DOCSLIB.ORG

Group Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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 [206] is the most appropriate if particle physics is the primary focus of interest. The book by G. Costa and G. Fogli [100] 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 [418]. Still a classic is [248]; 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 ◦gj ∈ G, (ii) a neutral element g0 ∈ G exists such that gi ◦ g0 = g0 ◦ gi = gi, (iii) for every gi exists an inverse −1 ∈ ◦ −1 −1 ◦ element gi G such that gi gi = g0 = gi gi, (iv) the operation is associative: gi ◦ (gj ◦ gk)=(gi ◦ gj) ◦ gk. The closure property can be expressed K. Sundermeyer, Symmetries in Fundamental Physics, 493 Fundamental Theories of Physics 176, DOI 10.1007/978-94-007-7642-5, c Springer Science+Business Media B.V. 2014 494 A Group Theory Table A.1 The group D4 1 i -1 -i C Ci -C -Ci 1 1 i -1 -i C Ci -C -Ci i i -1 -i 1 -Ci C Ci -C -1 -1 -i 1 i -C -Ci C Ci -i -i 1 i -1 Ci -C -Ci C C C Ci -C -Ci 1 i -1 -i Ci Ci -C -Ci C -i 1 i -1 -C -C -Ci C Ci -1 -i 1 i -Ci -Ci C Ci -C i -1 -i 1 by considering the operation as a mapping ◦ : G × G → G. There is indeed only one neutral element and the inverse of a group element is unique, as easily can be shown. A group is called finite if the number of its elements (also called the order of the group) is finite. An example of a finite group is the rotations of a square by ◦ 90 (in mathematics and crystallography called C4; a group of order 4). Examples of infinite groups are the integers with respect to addition (countably infinite) and the symmetry group of the circle, characterized by a parameter ϕ [0 ≤ ϕ<2π] (continuous group). The center of a group is defined as the set of those elements of G which commute with all other elements of the group: Cent(G)={c ∈ G | cg = gc for all g ∈ G}. Example: From the multiplication table of D4 (see Table A.1; more about this group later), one can see that its center is given by the set {1, −1}. 1 A group is called Abelian if the operation is commutative: gi ◦ gj = gj ◦ gi. Obviously the center of an Abelian group is the group itself. The symmetry group C4 of the square is Abelian; the analogous group of the cube is an example of a non–Abelian group. In physics we often deal with groups (or subgroups of) GL(n, F). This is the set of non–singular n × n matrices with elements from the field F which are in the majority of cases real numbers R or complex numbers C. GL(n, F) is in general a non–Abelian group with matrix multiplication as the group operation. The restriction to non–singular matrices is necessary because otherwise the inverse would not be defined. A subset of G, which itself is a group, is called a subgroup of G. By this definition, every group has two trivial subgroups, namely itself (G ⊂ G) and the set consisting of the neutral element ({g0}⊂G). Although in the context of symmetries in fundamental physics, continuous group are the most important ones, many notions from group theory will subsequently 1 This term derives from the name of the Norwegian mathematician Niels HenrikAbel (1802–1829). He is treated with some disrespect by the fact that contemporary theoreticians use his name in a negated form, by talking for instance of “non–Abelian” gauge theories. A.1 Basics 495 i 1 i––1 i b –1 1 C C Ci –C –Ci a –i c Fig. A.1 Symmetries of the square be explained on a finite group. This is the group D4, defined by its multiplication table above. The notation of its elements is taken from [400]. D4 is a group of order 8, its neutral element is (1), and it is non–Abelian (for instance (Ci)C = −i, but C(Ci)=i). This abstract group is realized by 90◦-rotations of the square and reflections along its diagonals; see Fig. A.1. Here the vertices of the square are represented by the points 1, i, –1, –i. The group D4 has various nontrivial subgroups; these are ◦ • order 2: rotations by 180 realized by C2 = {1, −1}, and four reflections realized by {1,C}, {1,Ci}, {1, −C}, {1, −Ci}. ◦ • order 4: rotations by 90 realized by C4 = {1,i,−1, −i}, reflections along diagonals, that is {1, −1,C,−C}, and reflections along the two parallels to the edges of the square: {1, −1,Ci,−Ci}. Field A field is a set F with elements fi together with two operations +: F × F → F ∗ : F × F → F with the properties that (i) F is an Abelian group under the operation + with f0 as the identity, (ii) F \{f0} is a group under ∗, and (iii) there is a distributive law fi ∗ (fj + fk)=fi ∗ fj + fi ∗ fk, (fi + fj) ∗ fk = fi ∗ fk + fj ∗ fk. Prominent examples of fields are the real and the complex numbers under the usual addition and multiplication operations. Further fields are known, as for instance quaternions (building a 4–dimensional number system and having some link to Dirac matrices), octonions, sedenions, ... the latter bearing a relation to the exceptional Lie groups. 496 A Group Theory Vector Space A (linear) vector space consists of a set V with elements vi and a field F together with two operations +: V × V → V: F × V → V with the properties that (i) (V,+) is an Abelian group, (ii) the operation fulfills fi (fj vk)=(fi ∗ fj) vk 1 vi = vi = vi 1 fi (vj + vk)=fi vj + fi vk, (fi + fj) vk = fi vk + fj vk. Further definitions: • ∈ F A set of vectors vk is called linearly dependent iff fj exist such that fk vk ≡ 0. k • The dimension D of a vector space is defined as the maximal number of linearly independent vectors. • In every vector space of dimension D one can find a set of basis vectors ek which are linearly independent and which make it possible to expand any vector D as v = fjej. Prominent examples of vector spaces are the Euclidean space, the solution space of linear differential and integral operators, the sets of N × M–matrices, and Hilbert spaces. Algebra A (linear) algebra A consists of a set V with elements vi, a field F together with three operations +,,. The algebra is a vector space with respect to (V,F, +,). The is a mapping : V × V → V with (vi + vj)vk = vivk + vjvk vi2(vj + vk)=vi2vj + vi2vk. Prominent examples are Lie algebras, which aside from the algebra properties additionally fulfill vi2(avj + bvk)=avi2vj + bvi2vj,a,b∈ F vi2vj = −vj2vi vi2(vj2vk)=(vi2vj)2vk + vj2(vi2vk), A.1 Basics 497 where the latter relation is the Jacobi identity, as it is known in physics for instance for the Poisson brackets or the generators of bosonic/Grassmann-even2 symmetry transformations. A.1.2 Mapping of Groups Group Homomorphism Let f be a group mapping f : G → Gˆ . If the mapping respects the group structure, that is if ◦ ◦ f(g1) Gˆ f(g2)=f(g1 G g2), (where ◦F denotes the operation in the respective group) it is called a (group) homo- morphism. The definition of a homomorphism can be generalized to fields, vector spaces, and algebras as structure–preserving maps between two of these algebraic structures. The kernel of the homomorphism f is defined as the subset of G for which each element is mapped to the neutral element in Gˆ : Kerf := {g ∈ G | f(g)=ˆg0 ∈ Gˆ }. The definition of homomorphism does not exclude that f(g1)=f(g2) for g1 = g2 (a so-called “many-to-one” relation). For instance, mapping all elements of an ar- bitrary group to the real number 1 is a homomorphism between the group and the group of real numbers. A “one-to-one” mapping is called an isomorphism. Isomor- phic groups are not distinguishable in their mathematical structure; one denotes ∼ G = Gˆ .If G = Gˆ the map f is called an automorphism.Aninner automorphism −1 is a mapping fh such that fh → g hg. For finite groups an isomorphism between two groups is easily detected in that both groups have identical multiplication tables.
Load more

Recommended publications
  • Classification of Finite Abelian Groups
    Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k.
    [Show full text]
  • Mathematics 310 Examination 1 Answers 1. (10 Points) Let G Be A
    Mathematics 310 Examination 1 Answers 1. (10 points) Let G be a group, and let x be an element of G. Finish the following definition: The order of x is ... Answer: . the smallest positive integer n so that xn = e. 2. (10 points) State Lagrange’s Theorem. Answer: If G is a finite group, and H is a subgroup of G, then o(H)|o(G). 3. (10 points) Let ( a 0! ) H = : a, b ∈ Z, ab 6= 0 . 0 b Is H a group with the binary operation of matrix multiplication? Be sure to explain your answer fully. 2 0! 1/2 0 ! Answer: This is not a group. The inverse of the matrix is , which is not 0 2 0 1/2 in H. 4. (20 points) Suppose that G1 and G2 are groups, and φ : G1 → G2 is a homomorphism. (a) Recall that we defined φ(G1) = {φ(g1): g1 ∈ G1}. Show that φ(G1) is a subgroup of G2. −1 (b) Suppose that H2 is a subgroup of G2. Recall that we defined φ (H2) = {g1 ∈ G1 : −1 φ(g1) ∈ H2}. Prove that φ (H2) is a subgroup of G1. Answer:(a) Pick x, y ∈ φ(G1). Then we can write x = φ(a) and y = φ(b), with a, b ∈ G1. Because G1 is closed under the group operation, we know that ab ∈ G1. Because φ is a homomorphism, we know that xy = φ(a)φ(b) = φ(ab), and therefore xy ∈ φ(G1). That shows that φ(G1) is closed under the group operation.
    [Show full text]
  • The Unitary Representations of the Poincaré Group in Any Spacetime
    The unitary representations of the Poincar´e group in any spacetime dimension Xavier Bekaert a and Nicolas Boulanger b a Institut Denis Poisson, Unit´emixte de Recherche 7013, Universit´ede Tours, Universit´ed’Orl´eans, CNRS, Parc de Grandmont, 37200 Tours (France) [email protected] b Service de Physique de l’Univers, Champs et Gravitation Universit´ede Mons – UMONS, Place du Parc 20, 7000 Mons (Belgium) [email protected] An extensive group-theoretical treatment of linear relativistic field equa- tions on Minkowski spacetime of arbitrary dimension D > 2 is presented in these lecture notes. To start with, the one-to-one correspondence be- tween linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representa- tions reduces the problem of classifying the representations of the Poincar´e group ISO(D 1, 1) to the classification of the representations of the sta- − bility subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the “helicity” and the “infinite-spin” representations) may be performed via the well-known representation theory of the orthogonal groups O(n) (with D 4 <n<D ). Finally, covariant field equations are given for each − unitary irreducible representation of the Poincar´egroup with non-negative arXiv:hep-th/0611263v2 13 Jun 2021 mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal groups in terms of Young diagrams.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Two-Dimensional Rotational Kinematics Rigid Bodies
    Rigid Bodies A rigid body is an extended object in which the Two-Dimensional Rotational distance between any two points in the object is Kinematics constant in time. Springs or human bodies are non-rigid bodies. 8.01 W10D1 Rotation and Translation Recall: Translational Motion of of Rigid Body the Center of Mass Demonstration: Motion of a thrown baton • Total momentum of system of particles sys total pV= m cm • External force and acceleration of center of mass Translational motion: external force of gravity acts on center of mass sys totaldp totaldVcm total FAext==mm = cm Rotational Motion: object rotates about center of dt dt mass 1 Main Idea: Rotation of Rigid Two-Dimensional Rotation Body Torque produces angular acceleration about center of • Fixed axis rotation: mass Disc is rotating about axis τ total = I α passing through the cm cm cm center of the disc and is perpendicular to the I plane of the disc. cm is the moment of inertial about the center of mass • Plane of motion is fixed: α is the angular acceleration about center of mass cm For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating Rotational Kinematics Fixed Axis Rotation: Angular for Fixed Axis Rotation Velocity Angle variable θ A point like particle undergoing circular motion at a non-constant speed has SI unit: [rad] dθ ω ≡≡ω kkˆˆ (1)An angular velocity vector Angular velocity dt SI unit: −1 ⎣⎡rad⋅ s ⎦⎤ (2) an angular acceleration vector dθ Vector: ω ≡ Component dt dθ ω ≡ magnitude dt ω >+0, direction kˆ direction ω < 0, direction − kˆ 2 Fixed Axis Rotation: Angular Concept Question: Angular Acceleration Speed 2 ˆˆd θ Object A sits at the outer edge (rim) of a merry-go-round, and Angular acceleration: α ≡≡α kk2 object B sits halfway between the rim and the axis of rotation.
    [Show full text]
  • Matrix Lie Groups
    Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2).
    [Show full text]
  • SPINORS and SPACE–TIME ANISOTROPY
    Sergiu Vacaru and Panayiotis Stavrinos SPINORS and SPACE{TIME ANISOTROPY University of Athens ————————————————— c Sergiu Vacaru and Panyiotis Stavrinos ii - i ABOUT THE BOOK This is the first monograph on the geometry of anisotropic spinor spaces and its applications in modern physics. The main subjects are the theory of grav- ity and matter fields in spaces provided with off–diagonal metrics and asso- ciated anholonomic frames and nonlinear connection structures, the algebra and geometry of distinguished anisotropic Clifford and spinor spaces, their extension to spaces of higher order anisotropy and the geometry of gravity and gauge theories with anisotropic spinor variables. The book summarizes the authors’ results and can be also considered as a pedagogical survey on the mentioned subjects. ii - iii ABOUT THE AUTHORS Sergiu Ion Vacaru was born in 1958 in the Republic of Moldova. He was educated at the Universities of the former URSS (in Tomsk, Moscow, Dubna and Kiev) and reveived his PhD in theoretical physics in 1994 at ”Al. I. Cuza” University, Ia¸si, Romania. He was employed as principal senior researcher, as- sociate and full professor and obtained a number of NATO/UNESCO grants and fellowships at various academic institutions in R. Moldova, Romania, Germany, United Kingdom, Italy, Portugal and USA. He has published in English two scientific monographs, a university text–book and more than hundred scientific works (in English, Russian and Romanian) on (super) gravity and string theories, extra–dimension and brane gravity, black hole physics and cosmolgy, exact solutions of Einstein equations, spinors and twistors, anistoropic stochastic and kinetic processes and thermodynamics in curved spaces, generalized Finsler (super) geometry and gauge gravity, quantum field and geometric methods in condensed matter physics.
    [Show full text]
  • Council for Innovative Research Peer Review Research Publishing System
    ISSN 2347-3487 Einstein's gravitation is Einstein-Grossmann's equations Alfonso Leon Guillen Gomez Independent scientific researcher, Bogota, Colombia E-mail: [email protected] Abstract While the philosophers of science discuss the General Relativity, the mathematical physicists do not question it. Therefore, there is a conflict. From the theoretical point view “the question of precisely what Einstein discovered remains unanswered, for we have no consensus over the exact nature of the theory's foundations. Is this the theory that extends the relativity of motion from inertial motion to accelerated motion, as Einstein contended? Or is it just a theory that treats gravitation geometrically in the spacetime setting?”. “The voices of dissent proclaim that Einstein was mistaken over the fundamental ideas of his own theory and that their basic principles are simply incompatible with this theory. Many newer texts make no mention of the principles Einstein listed as fundamental to his theory; they appear as neither axiom nor theorem. At best, they are recalled as ideas of purely historical importance in the theory's formation. The very name General Relativity is now routinely condemned as a misnomer and its use often zealously avoided in favour of, say, Einstein's theory of gravitation What has complicated an easy resolution of the debate are the alterations of Einstein's own position on the foundations of his theory”, (Norton, 1993) [1]. Of other hand from the mathematical point view the “General Relativity had been formulated as a messy set of partial differential equations in a single coordinate system. People were so pleased when they found a solution that they didn't care that it probably had no physical significance” (Hawking and Penrose, 1996) [2].
    [Show full text]
  • Relativity Without Tears
    Vol. 39 (2008) ACTA PHYSICA POLONICA B No 4 RELATIVITY WITHOUT TEARS Z.K. Silagadze Budker Institute of Nuclear Physics and Novosibirsk State University 630 090, Novosibirsk, Russia (Received December 21, 2007) Special relativity is no longer a new revolutionary theory but a firmly established cornerstone of modern physics. The teaching of special relativ- ity, however, still follows its presentation as it unfolded historically, trying to convince the audience of this teaching that Newtonian physics is natural but incorrect and special relativity is its paradoxical but correct amend- ment. I argue in this article in favor of logical instead of historical trend in teaching of relativity and that special relativity is neither paradoxical nor correct (in the absolute sense of the nineteenth century) but the most nat- ural and expected description of the real space-time around us valid for all practical purposes. This last circumstance constitutes a profound mystery of modern physics better known as the cosmological constant problem. PACS numbers: 03.30.+p Preface “To the few who love me and whom I love — to those who feel rather than to those who think — to the dreamers and those who put faith in dreams as in the only realities — I offer this Book of Truths, not in its character of Truth-Teller, but for the Beauty that abounds in its Truth; constituting it true. To these I present the composition as an Art-Product alone; let us say as a Romance; or, if I be not urging too lofty a claim, as a Poem. What I here propound is true: — therefore it cannot die: — or if by any means it be now trodden down so that it die, it will rise again ‘to the Life Everlasting’.
    [Show full text]
  • SEMISIMPLE WEAKLY SYMMETRIC PSEUDO–RIEMANNIAN MANIFOLDS 1. Introduction There Have Been a Number of Important Extensions of Th
    SEMISIMPLE WEAKLY SYMMETRIC PSEUDO{RIEMANNIAN MANIFOLDS ZHIQI CHEN AND JOSEPH A. WOLF Abstract. We develop the classification of weakly symmetric pseudo{riemannian manifolds G=H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G=H and the signature of the invariant pseudo{riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature (n − 1; 1) and trans{lorentzian signature (n − 2; 2). 1. Introduction There have been a number of important extensions of the theory of riemannian symmetric spaces. Weakly symmetric spaces, introduced by A. Selberg [8], play key roles in number theory, riemannian geometry and harmonic analysis. See [11]. Pseudo{riemannian symmetric spaces, including semisimple symmetric spaces, play central but complementary roles in number theory, differential geometry and relativity, Lie group representation theory and harmonic analysis. Much of the activity there has been on the Lorentz cases, which are of particular interest in physics. Here we study the classification of weakly symmetric pseudo{ riemannian manifolds G=H where G is a semisimple Lie group and H is a reductive subgroup. We apply the result to obtain explicit listings for metrics of riemannian (Table 5.1), lorentzian (Table 5.2) and trans{ lorentzian (Table 5.3) signature. The pseudo{riemannian symmetric case was analyzed extensively in the classical paper of M. Berger [1]. Our analysis in the weakly symmetric setting starts with the classifications of Kr¨amer[6], Brion [2], Mikityuk [7] and Yakimova [16, 17] for the weakly symmetric riemannian manifolds.
    [Show full text]
  • Chapter 25 Finite Simple Groups
    Chapter 25 Finite Simple Groups Chapter 25 Finite Simple Groups Historical Background Definition A group is simple if it has no nontrivial proper normal subgroup. The definition was proposed by Galois; he showed that An is simple for n ≥ 5 in 1831. It is an important step in showing that one cannot express the solutions of a quintic equation in radicals. If possible, one would factor a group G as G0 = G, find a normal subgroup G1 of maximum order to form G0/G1. Then find a maximal normal subgroup G2 of G1 and get G1/G2, and so on until we get the composition factors: G0/G1,G1/G2,...,Gn−1/Gn, with Gn = {e}. Jordan and Hölder proved that these factors are independent of the choices of the normal subgroups in the process. Jordan in 1870 found four infinite series including: Zp for a prime p, SL(n, Zp)/Z(SL(n, Zp)) except when (n, p) = (2, 2) or (2, 3). Between 1982-1905, Dickson found more infinite series; Miller and Cole showed that 5 (sporadic) groups constructed by Mathieu in 1861 are simple. Chapter 25 Finite Simple Groups In 1950s, more infinite families were found, and the classification project began. Brauer observed that the centralizer has an order 2 element is important; Feit-Thompson in 1960 confirmed the 1900 conjecture that non-Abelian simple group must have even order. From 1966-75, 19 new sporadic groups were found. Thompson developed many techniques in the N-group paper. Gorenstein presented an outline for the classification project in a lecture series at University of Chicago in 1972.
    [Show full text]