From Loop Groups to 2-Groups
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Orthogonal Symmetric Affine Kac-Moody Algebras
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 10, October 2015, Pages 7133–7159 http://dx.doi.org/10.1090/tran/6257 Article electronically published on April 20, 2015 ORTHOGONAL SYMMETRIC AFFINE KAC-MOODY ALGEBRAS WALTER FREYN Abstract. Riemannian symmetric spaces are fundamental objects in finite dimensional differential geometry. An important problem is the construction of symmetric spaces for generalizations of simple Lie groups, especially their closest infinite dimensional analogues, known as affine Kac-Moody groups. We solve this problem and construct affine Kac-Moody symmetric spaces in a series of several papers. This paper focuses on the algebraic side; more precisely, we introduce OSAKAs, the algebraic structures used to describe the connection between affine Kac-Moody symmetric spaces and affine Kac-Moody algebras and describe their classification. 1. Introduction Riemannian symmetric spaces are fundamental objects in finite dimensional dif- ferential geometry displaying numerous connections with Lie theory, physics, and analysis. The search for infinite dimensional symmetric spaces associated to affine Kac-Moody algebras has been an open question for 20 years, since it was first asked by C.-L. Terng in [Ter95]. We present a complete solution to this problem in a series of several papers, dealing successively with the functional analytic, the algebraic and the geometric aspects. In this paper, building on work of E. Heintze and C. Groß in [HG12], we introduce and classify orthogonal symmetric affine Kac- Moody algebras (OSAKAs). OSAKAs are the central objects in the classification of affine Kac-Moody symmetric spaces as they provide the crucial link between the geometric and the algebraic side of the theory. -
Lecture 2: Spaces of Maps, Loop Spaces and Reduced Suspension
LECTURE 2: SPACES OF MAPS, LOOP SPACES AND REDUCED SUSPENSION In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there will be a short digression on spaces of maps between (pointed) spaces and the relevant topologies. To be a bit more specific, one aim is to see that given a pointed space (X; x0), then there is an entire pointed space of loops in X. In order to obtain such a loop space Ω(X; x0) 2 Top∗; we have to specify an underlying set, choose a base point, and construct a topology on it. The underlying set of Ω(X; x0) is just given by the set of maps 1 Top∗((S ; ∗); (X; x0)): A base point is also easily found by considering the constant loop κx0 at x0 defined by: 1 κx0 :(S ; ∗) ! (X; x0): t 7! x0 The topology which we will consider on this set is a special case of the so-called compact-open topology. We begin by introducing this topology in a more general context. 1. Function spaces Let K be a compact Hausdorff space, and let X be an arbitrary space. The set Top(K; X) of continuous maps K ! X carries a natural topology, called the compact-open topology. It has a subbasis formed by the sets of the form B(T;U) = ff : K ! X j f(T ) ⊆ Ug where T ⊆ K is compact and U ⊆ X is open. Thus, for a map f : K ! X, one can form a typical basis open neighborhood by choosing compact subsets T1;:::;Tn ⊆ K and small open sets Ui ⊆ X with f(Ti) ⊆ Ui to get a neighborhood Of of f, Of = B(T1;U1) \ ::: \ B(Tn;Un): One can even choose the Ti to cover K, so as to `control' the behavior of functions g 2 Of on all of K. -
Arxiv:Math/0607130V2
TWISTED LOOP GROUPS AND THEIR AFFINE FLAG VARIETIES G. PAPPAS* AND M. RAPOPORT Introduction Loop groups are familiar objects in several branches of mathematics. Let us mention here three variants. The first variant is differential-geometric in nature. One starts with a Lie group G (e.g., a compact Lie group or its complexification). The associated loop group is then the group of (C0-, or C1-, or C∞-) maps of S1 into G, cf. [P-S] and the literature cited there. A twisted version arises from an automorphism α of G. The associated twisted loop group is the group of maps γ : R → G such that γ(θ + 2π) = α(γ(θ)) . The second variant is algebraic and arises in the context of Kac-Moody algebras. Here one constructs an infinite-dimensional algebraic group variety with Lie algebra equal or closely related to a given Kac-Moody algebra. (This statement is an oversimplification and the situation is in fact more complicated: there exist various constructions at a formal, a minimal, and a maximal level which produce infinite-dimensional groups with Lie algebras closely related to the given Kac-Moody Lie algebra, see [Ma2], also [T2], [T3] and the literature cited there). The third variant is algebraic-geometric in nature and is our main concern in this paper. Let us recall the basic definitions in the untwisted case. Let k be a field and let G0 be an algebraic group over Spec (k). We consider the functor LG0 on the category of k-algebras, R 7→ LG0(R)= G0(R((t))). -
Critical Values of Homology Classes of Loops and Positive Curvature
CRITICAL VALUES OF HOMOLOGY CLASSES OF LOOPS AND POSITIVE CURVATURE HANS-BERT RADEMACHER Abstract. We study compact and simply-connected Riemannian man- ifolds (M,g) with positive sectional curvature K ≥ 1. For a non-trivial homology class of lowest positive dimension in the space of loops based at a point p ∈ M or in the free loop space one can define a critical length crlp (M,g) resp. crl (M,g) . Then crlp (M,g) equals the length of a geodesic loop with base point p and crl (M,g) equals the length of a closed geodesic. This is the idea of the proof of the existence of a closed geodesic of positive length presented by Birkhoff in case of a sphere and by Lusternik & Fet in the general case. It is the main result of the paper that the numbers crlp (M,g) resp. crl (M,g) attain its maximal value 2π only for the round metric on the n-sphere. Under the additional assumption K ≤ 4 this result for crl (M,g) follows from results by Sugimoto in even dimensions and Ballmann, Thorbergsson & Ziller in odd dimensions. 1. Introduction For a compact Riemannian manifold (M, g) let = γ : [0, 1] M { −→ M ; γ absolutely continuous , 1 γ′ 2(t) dt < be the manifold of paths 0 k k ∞} (H1-paths) on M. We considerR in the sequel the following subspaces: the free loop space Λ = ΛM = γ ; γ(0) = γ(1) and for p,q M the space of paths between p and q :{ Ω ∈= M Ω M = γ } ; γ(0) = p,∈ γ(1) = q . -
Loop Groups and Diffeomorphism Groups of the Circle As Colimits Arxiv
Loop groups and diffeomorphism groups of the circle as colimits Andre´ Henriques University of Oxford, Mathematical Institute [email protected] Abstract 1 In this paper, we show that loop groups and the universal cover of Diff+(S ) can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of S1. Analogous results hold for based loop groups and for the based diffeomorphism group of S1. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie al- gebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group. Contents 1 Introduction and statement of results2 2 Colimits and central extensions4 2.1 Loop groups . .5 2.1.1 The free loop group . .5 2.1.2 The based loop group . 11 2.2 Diffeomorphism groups . 12 2.2.1 Diff(S1) and its universal cover . 12 arXiv:1706.08471v3 [math-ph] 27 Jan 2019 2.2.2 The based diffeomorphism group . 18 3 Application: representations of loop group conformal nets 19 3.1 Loop group conformal nets and affine Lie algebras . 19 3.1.1 Loop group conformal nets . 19 3.1.2 Affine Lie algebras . 20 3.2 The comparison functors k k k Rep(AG;k) ! Rep (LG) and Rep (LG) ! Rep (g^)............ 21 3.3 The based loop group and its representations . -
Homotopy and the Fundamental Group
CHAPTER 2 Homotopy and the Fundamental Group 1. Homotopy Denote I = [0; 1]. Let f; g : X ! Y be maps of topological spaces. A homotopy from f to g is a map H : X×I ! Y such that for all x 2 X, H(x; 0) = f(x) and H(x; 1) = g(x). Thus, on the `bottom' edge, H agrees with f and on the `top' edge it agrees with g. The intermediate maps H(−; t) for 0 < t < 1 may be thought of a 1-parameter family of maps through which f is continuously deformed into g. We say f is homotopic to g (f ∼ g) if there is a homotopy from f to g. Example 2.1. Let X = S1 and Y = S1 × I (a cylinder of radius 1 and height 1). Define H(x; t) = (x; t). (What are f and g?) Example 2.2. Let X = S1 and Y = D2. Let f be the inclusion of S1 in D2 and let g map S1 to the center of D2. Then H(x; t) = (1−t)x defines a homotopy of f to g. Example 2.3. Let X = Y = Rn. Let f be the identity map, and let g be defined by g(x) = 0 for all x. Define H(x; t) = (1 − t)x. Note that the same argument would work for any point with a slightly different H. If the identity map of a space is homotopic to a constant map (as in Example 2.3), we say the space is contractible. It is also useful to have a relative version of this definition. -
Jhep10(2019)039
Published for SISSA by Springer Received: July 4, 2019 Accepted: September 3, 2019 Published: October 4, 2019 The Maxwell group in 2+1 dimensions and its JHEP10(2019)039 infinite-dimensional enhancements Patricio Salgado-Rebolledo Instituto de F´ısica, Pontificia Universidad Cat´olica de Valpara´ıso, Casilla 4059, Valpara´ıso, Chile E-mail: [email protected] Abstract: The Maxwell group in 2+1 dimensions is given by a particular extension of a semi-direct product. This mathematical structure provides a sound framework to study different generalizations of the Maxwell symmetry in three space-time dimensions. By giv- ing a general definition of extended semi-direct products, we construct infinite-dimensional enhancements of the Maxwell group that enlarge the ISL(2, R) Kac-Moody group and the [ BMS3 group by including non-commutative supertranslations. The coadjoint representa- tion in each case is defined, and the corresponding geometric actions on coadjoint orbits are presented. These actions lead to novel Wess-Zumino terms that naturally realize the aforementioned infinite-dimensional symmetries. We briefly elaborate on potential appli- cations in the contexts of three-dimensional gravity, higher-spin symmetries, and quantum Hall systems. Keywords: Chern-Simons Theories, Gauge Symmetry, Global Symmetries, Sigma Models ArXiv ePrint: 1905.09421 Open Access, c The Authors. 3 https://doi.org/10.1007/JHEP10(2019)039 Article funded by SCOAP . Contents 1 Introduction 1 2 Extending semi-direct products 4 2.1 Extended semi-direct product groups -
Lectures on Loop Groups
LECTURES ON LOOP GROUPS NITU KITCHLOO UCSD AND JOHNS HOPKINS UNIV. Date: June 14, 2018. 1 1. Lecture I, Background and Overview Complex and Unitary forms: We set up notation by letting K denote a compact, simply connected, simple Lie group. For example K = SU(n). By G we shall mean the complexification of K. So, up to isomorphism, G is the unique simple complex algebraic group which includes K as a maximal compact subgroup. So in our example G = SLn(C). In fact, there is an anti-linear involution σ on G called the Cartan involution whose fixed ∗ −1 points is K. In our example of SLn(C), this involution is given by σ(A) = (A ) . Algebraic and smooth Loop groups: We define two types of loop groups LsmK and LalgK. The group LsmK is the (pointwise) 1 group of smooth maps from S to K in the compact open topology. The group LalgK may be defined as the subgroup of maps with finite Fourier expansions under all unitary rep- resentations of K. However, the topology on LalgK is given by the direct limit topology induced by the compact sub-spaces of LsmK that consist of maps of increasing degree under some faithful unitary representation of K. Alternatively, to define LalgK, we may mimic the above story by starting with the complex −1 ∗ form LalgG = G(C[z; z ]) of C -valued points in G, and then take fixed points under the ∗ (loop) Cartan involution σ^(A)(z) = σfA(σ1(z))g, where σ1 is the automorphism of C sending z to z−1. -
Fundamental Group and Covering Space
Fundamental Group and Covering Space David Gu Yau Mathematics Science Center Tsinghua University [email protected] July 5, 2020 David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 1 / 59 Algebraic Topology: Fundamental Group David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 2 / 59 Orientability-M¨obiusBand Figure: Escher. Ants David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 3 / 59 Surface Genus Topological Sphere Topological Torus Figure: How to differentiate the above two surfaces. David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 4 / 59 Key Idea Figure: Check whether all loops on the surface can shrink to a point. All oriented compact surfaces can be classified by their genus g and number of boundaries b. Therefore, we use (g; b) to represent the topological type of an oriented surface S. David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 5 / 59 Application Figure: Handle detection by finding the handle loops and the tunnel loops. David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 6 / 59 Application Figure: Topological Denoise in medical imaging. David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 7 / 59 Surface Topology Philosophy Associate groups with manifolds, study the topology by analyzing the group structures. C1 = Topological Spaces; Homeomorphisms f g C2 = Groups; Homomorphisms f g C1 C2 ! Functor between categories. David Gu (Stony Brook University) Computational Conformal Geometry July 5, 2020 8 / 59 Fundamental group Suppose q is a base point, all the oriented closed curves (loops) through q can be classified by homotopy. -
On Loop Spaces of Configuration Spaces 1
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 354, Number 5, Pages 1705{1748 S 0002-9947(02)02948-3 Article electronically published on January 11, 2002 ON LOOP SPACES OF CONFIGURATION SPACES F. R. COHEN AND S. GITLER Abstract. This article gives an analysis of topological and homological prop- erties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as \twistings" between the factors. The main homological results are given in terms of extensions of the \infinitesimal braid relations" or \universal Yang- Baxter Lie relations". 1. Introduction The purpose of this article is to describe properties of loop spaces of classical configuration spaces. The referee of this paper has requested that additional back- ground as well as motivation for this paper be included. This additional informa- tion, as well as further applications and problems, will be given in the Introduction andsection2ofthispaper. The classical configuration space of a space M, F (M;k), is the space of ordered k-tuples of distinct points in the product space M k.IncaseM is a manifold, the homology of the loop space for many configuration spaces is determined here. In contrast, the homology of the configuration space itself is not yet well-understood. For example, if M is given by a product of at least two spheres of dimension greater than zero, the integer homology of the configuration space is not understood at present. Some cases for which the homology is understood are given when M = N ×R1 for a manifold N [4, 9], while some beautiful general results in characteristic zero are given in [23, 29]. -
Thom Prospectra for Loopgroup Representations
THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS NITU KITCHLOO AND JACK MORAVA Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new local- ization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman. Introduction If P → M is a principal bundle with structure group G then LP → LM is a principal bundle with structure group LG = Maps(S1, G), We will assume in the rest of the paper that the frame bundle of M has been refined to have structure group G, and that the tangent bundle of M is thus defined via some representation V of G. It follows that the tangent bundle of LM is defined by the representation LV of LG. The circle group T acts on all these spaces by rotation of loops. This is a report on the beginnings of a theory of differential topology for such objects. Note that if we want the structure group LG to be connected, we need G to be 1-connected; thus SU(n) is preferable to U(n). -
Chapter 4: Introduction to Representation Theory
Preprint typeset in JHEP style - HYPER VERSION Chapter 4: Introduction to Representation Theory Gregory W. Moore Abstract: BASED MOSTLY ON GTLECT4 FROM 2009. BUT LOTS OF MATERIAL HAS BEEN IMPORTED AND REARRANGED FROM MATHMETHODS 511, 2014 AND GMP 2010 ON ASSOCIATED BUNDLES. SOMETHING ABOUT SU(2) REPS AND INDUCED REPS NOW RESTORED. BECAUSE OF IMPORTS THERE IS MUCH REDUNDANCY. THIS CHAPTER NEEDS A LOT OF WORK. April 27, 2018 -TOC- Contents 1. Symmetries of physical systems 3 2. Basic Definitions 4 2.1 Representation of a group 4 2.2 Matrix Representations 5 2.3 Examples 6 2.3.1 The fundamental representation of a matrix Lie group 6 2.3.2 The determinant representation 6 2.3.3 A representation of the symmetric groups Sn 6 2.3.4 Z and Z2 7 2.3.5 The Heisenberg group 7 3. Unitary Representations 8 3.1 Invariant Integration 9 3.2 Unitarizable Representations 10 3.3 Unitary representations and the Schr¨odingerequation 11 4. Projective Representations and Central Extensions 12 5. Induced Group Actions On Function Spaces 13 6. The regular representation 14 6.1 Matrix elements as functions on G 16 6.2 RG as a unitary rep: Invariant integration on the group 17 6.3 A More Conceptual Description 17 7. Reducible and Irreducible representations 18 7.1 Definitions 18 7.2 Reducible vs. Completely reducible representations 21 8. Schur's Lemmas 22 9. Orthogonality relations for matrix elements 23 10. Decomposition of the Regular Representation 25 11. Fourier Analysis as a branch of Representation Theory 31 11.1 The irreducible representations of abelian groups 31 11.2 The character group 31 11.3 Fourier duality 33 11.4 The Poisson summation formula 36 { 1 { 11.5 Application: Bloch's Theorem in Solid State Physics 37 11.6 The Heisenberg group extension of S^ × S for an abelian group S 39 12.