The Many-Particle States . 112 3.4.2 The Lorentz Covariance Properties . 116 3.5 Special Distributions . 129 3.5.1 Euclidean Formulation . 133 3.6 Problems ............................137 4 The Spinor Field 151 4.1 The Dirac Equation . -
A NEW APPROACH to RANK ONE LINEAR ALGEBRAIC GROUPS An
A NEW APPROACH TO RANK ONE LINEAR ALGEBRAIC GROUPS DANIEL ALLCOCK Abstract. One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material before one can even characterize PGL2. Our goal here is to show how to develop the root system, etc., using only the completeness of the flag variety, its immediate consequences, and some facts about solvable groups. In particular, one can skip over the usual analysis of Cartan subgroups, the fact that G is the union of its Borel subgroups, the connectedness of torus centralizers, and the normalizer theorem (that Borel subgroups are self-normalizing). The main idea is a new approach to the structure of rank 1 groups; the key step is lemma 5. All algebraic geometry is over a fixed algebraically closed field. G always denotes a connected linear algebraic group with Lie algebra g, T a maximal torus, and B a Borel subgroup containing it. We assume the structure theory for connected solvable groups, and the completeness of the flag variety G/B and some of its consequences. Namely: that all Borel subgroups (resp. maximal tori) are conjugate; that G is nilpotent if one of its Borel subgroups is; that CG(T )0 lies in every Borel subgroup containing T ; and that NG(B) contains B of finite index and (therefore) is self-normalizing. -
1.6 Covering Spaces 1300Y Geometry and Topology
1.6 Covering spaces 1300Y Geometry and Topology 1.6 Covering spaces Consider the fundamental group π1(X; x0) of a pointed space. It is natural to expect that the group theory of π1(X; x0) might be understood geometrically. For example, subgroups may correspond to images of induced maps ι∗π1(Y; y0) −! π1(X; x0) from continuous maps of pointed spaces (Y; y0) −! (X; x0). For this induced map to be an injection we would need to be able to lift homotopies in X to homotopies in Y . Rather than consider a huge category of possible spaces mapping to X, we restrict ourselves to a category of covering spaces, and we show that under some mild conditions on X, this category completely encodes the group theory of the fundamental group. Definition 9. A covering map of topological spaces p : X~ −! X is a continuous map such that there S −1 exists an open cover X = α Uα such that p (Uα) is a disjoint union of open sets (called sheets), each homeomorphic via p with Uα. We then refer to (X;~ p) (or simply X~, abusing notation) as a covering space of X. Let (X~i; pi); i = 1; 2 be covering spaces of X. A morphism of covering spaces is a covering map φ : X~1 −! X~2 such that the diagram commutes: φ X~1 / X~2 AA } AA }} p1 AA }}p2 A ~}} X We will be considering covering maps of pointed spaces p :(X;~ x~0) −! (X; x0), and pointed morphisms between them, which are defined in the obvious fashion. -
Group Theory
Appendix A Group Theory This appendix is a survey of only those topics in group theory that are needed to understand the composition of symmetry transformations and its consequences for fundamental physics. It is intended to be self-contained and covers those topics that are needed to follow the main text. Although in the end this appendix became quite long, a thorough understanding of group theory is possible only by consulting the appropriate literature in addition to this appendix. In order that this book not become too lengthy, proofs of theorems were largely omitted; again I refer to other monographs. From its very title, the book by H. Georgi [211] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [102] is written in the same spirit. Both books also cover the necessary group theory for grand unification ideas. A very comprehensive but also rather dense treatment is given by [428]. Still a classic is [254]; it contains more about the treatment of dynamical symmetries in quantum mechanics. A.1 Basics A.1.1 Definitions: Algebraic Structures From the structureless notion of a set, one can successively generate more and more algebraic structures. Those that play a prominent role in physics are defined in the following. Group A group G is a set with elements gi and an operation ◦ (called group multiplication) with the properties that (i) the operation is closed: gi ◦ g j ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi , (iii) for every gi exists an −1 ∈ ◦ −1 = = −1 ◦ inverse element gi G such that gi gi g0 gi gi , (iv) the operation is associative: gi ◦ (g j ◦ gk) = (gi ◦ g j ) ◦ gk. -
Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Functional Analysis 194, 347–409 (2002) doi:10.1006/jfan.2002.3942 Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups Helge Glo¨ ckner1 Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918 E-mail: [email protected] Communicated by L. Gross Received September 4, 2001; revised November 15, 2001; accepted December 16, 2001 We characterize the existence of Lie group structures on quotient groups and the existence of universal complexifications for the class of Baker–Campbell–Hausdorff (BCH–) Lie groups, which subsumes all Banach–Lie groups and ‘‘linear’’ direct limit r r Lie groups, as well as the mapping groups CK ðM; GÞ :¼fg 2 C ðM; GÞ : gjM=K ¼ 1g; for every BCH–Lie group G; second countable finite-dimensional smooth manifold M; compact subset SK of M; and 04r41: Also the corresponding test function r r # groups D ðM; GÞ¼ K CK ðM; GÞ are BCH–Lie groups. 2002 Elsevier Science (USA) 0. INTRODUCTION It is a well-known fact in the theory of Banach–Lie groups that the topological quotient group G=N of a real Banach–Lie group G by a closed normal Lie subgroup N is a Banach–Lie group provided LðNÞ is complemented in LðGÞ as a topological vector space [7, 30]. Only recently, it was observed that the hypothesis that LðNÞ be complemented can be omitted [17, Theorem II.2]. This strengthened result is then used in [17] to characterize those real Banach–Lie groups which possess a universal complexification in the category of complex Banach–Lie groups. -
Arxiv:Math/0511624V1
Automorphism groups of polycyclic-by-finite groups and arithmetic groups Oliver Baues ∗ Fritz Grunewald † Mathematisches Institut II Mathematisches Institut Universit¨at Karlsruhe Heinrich-Heine-Universit¨at D-76128 Karlsruhe D-40225 D¨usseldorf May 18, 2005 Abstract We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homo- topy theory. 2000 Mathematics Subject Classification: Primary 20F28, 20G30; Secondary 11F06, 14L27, 20F16, 20F34, 22E40, 55P10 arXiv:math/0511624v1 [math.GR] 25 Nov 2005 ∗e-mail: [email protected] †e-mail: [email protected] 1 Contents 1 Introduction 4 1.1 Themainresults ........................... 4 1.2 Outlineoftheproofsandmoreresults . 6 1.3 Cohomologicalrepresentations . 8 1.4 Applications to the groups of homotopy self-equivalences of spaces 9 2 Prerequisites on linear algebraic groups and arithmetic groups 12 2.1 Thegeneraltheory .......................... 12 2.2 Algebraicgroupsofautomorphisms. 15 3 The group of automorphisms of a solvable-by-finite linear algebraic group 17 3.1 The algebraic structure of Auta(H)................ -
Arxiv:1906.01274V1 [Math.NT] 4 Jun 2019 Rpsto .] Aqe Ishmefas Rvdsoepofi I Yale 1.4.1]
CHOW’S THEOREM FOR SEMI-ABELIAN VARIETIES AND BOUNDS FOR SPLITTING FIELDS OF ALGEBRAIC TORI CHIA-FU YU Abstract. A theorem of Chow concerns homomorphisms of two abelian vari- eties under a primary field extension base change. In this paper we generalize Chow’s theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori. 1. Introduction Let k be a field, k¯ an algebraic closure of k, and ks the separable closure of k in k¯. A connected algebraic k-group T is an algebraic torus if there is a k¯-isomorphism d T ⊗k k¯ ≃ (Gm) ⊗k k of algebraic groups for some integer d ≥ 0. We say T splits d over a field extension K of k if there is a K-isomorphism T ⊗k K ≃ (Gm) ⊗k K. This paper is motivated from the following fundamental result. Theorem 1.1. Any algebraic k-torus T splits over ks. In other words, T splits over a finite separable field extension of k. This theorem is well known and it is stated and proved in the literature several times. Surprisingly, different authors choose their favorite proofs which are all quite different. As far as we know, the first proof is given by Takashi Ono [40, Proposition 1.2.1]. Armand Borel gives a different proof in his book Linear Algebraic Groups; see [3, Proposition 8.11]. In the second edition of his book Linear Algebraic Groups [49], T.A. -
Unitary Group - Wikipedia
Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group Unitary group In mathematics, the unitary group of degree n, denoted U( n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL( n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group. In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group. The unitary group U( n) is a real Lie group of dimension n2. The Lie algebra of U( n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator. The general unitary group (also called the group of unitary similitudes ) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix. Contents Properties Topology Related groups 2-out-of-3 property Special unitary and projective unitary groups G-structure: almost Hermitian Generalizations Indefinite forms Finite fields Degree-2 separable algebras Algebraic groups Unitary group of a quadratic module Polynomial invariants Classifying space See also Notes References Properties Since the determinant of a unitary matrix is a complex number with norm 1, the determinant gives a group 1 of 7 2/23/2018, 10:13 AM Unitary group - Wikipedia https://en.wikipedia.org/wiki/Unitary_group homomorphism The kernel of this homomorphism is the set of unitary matrices with determinant 1. -
Material on Algebraic and Lie Groups
2 Lie groups and algebraic groups. 2.1 Basic Definitions. In this subsection we will introduce the class of groups to be studied. We first recall that a Lie group is a group that is also a differentiable manifold 1 and multiplication (x, y xy) and inverse (x x ) are C1 maps. An algebraic group is a group7! that is also an algebraic7! variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we first consider GL(n, C) as an affi ne algebraic group. Here Mn(C) denotes the space of n n matrices and GL(n, C) = g Mn(C) det(g) =) . Now Mn(C) is given the structure nf2 2 j 6 g of affi ne space C with the coordinates xij for X = [xij] . This implies that GL(n, C) is Z-open and as a variety is isomorphic with the affi ne variety 1 Mn(C) det . This implies that (GL(n, C)) = C[xij, det ]. f g O Lemma 1 If G is an algebraic group over an algebraically closed field, F , then every point in G is smooth. Proof. Let Lg : G G be given by Lgx = gx. Then Lg is an isomorphism ! 1 1 of G as an algebraic variety (Lg = Lg ). Since isomorphisms preserve the set of smooth points we see that if x G is smooth so is every element of Gx = G. 2 Proposition 2 If G is an algebraic group over an algebraically closed field F then the Z-connected components Proof. -
The Frattini Module and P -Automorphisms of Free Pro-P Groups
Publ. RIMS, Kyoto Univ. (), The Frattini module and p!-automorphisms of free pro-p groups. By Darren Semmen ∗ Abstract If a non-trivial subgroup A of the group of continuous automorphisms of a non- cyclic free pro-p group F has finite order, not divisible by p, then the group of fixed points FixF (A) has infinite rank. The semi-direct product F >!A is the universal p-Frattini cover of a finite group G, and so is the projective limit of a sequence of finite groups starting with G, each a canonical group extension of its predecessor by the Frattini module. Examining appearances of the trivial simple module 1 in the Frattini module’s Jordan-H¨older series arose in investigations ([FK97], [BaFr02] and [Sem02]) of modular towers. The number of these appearances prevents FixF (A) from having finite rank. For any group A of automorphisms of a group Γ, the set of fixed points FixΓ(A) := {g ∈ Γ | α(g) = g, ∀α ∈ A} of Γ under the action of A is a subgroup of Γ. Nielsen [N21] and, for the infinite rank case, Schreier [Schr27] showed that any subgroup of a free discrete group will be free. Tate (cf. [Ser02, I.§4.2, Cor. 3a]) extended this to free pro-p groups. In light of this, it is natural to ask for a free group F , what is the rank of FixF (A)? When F is a free discrete group and A is finite, Dyer and Scott [DS75] demonstrated that FixF (A) is a free factor of F , i.e.