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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. -
An Overview of Topological Groups: Yesterday, Today, Tomorrow
axioms Editorial An Overview of Topological Groups: Yesterday, Today, Tomorrow Sidney A. Morris 1,2 1 Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia; [email protected]; Tel.: +61-41-7771178 2 Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia Academic Editor: Humberto Bustince Received: 18 April 2016; Accepted: 20 April 2016; Published: 5 May 2016 It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin’s book “Fourier Analysis on Groups” [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is “Introduction to Topological Groups” [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book “Topological Groups” [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: “This book belongs to that rare category of mathematical works that can truly be called classical - books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians”. The final book I mention from my graduate studies days is “Topological Transformation Groups” [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert’s fifth problem as well as a structure theory for locally compact non-abelian groups. -
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+. -
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. -
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). -
Topological Groups
Topological Groups Siyan Daniel Li-Huerta September 8, 2020 Definition A topological group is a group G with a topological space structure such that The multiplication map µ : G × G ! G is continuous, The inverse map ι : G ! G is continuous. Example Any group G with the discrete topology, The group R under addition with the Euclidean topology, × The group C under multiplication with the Euclidean topology, 1 × × The subgroup S = fz 2 C j jzj = 1g ⊂ C , The group GLn(R) = fA 2 Matn×n(R) j det A 6= 0g under multiplication with the Euclidean topology. Remark Any finite Hausdorff topological group G must be discrete. 2 / 9 Let G be a topological group, and let g be in G. Then left translation (i.e. multiplication) by g equals the composite (g;id) µ G / fgg × G / G; so it's continuous. Its inverse is left translation by g −1, so it's even a homeomorphism. The same holds for right translation. Definition Let X be a topological space. We say X is homogeneous if, for every x and y in X , there exists a homeomorphism f : X ! X such that f (x) = y. Example Any topological group G is homogeneous (using left translation), n n+1 The n-sphere S = f~v 2 R j k~vk= 1g is homogeneous (using rotations). Thus topological groups are very special topological spaces: they look \the same" around every point. So it often suffices to study neighborhoods of one point. Let's pick the point 1! 3 / 9 Proposition Let G be a topological group, and let U be a neighborhood of 1. -
A Linear Algebraic Group? Skip Garibaldi
WHATIS... ? a Linear Algebraic Group? Skip Garibaldi From a marketing perspective, algebraic groups group over F, a Lie group, or a topological group is are poorly named.They are not the groups you met a “group object” in the category of affine varieties as a student in abstract algebra, which I will call over F, smooth manifolds, or topological spaces, concrete groups for clarity. Rather, an algebraic respectively. For algebraic groups, this implies group is the analogue in algebra of a topological that the set G(K) is a concrete group for each field group (fromtopology) or a Liegroup (from analysis K containing F. and geometry). Algebraic groups provide a unifying language Examples for apparently different results in algebra and The basic example is the general linear group number theory. This unification can not only sim- GLn for which GLn(K) is the concrete group of plify proofs, it can also suggest generalizations invertible n-by-n matrices with entries in K. It is the and bring new tools to bear, such as Galois co- collection of solutions (t, X)—with t ∈ K and X an homology, Steenrod operations in Chow theory, n-by-n matrix over K—to the polynomial equation etc. t ·det X = 1. Similar reasoning shows that familiar matrix groups such as SLn, orthogonal groups, Definitions and symplectic groups can be viewed as linear A linear algebraic group over a field F is a smooth algebraic groups. The main difference here is that affine variety over F that is also a group, much instead of viewing them as collections of matrices like a topological group is a topological space over F, we view them as collections of matrices that is also a group and a Lie group is a smooth over every extension K of F. -
Smooth Actions of the Circle Group on Exotic Spheres
Pacific Journal of Mathematics SMOOTH ACTIONS OF THE CIRCLE GROUP ON EXOTIC SPHERES V. J. JOSEPH Vol. 95, No. 2 October 1981 PACIFIC JOURNAL OF MATHEMATICS Vol. 95, No. 2, 1981 SMOOTH ACTIONS OF THE CIRCLE GROUP ON EXOTIC SPHERES VAPPALA J. JOSEPH Recent work of Schultz translates the question of which exotic spheres Sn admit semif ree circle actions with λ>dimen- sional fixed point set entirely to problems in homotopy theory provided the spheres bound spin manifolds. In this article we study circle actions on homotopy spheres not bounding spin manifolds and prove, in particular, that the spin boundary hypothesis can be dropped if (n—k) is not divisible by 128. It is also proved that any ordinary sphere can be realized as the fixed point set of such a circle action on a homotopy sphere which is not a spin boundary; some of these actions are not necessarily semi-free. This extends earlier results obtained by Bredon and Schultz. The Adams conjecture, its consequences regarding splittings of certain classifying spaces and standard results of simply-connected surgery are used to construct the actions. The computations involved relate to showing that certain surgery obstructions vanish. I* Introduction* Results due to Schultz give a purely homo- topy theoretic characterization of those homotopy (n + 2&)-spheres admitting smooth semi-free circle actions with ^-dimensional fixed point sets provided one limits attention to exotic spheres bounding spin manifolds. The method of proof is similar to that described in [14] for actions of prime order cyclic groups; a detailed account will appear in [21]. -
Introduction to Fourier Methods
Appendix L Introduction to Fourier Methods L.1. The Circle Group and its Characters We will denote by S the unit circle in the complex plane: S = {(x, y) ∈ R2 | x2 + y2 =1} = {(cos θ, sin θ) ∈ R2 | 0 ≤ θ< 2π} = {z ∈ C | |z| =1} = {eiθ | 0 ≤ θ< 2π}. We note that S is a group under multiplication, i.e., if z1 and z2 are in S then so is z1z2. and if z = x + yi ∈ S thenz ¯ = x − yi also is in S with zz¯ =zz ¯ =1, soz ¯ = z−1 = 1/z. In particular then, given any function f defined on S, and any z0 in S, we can define another function fz0 on S (called f translated by z0), by fz0 (z) = f(zz0). The set of piecewise continuous complex-valued functions on S will be denoted by H(S). The map κ : t 7→ eit is a continuous group homomorphism of the additive group of real numbers, R, onto S, and so it induces a homeomorphism and group isomorphism [t] 7→ eit of the compact quotient group K := R/2πZ with S. Here [t] ∈ K of course denotes the coset of the real number t, i.e., the set of all real numbers s that differ from t by an integer multiple of 2π, and we note that we can always choose a unique representative of [t] in [0, 2π). K is an additive model for S, and this makes it more convenient for some purposes. A function on K is clearly the “same thing” as a 2π-periodic function on R.