DOCSLIB.ORG

Chapter 18 Manifolds Arising from Group Actions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Chapter 18 Manifolds Arising from Group Actions 18.1 Proper Maps We saw in Chapter 3 that many manifolds arise from a group action. The scenario is that we have a smooth action ': G M M of a Lie group G acting on a manifold M (recall⇥ that! an action ' is smooth if it is a smooth map). If G acts transitively on M,thenforanypointx M, 2 if Gx is the stabilizer of x,thenweknowfromProposi- tion ?? that G/Gx is di↵eomorphic to M and that the projection ⇡ : G G/G is a submersion. ! x 823 824 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS If the action is not transitive, then we consider the orbit space M/G of orbits G x. · However, in general, M/G is not even Hausdor↵. It is thus desirable to look for sufficient conditions that ensure that M/G is Hausdor↵. Asufficientconditioncanbegivenusingthenotionofa proper map. Before we go any further, let us observe that the case where our action is transitive is subsumed by the more general situation of an orbit space. Indeed, if our action is transitive, for any x M,weknow 2 that the stabilizer H = Gx of x is a closed subgroup of G. 18.1. PROPER MAPS 825 Then, we can consider the right action G H G of H on G given by ⇥ ! g h = gh, g G, h H. · 2 2 The orbits of this (right) action are precisely the left cosets gH of H. Therefore, the set of left cosets G/H (the homogeneous space induced by the action : G M M)istheset of orbits of the right action G· H⇥ G!. ⇥ ! Observe that we have a transitive left action of G on the space G/H of left cosets, given by g g H = g g H. 1 · 2 1 2 The stabilizer of 1H is obviously H itself. 826 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Thus, we recover the original transitive left action of G on M = G/H. Now, it turns out that an action of the form G H G, where H is a closed subgroup of a Lie group⇥ G!,isa special case of a free and proper action M G G,in which case the orbit space M/G is a manifold,⇥ ! and the projection ⇡ : G M/G is a submersion. ! Let us now define proper maps. Definition 18.1. If X and Y are two Hausdor↵topo- logical spaces,1 acontinuousmap': X Y is proper i↵for every topological space Z,themap! ' id: X Z Y Z is a closed map (recall that f is⇥ a closed⇥ map! i↵the⇥ image of any closed set by f is a closed set). If we let Z be a one-point space, we see that a proper map is closed. 1It is not necessary to assume that X and Y are Hausdor↵but, if X and/or Y are not Hausdor↵, we have to replace “compact” by “quasi-compact.” We have no need for this extra generality. 18.1. PROPER MAPS 827 If F is any closed subset of X,thentherestrictionof' to F is proper (see Bourbaki, General Topology [9], Chapter 1, Section 10). The following result can be shown (see Bourbaki, General Topology [9], Chapter 1, Section 10): Proposition 18.1. A continuous map ': X Y is 1 ! proper i↵ ' is closed and if '− (y) is compact for every y Y . 2 1 If ' is proper, it is easy to show that '− (K)iscompact in X whenever K is compact in Y . Moreover, if Y is also locally compact, then we have the following result (see Bourbaki, General Topology [9], Chapter 1, Section 10). 828 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Proposition 18.2. If Y is locally compact, a map 1 ': X Y is a proper map i↵ '− (K) is compact in X whenever! K is compact in Y In particular, this is true if Y is a manifold since manifolds are locally compact. This explains why Lee [35] (Chapter 9) takes the property stated in Proposition 18.2 as the definition of a proper map (because he only deals with manifolds). Finally, we can define proper actions. 18.2. PROPER AND FREE ACTIONS 829 18.2 Proper and Free Actions Definition 18.2. Given a Hausdor↵topological group G and a topological space M,aleftaction : G M M is proper if it is continuous and if the map· ⇥ ! ✓ : G M M M, (g, x) (g x, x) ⇥ ! ⇥ 7! · is proper. If H is a closed subgroup of G and if : G M M is aproperaction,thentherestrictionofthisactionto· ⇥ ! H is also proper. If we let M = G,thenG acts on itself by left translation, and the map ✓ : G G G G given by ✓(g, x)= (gx, x)isahomeomorphism,soitisproper.⇥ ! ⇥ 830 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS It follows that the action : H G G of a closed subgroup H of G on G (given· by⇥ (h, g)! hg)isproper. 7! The same is true for the right action of H on G. As desired, proper actions yield Hausdor↵orbit spaces. Proposition 18.3. If the action : G M M is proper (where G is Hausdor↵), then· the⇥ orbit! space M/G is Hausdor↵. Furthermore, M is also Haus- dor↵. We also have the following properties (see Bourbaki, Gen- eral Topology [9], Chapter 3, Section 4). 18.2. PROPER AND FREE ACTIONS 831 Proposition 18.4. Let : G M M be a proper action, with G Hausdor↵·. For⇥ any x! M, let G x be 2 · the orbit of x and let Gx be the stabilizer of x. Then: (a) The map g g x is a proper map from G to M. 7! · (b) Gx is compact. (c) The canonical map from G/Gx to G x is a home- omorphism. · (d) The orbit G x is closed in M. · If G is locally compact, we have the following character- ization of being proper (see Bourbaki, General Topology [9], Chapter 3, Section 4). Proposition 18.5. If G and M are Hausdor↵and G is locally compact, then the action : G M M is proper i↵for all x, y M, there exist· some⇥ open! sets, V and V in M, with2x V and y V , so that the x y 2 x 2 y closure K of the set K = g G Vx g Vy = , is compact in G. { 2 | · \ 6 ;} 832 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS In particular, if G has the discrete topology, the above condition holds i↵the sets g G Vx g Vy = are finite. { 2 | · \ 6 ;} Also, if G is compact, then K is automatically compact, so every compact group acts properly. If M is locally compact, we have the following character- ization of being proper (see Bourbaki, General Topology [9], Chapter 3, Section 4). Proposition 18.6. Let : G M M be a con- tinuous action, with G and· M⇥ Hausdor! ↵. For any compact subset K of M we have: (a) The set G = g G g K K = is closed. K { 2 | · \ 6 ;} (b) If M is locally compact, then the action is proper i↵ GK is compact for every compact subset K of M. 18.2. PROPER AND FREE ACTIONS 833 In the special case where G is discrete (and M is locally compact), condition (b) says that the action is proper i↵ GK is finite. Remark: If G is a Hausdor↵topological group and if H is a subgroup of G,thenitcanbeshownthattheaction of G on G/H ((g1,g2H) g1g2H)isproperi↵H is compact in G. 7! Definition 18.3. An action : G M M is free if for all g G and all x M,if· g =1then⇥ !g x = x. 2 2 6 · 6 An equivalent way to state that an action : G M M · ⇥ ! is free is as follows. For every g G,letLg : M M be the di↵eomorphism of M given2 by ! L (x)=g x, x M. g · 2 Then, the action : G M M is free i↵for all g G, if g =1thenL has· no⇥ fixed! point. 2 6 g 834 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS Another equivalent statement is that for every x M, the stabilizer G of x is reduced to the trivial group2 1 . x { } If H is a subgroup of G,obviouslyH acts freely on G (by multiplication on the left or on the right). Before stating the main results of this section, observe that in the definition of a fibre bundle (Definition ??), the local trivialization maps are of the form 1 ' : ⇡− (U ) U F, ↵ ↵ ! ↵ ⇥ where the fibre F appears on the right. In particular, for a principal fibre bundle ⇠,thefibreF is equal to the structure group G,andthisisthereason why G acts on the right on the total space E of ⇠ (see Proposition ??). 18.2. PROPER AND FREE ACTIONS 835 To be more precise, we could call such bundles right bun- dles. We can also define left bundles as bundles whose local trivialization maps are of the form 1 ' : ⇡− (U ) F U . ↵ ↵ ! ⇥ ↵ Then, if ⇠ is a left principal bundle, the group G acts on E on the left. Duistermaat and Kolk [19] address this issue at the end of their Appendix B, and prove the theorem stated below (Chapter 1, Section 11). Beware that in Duistermaat and Kolk [19], this theorem is stated for left bundles,whichisadeparturefromthe prevalent custom of dealing with right bundles. 836 CHAPTER 18. MANIFOLDS ARISING FROM GROUP ACTIONS However, the weaker version that does not mention princi- pal bundles is usually stated for left actions; for instance, in Lee [35] (Chapter 9, Theorem 9.16).
Recommended publications
  • The Grassmann Manifold

    The Grassmann Manifold

    The Grassmann Manifold 1. For vector spaces V and W denote by L(V; W ) the vector space of linear maps from V to W . Thus L(Rk; Rn) may be identified with the space Rk£n of k £ n matrices. An injective linear map u : Rk ! V is called a k-frame in V . The set k n GFk;n = fu 2 L(R ; R ): rank(u) = kg of k-frames in Rn is called the Stiefel manifold. Note that the special case k = n is the general linear group: k k GLk = fa 2 L(R ; R ) : det(a) 6= 0g: The set of all k-dimensional (vector) subspaces ¸ ½ Rn is called the Grassmann n manifold of k-planes in R and denoted by GRk;n or sometimes GRk;n(R) or n GRk(R ). Let k ¼ : GFk;n ! GRk;n; ¼(u) = u(R ) denote the map which assigns to each k-frame u the subspace u(Rk) it spans. ¡1 For ¸ 2 GRk;n the fiber (preimage) ¼ (¸) consists of those k-frames which form a basis for the subspace ¸, i.e. for any u 2 ¼¡1(¸) we have ¡1 ¼ (¸) = fu ± a : a 2 GLkg: Hence we can (and will) view GRk;n as the orbit space of the group action GFk;n £ GLk ! GFk;n :(u; a) 7! u ± a: The exercises below will prove the following n£k Theorem 2. The Stiefel manifold GFk;n is an open subset of the set R of all n £ k matrices. There is a unique differentiable structure on the Grassmann manifold GRk;n such that the map ¼ is a submersion.
  • Cheap Orthogonal Constraints in Neural Networks: a Simple Parametrization of the Orthogonal and Unitary Group

    Cheap Orthogonal Constraints in Neural Networks: a Simple Parametrization of the Orthogonal and Unitary Group

    Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group Mario Lezcano-Casado 1 David Mart´ınez-Rubio 2 Abstract Recurrent Neural Networks (RNNs). In RNNs the eigenval- ues of the gradient of the recurrent kernel explode or vanish We introduce a novel approach to perform first- exponentially fast with the number of time-steps whenever order optimization with orthogonal and uni- the recurrent kernel does not have unitary eigenvalues (Ar- tary constraints. This approach is based on a jovsky et al., 2016). This behavior is the same as the one parametrization stemming from Lie group theory encountered when computing the powers of a matrix, and through the exponential map. The parametrization results in very slow convergence (vanishing gradient) or a transforms the constrained optimization problem lack of convergence (exploding gradient). into an unconstrained one over a Euclidean space, for which common first-order optimization meth- In the seminal paper (Arjovsky et al., 2016), they note that ods can be used. The theoretical results presented unitary matrices have properties that would solve the ex- are general enough to cover the special orthogo- ploding and vanishing gradient problems. These matrices nal group, the unitary group and, in general, any form a group called the unitary group and they have been connected compact Lie group. We discuss how studied extensively in the fields of Lie group theory and Rie- this and other parametrizations can be computed mannian geometry. Optimization methods over the unitary efficiently through an implementation trick, mak- and orthogonal group have found rather fruitful applications ing numerically complex parametrizations usable in RNNs in recent years (cf.
  • Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19Th, 2020

    Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19Th, 2020

    Math 5111 (Algebra 1) Lecture #14 of 24 ∼ October 19th, 2020 Group Isomorphism Theorems + Group Actions The Isomorphism Theorems for Groups Group Actions Polynomial Invariants and An This material represents x3.2.3-3.3.2 from the course notes. Quotients and Homomorphisms, I Like with rings, we also have various natural connections between normal subgroups and group homomorphisms. To begin, observe that if ' : G ! H is a group homomorphism, then ker ' is a normal subgroup of G. In fact, I proved this fact earlier when I introduced the kernel, but let me remark again: if g 2 ker ', then for any a 2 G, then '(aga−1) = '(a)'(g)'(a−1) = '(a)'(a−1) = e. Thus, aga−1 2 ker ' as well, and so by our equivalent properties of normality, this means ker ' is a normal subgroup. Thus, we can use homomorphisms to construct new normal subgroups. Quotients and Homomorphisms, II Equally importantly, we can also do the reverse: we can use normal subgroups to construct homomorphisms. The key observation in this direction is that the map ' : G ! G=N associating a group element to its residue class / left coset (i.e., with '(a) = a) is a ring homomorphism. Indeed, the homomorphism property is precisely what we arranged for the left cosets of N to satisfy: '(a · b) = a · b = a · b = '(a) · '(b). Furthermore, the kernel of this map ' is, by definition, the set of elements in G with '(g) = e, which is to say, the set of elements g 2 N. Thus, kernels of homomorphisms and normal subgroups are precisely the same things.
  • An Overview of Topological Groups: Yesterday, Today, Tomorrow

    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.
  • The Fundamental Groupoid of the Quotient of a Hausdorff

    The Fundamental Groupoid of the Quotient of a Hausdorff

    The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action Ronald Brown,∗ Philip J. Higgins,† Mathematics Division, Department of Mathematical Sciences, School of Informatics, Science Laboratories, University of Wales, Bangor South Rd., Gwynedd LL57 1UT, U.K. Durham, DH1 3LE, U.K. October 23, 2018 University of Wales, Bangor, Maths Preprint 02.25 Abstract The main result is that the fundamental groupoidof the orbit space of a discontinuousaction of a discrete groupon a Hausdorffspace which admits a universal coveris the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author’s book on Topology [3]. This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections. Since the book is out of print, and the result seems not well known, we now advertise it here. It is hoped that this account will also allow wider views of these results, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. The Exercises should be regarded as further propositions for which we leave the proofs to the reader.
  • Building Deep Networks on Grassmann Manifolds

    Building Deep Networks on Grassmann Manifolds

    Building Deep Networks on Grassmann Manifolds Zhiwu Huangy, Jiqing Wuy, Luc Van Goolyz yComputer Vision Lab, ETH Zurich, Switzerland zVISICS, KU Leuven, Belgium fzhiwu.huang, jiqing.wu, [email protected] Abstract The popular applications of Grassmannian data motivate Learning representations on Grassmann manifolds is popular us to build a deep neural network architecture for Grassman- in quite a few visual recognition tasks. In order to enable deep nian representation learning. To this end, the new network learning on Grassmann manifolds, this paper proposes a deep architecture is designed to accept Grassmannian data di- network architecture by generalizing the Euclidean network rectly as input, and learns new favorable Grassmannian rep- paradigm to Grassmann manifolds. In particular, we design resentations that are able to improve the final visual recog- full rank mapping layers to transform input Grassmannian nition tasks. In other words, the new network aims to deeply data to more desirable ones, exploit re-orthonormalization learn Grassmannian features on their underlying Rieman- layers to normalize the resulting matrices, study projection nian manifolds in an end-to-end learning architecture. In pooling layers to reduce the model complexity in the Grass- summary, two main contributions are made by this paper: mannian context, and devise projection mapping layers to re- spect Grassmannian geometry and meanwhile achieve Eu- • We explore a novel deep network architecture in the con- clidean forms for regular output layers. To train the Grass- text of Grassmann manifolds, where it has not been possi- mann networks, we exploit a stochastic gradient descent set- ble to apply deep neural networks.
  • GROUP ACTIONS 1. Introduction the Groups Sn, An, and (For N ≥ 3)

    GROUP ACTIONS 1. Introduction the Groups Sn, An, and (For N ≥ 3)

    GROUP ACTIONS KEITH CONRAD 1. Introduction The groups Sn, An, and (for n ≥ 3) Dn behave, by their definitions, as permutations on certain sets. The groups Sn and An both permute the set f1; 2; : : : ; ng and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If we label the vertices of the n-gon in a definite manner by the numbers from 1 to n then we can view Dn as a subgroup of Sn. For instance, the labeling of the square below lets us regard the 90 degree counterclockwise rotation r in D4 as (1234) and the reflection s across the horizontal line bisecting the square as (24). The rest of the elements of D4, as permutations of the vertices, are in the table below the square. 2 3 1 4 1 r r2 r3 s rs r2s r3s (1) (1234) (13)(24) (1432) (24) (12)(34) (13) (14)(23) If we label the vertices in a different way (e.g., swap the labels 1 and 2), we turn the elements of D4 into a different subgroup of S4. More abstractly, if we are given a set X (not necessarily the set of vertices of a square), then the set Sym(X) of all permutations of X is a group under composition, and the subgroup Alt(X) of even permutations of X is a group under composition. If we list the elements of X in a definite order, say as X = fx1; : : : ; xng, then we can think about Sym(X) as Sn and Alt(X) as An, but a listing in a different order leads to different identifications 1 of Sym(X) with Sn and Alt(X) with An.
  • Lie Group and Geometry on the Lie Group SL2(R)

    Lie Group and Geometry on the Lie Group SL2(R)

    INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR Lie group and Geometry on the Lie Group SL2(R) PROJECT REPORT – SEMESTER IV MOUSUMI MALICK 2-YEARS MSc(2011-2012) Guided by –Prof.DEBAPRIYA BISWAS Lie group and Geometry on the Lie Group SL2(R) CERTIFICATE This is to certify that the project entitled “Lie group and Geometry on the Lie group SL2(R)” being submitted by Mousumi Malick Roll no.-10MA40017, Department of Mathematics is a survey of some beautiful results in Lie groups and its geometry and this has been carried out under my supervision. Dr. Debapriya Biswas Department of Mathematics Date- Indian Institute of Technology Khargpur 1 Lie group and Geometry on the Lie Group SL2(R) ACKNOWLEDGEMENT I wish to express my gratitude to Dr. Debapriya Biswas for her help and guidance in preparing this project. Thanks are also due to the other professor of this department for their constant encouragement. Date- place-IIT Kharagpur Mousumi Malick 2 Lie group and Geometry on the Lie Group SL2(R) CONTENTS 1.Introduction ................................................................................................... 4 2.Definition of general linear group: ............................................................... 5 3.Definition of a general Lie group:................................................................... 5 4.Definition of group action: ............................................................................. 5 5. Definition of orbit under a group action: ...................................................... 5 6.1.The general linear
  • Definition 1.1. a Group Is a Quadruple (G, E, ⋆, Ι)

    Definition 1.1. a Group Is a Quadruple (G, E, ⋆, Ι)

    GROUPS AND GROUP ACTIONS. 1. GROUPS We begin by giving a definition of a group: Definition 1.1. A group is a quadruple (G, e, ?, ι) consisting of a set G, an element e G, a binary operation ?: G G G and a map ι: G G such that ∈ × → → (1) The operation ? is associative: (g ? h) ? k = g ? (h ? k), (2) e ? g = g ? e = g, for all g G. ∈ (3) For every g G we have g ? ι(g) = ι(g) ? g = e. ∈ It is standard to suppress the operation ? and write gh or at most g.h for g ? h. The element e is known as the identity element. For clarity, we may also write eG instead of e to emphasize which group we are considering, but may also write 1 for e where this is more conventional (for the group such as C∗ for example). Finally, 1 ι(g) is usually written as g− . Remark 1.2. Let us note a couple of things about the above definition. Firstly clo- sure is not an axiom for a group whatever anyone has ever told you1. (The reason people get confused about this is related to the notion of subgroups – see Example 1.8 later in this section.) The axioms used here are not “minimal”: the exercises give a different set of axioms which assume only the existence of a map ι without specifying it. We leave it to those who like that kind of thing to check that associa- tivity of triple multiplications implies that for any k N, bracketing a k-tuple of ∈ group elements (g1, g2, .
  • Covering Group Theory for Compact 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.
  • Stiefel Manifolds and Polygons

    Stiefel Manifolds and Polygons

    Stiefel Manifolds and Polygons Clayton Shonkwiler Department of Mathematics, Colorado State University; [email protected] Abstract Polygons are compound geometric objects, but when trying to understand the expected behavior of a large collection of random polygons – or even to formalize what a random polygon is – it is convenient to interpret each polygon as a point in some parameter space, essentially trading the complexity of the object for the complexity of the space. In this paper I describe such an interpretation where the parameter space is an abstract but very nice space called a Stiefel manifold and show how to exploit the geometry of the Stiefel manifold both to generate random polygons and to morph one polygon into another. Introduction The goal is to describe an identification of polygons with points in certain nice parameter spaces. For example, every triangle in the plane can be identified with a pair of orthonormal vectors in 3-dimensional 3 3 space R , and hence planar triangle shapes correspond to points in the space St2(R ) of all such pairs, which is an example of a Stiefel manifold. While this space is defined somewhat abstractly and is hard to visualize – for example, it cannot be embedded in Euclidean space of fewer than 5 dimensions [9] – it is easy to visualize and manipulate points in it: they’re just pairs of perpendicular unit vectors. Moreover, this is a familiar and well-understood space in the setting of differential geometry. For example, the group SO(3) of rotations of 3-space acts transitively on it, meaning that there is a rotation of space which transforms any triangle into any other triangle.
  • Representation Theory

    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.