On the Configuration Space Topology in General Relativity
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Representation Theory and Quantum Mechanics Tutorial a Little Lie Theory
Representation theory and quantum mechanics tutorial A little Lie theory Justin Campbell July 6, 2017 1 Groups 1.1 The colloquial usage of the words \symmetry" and \symmetrical" is imprecise: we say, for example, that a regular pentagon is symmetrical, but what does that mean? In the mathematical parlance, a symmetry (or more technically automorphism) of the pentagon is a (distance- and angle-preserving) transformation which leaves it unchanged. It has ten such symmetries: 2π 4π 6π 8π rotations through 0, 5 , 5 , 5 , and 5 radians, as well as reflection over any of the five lines passing through a vertex and the center of the pentagon. These ten symmetries form a set D5, called the dihedral group of order 10. Any two elements of D5 can be composed to obtain a third. This operation of composition has some important formal properties, summarized as follows. Definition 1.1.1. A group is a set G together with an operation G × G ! G; denoted by (x; y) 7! xy, satisfying: (i) there exists e 2 G, called the identity, such that ex = xe = g for all x 2 G, (ii) any x 2 G has an inverse x−1 2 G satisfying xx−1 = x−1x = e, (iii) for any x; y; z 2 G the associativity law (xy)z = x(yz) holds. To any mathematical (geometric, algebraic, etc.) object one can attach its automorphism group consisting of structure-preserving invertible maps from the object to itself. For example, the automorphism group of a regular pentagon is the dihedral group D5. -
Homotopy Theory and Circle Actions on Symplectic Manifolds
HOMOTOPY AND GEOMETRY BANACH CENTER PUBLICATIONS, VOLUME 45 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1998 HOMOTOPY THEORY AND CIRCLE ACTIONS ON SYMPLECTIC MANIFOLDS JOHNOPREA Department of Mathematics, Cleveland State University Cleveland, Ohio 44115, U.S.A. E-mail: [email protected] Introduction. Traditionally, soft techniques of algebraic topology have found much use in the world of hard geometry. In recent years, in particular, the subject of symplectic topology (see [McS]) has arisen from important and interesting connections between sym- plectic geometry and algebraic topology. In this paper, we will consider one application of homotopical methods to symplectic geometry. Namely, we shall discuss some aspects of the homotopy theory of circle actions on symplectic manifolds. Because this paper is meant to be accessible to both geometers and topologists, we shall try to review relevant ideas in homotopy theory and symplectic geometry as we go along. We also present some new results (e.g. see Theorem 2.12 and x5) which extend the methods reviewed earlier. This paper then serves two roles: as an exposition and survey of the homotopical ap- proach to symplectic circle actions and as a first step to extending the approach beyond the symplectic world. 1. Review of symplectic geometry. A manifold M 2n is symplectic if it possesses a nondegenerate 2-form ! which is closed (i.e. d! = 0). The nondegeneracy condition is equivalent to saying that !n is a true volume form (i.e. nonzero at each point) on M. Furthermore, the nondegeneracy of ! sets up an isomorphism between 1-forms and vector fields on M by assigning to a vector field X the 1-form iX ! = !(X; −). -
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. -
4. Groups of Permutations 1
4. Groups of permutations 1 4. Groups of permutations Consider a set of n distinguishable objects, fB1;B2;B3;:::;Bng. These may be arranged in n! different ways, called permutations of the set. Permu- tations can also be thought of as transformations of a given ordering of the set into other orderings. A convenient notation for specifying a given permutation operation is 1 2 3 : : : n ! , a1 a2 a3 : : : an where the numbers fa1; a2; a3; : : : ; ang are the numbers f1; 2; 3; : : : ; ng in some order. This operation can be interpreted in two different ways, as follows. Interpretation 1: The object in position 1 in the initial ordering is moved to position a1, the object in position 2 to position a2,. , the object in position n to position an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the positions of objects in the ordered set. Interpretation 2: The object labeled 1 is replaced by the object labeled a1, the object labeled 2 by the object labeled a2,. , the object labeled n by the object labeled an. In this interpretation, the numbers in the two rows of the permutation symbol refer to the labels of the objects in the ABCD ! set. The labels need not be numerical { for instance, DCAB is a well-defined permutation which changes BDCA, for example, into CBAD. Either of these interpretations is acceptable, but one interpretation must be used consistently in any application. The particular application may dictate which is the appropriate interpretation to use. Note that, in either interpre- tation, the order of the columns in the permutation symbol is irrelevant { the columns may be written in any order without affecting the result, provided each column is kept intact. -
PERMUTATION GROUPS (16W5087)
PERMUTATION GROUPS (16w5087) Cheryl E. Praeger (University of Western Australia), Katrin Tent (University of Munster),¨ Donna Testerman (Ecole Polytechnique Fed´ erale´ de Lausanne), George Willis (University of Newcastle) November 13 - November 18, 2016 1 Overview of the Field Permutation groups are a mathematical approach to analysing structures by studying the rearrangements of the elements of the structure that preserve it. Finite permutation groups are primarily understood through combinatorial methods, while concepts from logic and topology come to the fore when studying infinite per- mutation groups. These two branches of permutation group theory are not completely independent however because techniques from algebra and geometry may be applied to both, and ideas transfer from one branch to the other. Our workshop brought together researchers on both finite and infinite permutation groups; the participants shared techniques and expertise and explored new avenues of study. The permutation groups act on discrete structures such as graphs or incidence geometries comprising points and lines, and many of the same intuitions apply whether the structure is finite or infinite. Matrix groups are also an important source of both finite and infinite permutation groups, and the constructions used to produce the geometries on which they act in each case are similar. Counting arguments are important when studying finite permutation groups but cannot be used to the same effect with infinite permutation groups. Progress comes instead by using techniques from model theory and descriptive set theory. Another approach to infinite permutation groups is through topology and approx- imation. In this approach, the infinite permutation group is locally the limit of finite permutation groups and results from the finite theory may be brought to bear. -
Characterized Subgroups of the Circle Group
Characterized subgroups of the circle group Characterized subgroups of the circle group Anna Giordano Bruno (University of Udine) 31st Summer Conference on Topology and its Applications Leicester - August 2nd, 2016 (Topological Methods in Algebra and Analysis) Characterized subgroups of the circle group Introduction p-torsion and torsion subgroup Introduction Let G be an abelian group and p a prime. The p-torsion subgroup of G is n tp(G) = fx 2 G : p x = 0 for some n 2 Ng: The torsion subgroup of G is t(G) = fx 2 G : nx = 0 for some n 2 N+g: The circle group is T = R=Z written additively (T; +); for r 2 R, we denoter ¯ = r + Z 2 T. We consider on T the quotient topology of the topology of R and µ is the (unique) Haar measure on T. 1 tp(T) = Z(p ). t(T) = Q=Z. Characterized subgroups of the circle group Introduction Topologically p-torsion and topologically torsion subgroup Let now G be a topological abelian group. [Bracconier 1948, Vilenkin 1945, Robertson 1967, Armacost 1981]: The topologically p-torsion subgroup of G is n tp(G) = fx 2 G : p x ! 0g: The topologically torsion subgroup of G is G! = fx 2 G : n!x ! 0g: Clearly, tp(G) ⊆ tp(G) and t(G) ⊆ G!. [Armacost 1981]: 1 tp(T) = tp(T) = Z(p ); e¯ 2 T!, bute ¯ 62 t(T) = Q=Z. Problem: describe T!. Characterized subgroups of the circle group Introduction Topologically p-torsion and topologically torsion subgroup [Borel 1991; Dikranjan-Prodanov-Stoyanov 1990; D-Di Santo 2004]: + For every x 2 [0; 1) there exists a unique (cn)n 2 NN such that 1 X cn x = ; (n + 1)! n=1 cn < n + 1 for every n 2 N+ and cn < n for infinitely many n 2 N+. -
Binding Groups, Permutation Groups and Modules of Finite Morley Rank Alexandre Borovik, Adrien Deloro
Binding groups, permutation groups and modules of finite Morley rank Alexandre Borovik, Adrien Deloro To cite this version: Alexandre Borovik, Adrien Deloro. Binding groups, permutation groups and modules of finite Morley rank. 2019. hal-02281140 HAL Id: hal-02281140 https://hal.archives-ouvertes.fr/hal-02281140 Preprint submitted on 9 Sep 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. BINDING GROUPS, PERMUTATION GROUPS AND MODULES OF FINITE MORLEY RANK ALEXANDRE BOROVIK AND ADRIEN DELORO If one has (or if many people have) spent decades classifying certain objects, one is apt to forget just why one started the project in the first place. Predictions of the death of group theory in 1980 were the pronouncements of just such amne- siacs. [68, p. 4] Contents 1. Introduction and background 2 1.1. The classification programme 2 1.2. Concrete groups of finite Morley rank 3 2. Binding groups and bases 4 2.1. Binding groups 4 2.2. Bases and parametrisation 5 2.3. Sharp bounds for bases 6 3. Permutation groups of finite Morley rank 7 3.1. Highly generically multiply transitive groups 7 3.2. -
Solutions to Exercises for Mathematics 205A
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A | Part 6 Fall 2008 APPENDICES Appendix A : Topological groups (Munkres, Supplementary exercises following $ 22; see also course notes, Appendix D) Problems from Munkres, x 30, pp. 194 − 195 Munkres, x 26, pp. 170{172: 12, 13 Munkres, x 30, pp. 194{195: 18 Munkres, x 31, pp. 199{200: 8 Munkres, x 33, pp. 212{214: 10 Solutions for these problems will not be given. Additional exercises Notation. Let F be the real or complex numbers. Within the matrix group GL(n; F) there are certain subgroups of particular importance. One such subgroup is the special linear group SL(n; F) of all matrices of determinant 1. 0. Prove that the group SL(2; C) has no nontrivial proper normal subgroups except for the subgroup { Ig. [Hint: If N is a normal subgroup, show first that if A 2 N then N contains all matrices that are similar to A. Therefore the proof reduces to considering normal subgroups containing a Jordan form matrix of one of the following two types: α 0 " 1 ; 0 α−1 0 " Here α is a complex number not equal to 0 or 1 and " = 1. The idea is to show that if N contains one of these Jordan forms then it contains all such forms, and this is done by computing sufficiently many matrix products. Trial and error is a good way to approach this aspect of the problem.] SOLUTION. We shall follow the steps indicated in the hint. If N is a normal subgroup of SL(2; C) and A 2 N then N contains all matrices that are similar to A. -
The Classical Groups and Domains 1. the Disk, Upper Half-Plane, SL 2(R
(June 8, 2018) The Classical Groups and Domains Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ The complex unit disk D = fz 2 C : jzj < 1g has four families of generalizations to bounded open subsets in Cn with groups acting transitively upon them. Such domains, defined more precisely below, are bounded symmetric domains. First, we recall some standard facts about the unit disk, the upper half-plane, the ambient complex projective line, and corresponding groups acting by linear fractional (M¨obius)transformations. Happily, many of the higher- dimensional bounded symmetric domains behave in a manner that is a simple extension of this simplest case. 1. The disk, upper half-plane, SL2(R), and U(1; 1) 2. Classical groups over C and over R 3. The four families of self-adjoint cones 4. The four families of classical domains 5. Harish-Chandra's and Borel's realization of domains 1. The disk, upper half-plane, SL2(R), and U(1; 1) The group a b GL ( ) = f : a; b; c; d 2 ; ad − bc 6= 0g 2 C c d C acts on the extended complex plane C [ 1 by linear fractional transformations a b az + b (z) = c d cz + d with the traditional natural convention about arithmetic with 1. But we can be more precise, in a form helpful for higher-dimensional cases: introduce homogeneous coordinates for the complex projective line P1, by defining P1 to be a set of cosets u 1 = f : not both u; v are 0g= × = 2 − f0g = × P v C C C where C× acts by scalar multiplication. -
6 Permutation Groups
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 6 Permutation Groups Let S be a nonempty set and M(S) be the collection of all mappings from S into S. In this section, we will emphasize on the collection of all invertible mappings from S into S. The elements of this set will be called permutations because of Theorem 2.4 and the next definition. Definition 6.1 Let S be a nonempty set. A one-to-one mapping from S onto S is called a permutation. Consider the collection of all permutations on S. Then this set is a group with respect to composition. Theorem 6.1 The set of all permutations of a nonempty set S is a group with respect to composition. This group is called the symmetric group on S and will be denoted by Sym(S). Proof. By Theorem 2.4, the set of all permutations on S is just the set I(S) of all invertible mappings from S to S. According to Theorem 4.3, this set is a group with respect to composition. Definition 6.2 A group of permutations , with composition as the operation, is called a permutation group on S. Example 6.1 1. Sym(S) is a permutation group. 2. The collection L of all invertible linear functions from R to R is a permu- tation group with respect to composition.(See Example 4.4.) Note that L is 1 a proper subset of Sym(R) since we can find a function in Sym(R) which is not in L, namely, the function f(x) = x3. -
CS E6204 Lecture 5 Automorphisms
CS E6204 Lecture 5 Automorphisms Abstract An automorphism of a graph is a permutation of its vertex set that preserves incidences of vertices and edges. Under composition, the set of automorphisms of a graph forms what algbraists call a group. In this section, graphs are assumed to be simple. 1. The Automorphism Group 2. Graphs with Given Group 3. Groups of Graph Products 4. Transitivity 5. s-Regularity and s-Transitivity 6. Graphical Regular Representations 7. Primitivity 8. More Automorphisms of Infinite Graphs * This lecture is based on chapter [?] contributed by Mark E. Watkins of Syracuse University to the Handbook of Graph The- ory. 1 1 The Automorphism Group Given a graph X, a permutation α of V (X) is an automor- phism of X if for all u; v 2 V (X) fu; vg 2 E(X) , fα(u); α(v)g 2 E(X) The set of all automorphisms of a graph X, under the operation of composition of functions, forms a subgroup of the symmetric group on V (X) called the automorphism group of X, and it is denoted Aut(X). Figure 1: Example 2.2.3 from GTAIA. Notation The identity of any permutation group is denoted by ι. (JG) Thus, Watkins would write ι instead of λ0 in Figure 1. 2 Rigidity and Orbits A graph is asymmetric if the identity ι is its only automor- phism. Synonym: rigid. Figure 2: The smallest rigid graph. (JG) The orbit of a vertex v in a graph G is the set of all vertices α(v) such that α is an automorphism of G. -
Quasi P Or Not Quasi P? That Is the Question
Rose-Hulman Undergraduate Mathematics Journal Volume 3 Issue 2 Article 2 Quasi p or not Quasi p? That is the Question Ben Harwood Northern Kentucky University, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Harwood, Ben (2002) "Quasi p or not Quasi p? That is the Question," Rose-Hulman Undergraduate Mathematics Journal: Vol. 3 : Iss. 2 , Article 2. Available at: https://scholar.rose-hulman.edu/rhumj/vol3/iss2/2 Quasi p- or not quasi p-? That is the Question.* By Ben Harwood Department of Mathematics and Computer Science Northern Kentucky University Highland Heights, KY 41099 e-mail: [email protected] Section Zero: Introduction The question might not be as profound as Shakespeare’s, but nevertheless, it is interesting. Because few people seem to be aware of quasi p-groups, we will begin with a bit of history and a definition; and then we will determine for each group of order less than 24 (and a few others) whether the group is a quasi p-group for some prime p or not. This paper is a prequel to [Hwd]. In [Hwd] we prove that (Z3 £Z3)oZ2 and Z5 o Z4 are quasi 2-groups. Those proofs now form a portion of Proposition (12.1) It should also be noted that [Hwd] may also be found in this journal. Section One: Why should we be interested in quasi p-groups? In a 1957 paper titled Coverings of algebraic curves [Abh2], Abhyankar conjectured that the algebraic fundamental group of the affine line over an algebraically closed field k of prime characteristic p is the set of quasi p-groups, where by the algebraic fundamental group of the affine line he meant the family of all Galois groups Gal(L=k(X)) as L varies over all finite normal extensions of k(X) the function field of the affine line such that no point of the line is ramified in L, and where by a quasi p-group he meant a finite group that is generated by all of its p-Sylow subgroups.