Unitary Group - Wikipedia
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
INTEGER POINTS and THEIR ORTHOGONAL LATTICES 2 to Remove the Congruence Condition
INTEGER POINTS ON SPHERES AND THEIR ORTHOGONAL LATTICES MENNY AKA, MANFRED EINSIEDLER, AND URI SHAPIRA (WITH AN APPENDIX BY RUIXIANG ZHANG) Abstract. Linnik proved in the late 1950’s the equidistribution of in- teger points on large spheres under a congruence condition. The congru- ence condition was lifted in 1988 by Duke (building on a break-through by Iwaniec) using completely different techniques. We conjecture that this equidistribution result also extends to the pairs consisting of a vector on the sphere and the shape of the lattice in its orthogonal complement. We use a joining result for higher rank diagonalizable actions to obtain this conjecture under an additional congruence condition. 1. Introduction A theorem of Legendre, whose complete proof was given by Gauss in [Gau86], asserts that an integer D can be written as a sum of three squares if and only if D is not of the form 4m(8k + 7) for some m, k N. Let D = D N : D 0, 4, 7 mod8 and Z3 be the set of primitive∈ vectors { ∈ 6≡ } prim in Z3. Legendre’s Theorem also implies that the set 2 def 3 2 S (D) = v Zprim : v 2 = D n ∈ k k o is non-empty if and only if D D. This important result has been refined in many ways. We are interested∈ in the refinement known as Linnik’s problem. Let S2 def= x R3 : x = 1 . For a subset S of rational odd primes we ∈ k k2 set 2 D(S)= D D : for all p S, D mod p F× . -
Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields. -
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. -
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 . -
Finite Resolutions of Modules for Reductive Algebraic Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA I@473488 (1986) Finite Resolutions of Modules for Reductive Algebraic Groups S. DONKIN School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England Communicafed by D. A. Buchsbaum Received January 22, 1985 INTRODUCTION A popular theme in recent work of Akin and Buchsbaum is the construc- tion of a finite left resolution of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a field or the integers. The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successivequotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic group theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. -
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; −). -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
1. INTRODUCTION 1.1. Introductory Remarks on Supersymmetry. 1.2
1. INTRODUCTION 1.1. Introductory remarks on supersymmetry. 1.2. Classical mechanics, the electromagnetic, and gravitational fields. 1.3. Principles of quantum mechanics. 1.4. Symmetries and projective unitary representations. 1.5. Poincar´esymmetry and particle classification. 1.6. Vector bundles and wave equations. The Maxwell, Dirac, and Weyl - equations. 1.7. Bosons and fermions. 1.8. Supersymmetry as the symmetry of Z2{graded geometry. 1.1. Introductory remarks on supersymmetry. The subject of supersymme- try (SUSY) is a part of the theory of elementary particles and their interactions and the still unfinished quest of obtaining a unified view of all the elementary forces in a manner compatible with quantum theory and general relativity. Supersymme- try was discovered in the early 1970's, and in the intervening years has become a major component of theoretical physics. Its novel mathematical features have led to a deeper understanding of the geometrical structure of spacetime, a theme to which great thinkers like Riemann, Poincar´e,Einstein, Weyl, and many others have contributed. Symmetry has always played a fundamental role in quantum theory: rotational symmetry in the theory of spin, Poincar´esymmetry in the classification of elemen- tary particles, and permutation symmetry in the treatment of systems of identical particles. Supersymmetry is a new kind of symmetry which was discovered by the physicists in the early 1970's. However, it is different from all other discoveries in physics in the sense that there has been no experimental evidence supporting it so far. Nevertheless an enormous effort has been expended by many physicists in developing it because of its many unique features and also because of its beauty and coherence1. -
Projective Unitary Representations of Infinite Dimensional Lie Groups
Projective unitary representations of infinite dimensional Lie groups Bas Janssens and Karl-Hermann Neeb January 5, 2015 Abstract For an infinite dimensional Lie group G modelled on a locally convex Lie algebra g, we prove that every smooth projective unitary represen- tation of G corresponds to a smooth linear unitary representation of a Lie group extension G] of G. (The main point is the smooth structure on G].) For infinite dimensional Lie groups G which are 1-connected, regular, and modelled on a barrelled Lie algebra g, we characterize the unitary g-representations which integrate to G. Combining these results, we give a precise formulation of the correspondence between smooth pro- jective unitary representations of G, smooth linear unitary representations of G], and the appropriate unitary representations of its Lie algebra g]. Contents 1 Representations of locally convex Lie groups 5 1.1 Smooth functions . 5 1.2 Locally convex Lie groups . 6 2 Projective unitary representations 7 2.1 Unitary representations . 7 2.2 Projective unitary representations . 7 3 Integration of Lie algebra representations 8 3.1 Derived representations . 9 3.2 Globalisation of Lie algebra representations . 10 3.2.1 Weak and strong topology on V . 11 3.2.2 Regular Lie algebra representations . 15 4 Projective unitary representations and central extensions 20 5 Smoothness of projective representations 25 5.1 Smoothness criteria . 25 5.2 Structure of the set of smooth rays . 26 1 6 Lie algebra extensions and cohomology 28 7 The main theorem 31 8 Covariant representations 33 9 Admissible derivations 36 9.1 Admissible derivations . -
Be the Integral Symplectic Group and S(G) Be the Set of All Positive Integers Which Can Occur As the Order of an Element in G
FINITE ORDER ELEMENTS IN THE INTEGRAL SYMPLECTIC GROUP KUMAR BALASUBRAMANIAN, M. RAM MURTY, AND KARAM DEO SHANKHADHAR Abstract For g 2 N, let G = Sp(2g; Z) be the integral symplectic group and S(g) be the set of all positive integers which can occur as the order of an element in G. In this paper, we show that S(g) is a bounded subset of R for all positive integers g. We also study the growth of the functions f(g) = jS(g)j, and h(g) = maxfm 2 N j m 2 S(g)g and show that they have at least exponential growth. 1. Introduction Given a group G and a positive integer m 2 N, it is natural to ask if there exists k 2 G such that o(k) = m, where o(k) denotes the order of the element k. In this paper, we make some observations about the collection of positive integers which can occur as orders of elements in G = Sp(2g; Z). Before we proceed further we set up some notations and briefly mention the questions studied in this paper. Let G = Sp(2g; Z) be the group of all 2g × 2g matrices with integral entries satisfying > A JA = J 0 I where A> is the transpose of the matrix A and J = g g . −Ig 0g α1 αk Throughout we write m = p1 : : : pk , where pi is a prime and αi > 0 for all i 2 f1; 2; : : : ; kg. We also assume that the primes pi are such that pi < pi+1 for 1 ≤ i < k. -
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). -
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+.