Lecture 2: Atlas Demo
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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. -
A Note on Presentation of General Linear Groups Over a Finite Field
Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. -
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. -
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. -
Abelian Varieties with Complex Multiplication and Modular Functions, by Goro Shimura, Princeton Univ
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 3, Pages 405{408 S 0273-0979(99)00784-3 Article electronically published on April 27, 1999 Abelian varieties with complex multiplication and modular functions, by Goro Shimura, Princeton Univ. Press, Princeton, NJ, 1998, xiv + 217 pp., $55.00, ISBN 0-691-01656-9 The subject that might be called “explicit class field theory” begins with Kro- necker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x)=exp(2πix), i.e., with x Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something2 similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. Nowadays we would add that the reciprocity law describing the Galois group of an abelian extension L/K in terms of ideals of K should also be given explicitly. After K = Q, the next case is that of an imaginary quadratic number field K, with the real torus R/Z replaced by an elliptic curve E with complex multiplication. (Kronecker knew what the result should be, although complete proofs were given only later, by Weber and Takagi.) For simplicity, let be the ring of integers in O K, and let A be an -ideal. Regarding A as a lattice in C, we get an elliptic curve O E = C/A with End(E)= ;Ehas complex multiplication, or CM,by .If j=j(A)isthej-invariant ofOE,thenK(j) is the Hilbert class field of K, i.e.,O the maximal abelian unramified extension of K. -
Orders on Computable Torsion-Free Abelian Groups
Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z. -
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. -
Semigroups and Monoids
S Luis Alonso-Ovalle // Contents Subgroups Semigroups and monoids Subgroups Groups. A group G is an algebra consisting of a set G and a single binary operation ◦ satisfying the following axioms: . ◦ is completely defined and G is closed under ◦. ◦ is associative. G contains an identity element. Each element in G has an inverse element. Subgroups. We define a subgroup G0 as a subalgebra of G which is itself a group. Examples: . The group of even integers with addition is a proper subgroup of the group of all integers with addition. The group of all rotations of the square h{I, R, R0, R00}, ◦i, where ◦ is the composition of the operations is a subgroup of the group of all symmetries of the square. Some non-subgroups: SEMIGROUPS AND MONOIDS . The system h{I, R, R0}, ◦i is not a subgroup (and not even a subalgebra) of the original group. Why? (Hint: ◦ closure). The set of all non-negative integers with addition is a subalgebra of the group of all integers with addition, because the non-negative integers are closed under addition. But it is not a subgroup because it is not itself a group: it is associative and has a zero, but . does any member (except for ) have an inverse? Order. The order of any group G is the number of members in the set G. The order of any subgroup exactly divides the order of the parental group. E.g.: only subgroups of order , , and are possible for a -member group. (The theorem does not guarantee that every subset having the proper number of members will give rise to a subgroup. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties .