TOPOLOGICAL GROUPS the Purpose of These Notes Is to Give A
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Arithmetic Equivalence and Isospectrality
ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY ANDREW V.SUTHERLAND ABSTRACT. In these lecture notes we give an introduction to the theory of arithmetic equivalence, a notion originally introduced in a number theoretic setting to refer to number fields with the same zeta function. Gassmann established a direct relationship between arithmetic equivalence and a purely group theoretic notion of equivalence that has since been exploited in several other areas of mathematics, most notably in the spectral theory of Riemannian manifolds by Sunada. We will explicate these results and discuss some applications and generalizations. 1. AN INTRODUCTION TO ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a number field (a finite extension of Q), and let OK be its ring of integers (the integral closure of Z in K). The Dedekind zeta function of K is defined by the Dirichlet series X s Y s 1 ζK (s) := N(I)− = (1 N(p)− )− I OK p − ⊆ where the sum ranges over nonzero OK -ideals, the product ranges over nonzero prime ideals, and N(I) := [OK : I] is the absolute norm. For K = Q the Dedekind zeta function ζQ(s) is simply the : P s Riemann zeta function ζ(s) = n 1 n− . As with the Riemann zeta function, the Dirichlet series (and corresponding Euler product) defining≥ the Dedekind zeta function converges absolutely and uniformly to a nonzero holomorphic function on Re(s) > 1, and ζK (s) extends to a meromorphic function on C and satisfies a functional equation, as shown by Hecke [25]. The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants of K, including its class number, and as with the Riemann zeta function, the zeros of ζK (s) are intimately related to the distribution of prime ideals in OK . -
7.2 Binary Operators Closure
last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers. -
3. Closed Sets, Closures, and Density
3. Closed sets, closures, and density 1 Motivation Up to this point, all we have done is define what topologies are, define a way of comparing two topologies, define a method for more easily specifying a topology (as a collection of sets generated by a basis), and investigated some simple properties of bases. At this point, we will start introducing some more interesting definitions and phenomena one might encounter in a topological space, starting with the notions of closed sets and closures. Thinking back to some of the motivational concepts from the first lecture, this section will start us on the road to exploring what it means for two sets to be \close" to one another, or what it means for a point to be \close" to a set. We will draw heavily on our intuition about n convergent sequences in R when discussing the basic definitions in this section, and so we begin by recalling that definition from calculus/analysis. 1 n Definition 1.1. A sequence fxngn=1 is said to converge to a point x 2 R if for every > 0 there is a number N 2 N such that xn 2 B(x) for all n > N. 1 Remark 1.2. It is common to refer to the portion of a sequence fxngn=1 after some index 1 N|that is, the sequence fxngn=N+1|as a tail of the sequence. In this language, one would phrase the above definition as \for every > 0 there is a tail of the sequence inside B(x)." n Given what we have established about the topological space Rusual and its standard basis of -balls, we can see that this is equivalent to saying that there is a tail of the sequence inside any open set containing x; this is because the collection of -balls forms a basis for the usual topology, and thus given any open set U containing x there is an such that x 2 B(x) ⊆ U. -
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. -
Covering Group Theory for Compact Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 161 (2001) 255–267 www.elsevier.com/locate/jpaa Covering group theory for compact groups Valera Berestovskiia, Conrad Plautb;∗ aDepartment of Mathematics, Omsk State University, Pr. Mira 55A, Omsk 77, 644077, Russia bDepartment of Mathematics, University of Tennessee, Knoxville, TN 37919, USA Received 27 May 1999; received in revised form 27 March 2000 Communicated by A. Carboni Abstract We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism 2 K : G → G in this category. The abelian topological group 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems. c 2001 Elsevier Science B.V. All rights reserved. MSC: Primary: 22C05; 22B05; secondary: 22KXX 1. Introduction and main results In [1] we developed a covering group theory and generalized notion of fundamental group that discards such traditional notions as local arcwise connectivity and local simple connectivity. The theory applies to a large category of “coverable” topological groups that includes, for example, all metrizable, connected, locally connected groups and even some totally disconnected groups. However, for locally compact groups, being coverable is equivalent to being arcwise connected [2]. Therefore, many connected compact groups (e.g. solenoids or the character group of ZN ) are not coverable. -
On Certain Relations for C-Closure Operations on an Ordered Semigroup
International Journal of Pure and Applied Mathematics Volume 92 No. 1 2014, 51-59 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: http://dx.doi.org/10.12732/ijpam.v92i1.4 ijpam.eu ON CERTAIN RELATIONS FOR C-CLOSURE OPERATIONS ON AN ORDERED SEMIGROUP Thawhat Changphas Department of Mathematics Faculty of Science Khon Kaen University Khon Kaen, 40002, THAILAND Abstract: In this paper, a relation for C-closure operations on an ordered semigroup is introduced, using this relation regular and simple ordered semi- groups are characterized. AMS Subject Classification: 06F05 Key Words: semigroup, ordered semigroup, ideal, regular ordered semigroup, simple ordered semigroup, C-closure operation 1. Preliminaries It is known that a semigroup S is regular if and only if it satisfies: A ∩ B = AB for all right ideals A and for all left ideals B of S. Using this property, Pondˇel´iˇcek [2] introduced a relation for C-closure operations on S, and studied some types of semigroups using the relation. The purpose of this paper is to extend Pondˇel´iˇcek’s results to ordered semigroups. In fact, we define a relation for C-closure operations on an ordered semigroup, and characterize regular and simple ordered semigroups using the relation. Firstly, let us recall some certain definitions and results which are in [2]. c 2014 Academic Publications, Ltd. Received: November 7, 2013 url: www.acadpubl.eu 52 T. Changphas Let S be a nonempty set. A mapping U:Su(S) → Su(S) (The symbol Su(S) stands for the set of all subsets of S) is called a C-closure operation on S if, for any A, B in Su(S), it satisfies: (i) U(∅) = ∅; (ii) A ⊆ B ⇒ U(A) ⊆ U(B); (iii) A ⊆ U(A); (iv) U(U(A)) = U(A). -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
Math 396. Interior, Closure, and Boundary We Wish to Develop Some Basic Geometric Concepts in Metric Spaces Which Make Precise C
Math 396. Interior, closure, and boundary We wish to develop some basic geometric concepts in metric spaces which make precise certain intuitive ideas centered on the themes of \interior" and \boundary" of a subset of a metric space. One warning must be given. Although there are a number of results proven in this handout, none of it is particularly deep. If you carefully study the proofs (which you should!), then you'll see that none of this requires going much beyond the basic definitions. We will certainly encounter some serious ideas and non-trivial proofs in due course, but at this point the central aim is to acquire some linguistic ability when discussing some basic geometric ideas in a metric space. Thus, the main goal is to familiarize ourselves with some very convenient geometric terminology in terms of which we can discuss more sophisticated ideas later on. 1. Interior and closure Let X be a metric space and A X a subset. We define the interior of A to be the set ⊆ int(A) = a A some Br (a) A; ra > 0 f 2 j a ⊆ g consisting of points for which A is a \neighborhood". We define the closure of A to be the set A = x X x = lim an; with an A for all n n f 2 j !1 2 g consisting of limits of sequences in A. In words, the interior consists of points in A for which all nearby points of X are also in A, whereas the closure allows for \points on the edge of A". -
SPACES with a COMPACT LIE GROUP of TRANSFORMATIONS A
SPACES WITH A COMPACT LIE GROUP OF TRANSFORMATIONS a. m. gleason Introduction. A topological group © is said to act on a topological space 1{ if the elements of © are homeomorphisms of 'R. onto itself and if the mapping (<r, p)^>a{p) of ©X^ onto %_ is continuous. Familiar examples include the rotation group acting on the Car- tesian plane and the Euclidean group acting on Euclidean space. The set ®(p) (that is, the set of all <r(p) where <r(E®) is called the orbit of p. If p and q are two points of fx, then ®(J>) and ©(g) are either identical or disjoint, hence 'R. is partitioned by the orbits. The topological structure of the partition becomes an interesting question. In the case of the rotations of the Cartesian plane we find that, excising the singularity at the origin, the remainder of space is fibered as a direct product. A similar result is easily established for a compact Lie group acting analytically on an analytic manifold. In this paper we make use of Haar measure to extend this result to the case of a compact Lie group acting on any completely regular space. The exact theorem is given in §3. §§1 and 2 are preliminaries, while in §4 we apply the main result to the study of the structure of topo- logical groups. These applications form the principal motivation of the entire study,1 and the author hopes to develop them in greater detail in a subsequent paper. 1. An extension theorem. In this section we shall prove an exten- sion theorem which is an elementary generalization of the well known theorem of Tietze. -
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. -
Representation Theory of Compact Groups - Lecture Notes up to 08 Mar 2017
Representation theory of compact groups - Lecture notes up to 08 Mar 2017 Alexander Yom Din March 9, 2017 These notes are not polished yet; They are not very organized, and might contain mistakes. 1 Introduction 1 ∼ ∼ × Consider the circle group S = R=Z = C1 - it is a compact group. One studies the space L2(S1) of square-integrable complex-valued fucntions on the circle. We have an action of S1 on L2(S1) given by (gf)(x) = f(g−1x): We can suggestively write ^ Z 2 1 L (S ) = C · δx; x2S1 where gδx = δgx: Thus, the "basis" of delta functions is "permutation", or "geometric". Fourier theory describes another, "spectral" basis. Namely, we consider the functions 2πinx χn : x (mod 1) 7! e where n 2 Z. The main claim is that these functions form a Hilbert basis for L2(S1), hence we can write ^ 2 1 M L (S ) = C · χn; n2Z where −1 gχn = χn (g)χn: In other words, we found a basis of eigenfunctions for the translation operators. 1 What we would like to do in the course is to generalize the above picture to more general compact groups. For a compact group G, one considers L2(G) with the action of G × G by translation on the left and on the right: −1 ((g1; g2)f)(x) = f(g1 xg2): When the group is no longer abelian, we will not be able to find enough eigen- functions for the translation operators to form a Hilbert basis. Eigenfunctions can be considered as spanning one-dimensional invariant subspaces, and what we will be able to find is "enough" finite-dimensional "irreducible" invariant sub- spaces (which can not be decomposed into smaller invariant subspaces).