1.3 Double Groupoids
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Density of Thin Film Billiard Reflection Pseudogroup in Hamiltonian Symplectomorphism Pseudogroup Alexey Glutsyuk
Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk To cite this version: Alexey Glutsyuk. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomor- phism pseudogroup. 2020. hal-03026432v2 HAL Id: hal-03026432 https://hal.archives-ouvertes.fr/hal-03026432v2 Preprint submitted on 6 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk∗yzx December 3, 2020 Abstract Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C1-smooth hypersurface γ ⊂ Rn+1 that is either a global strictly convex closed hypersurface, or a germ of hy- persurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deforma- tions. In the case, when γ is a global convex hypersurface, we show that the latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case, when γ is a germ. -
Double Groupoids and Crossed Modules Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 17, No 4 (1976), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES RONALD BROWN CHRISTOPHER B. SPENCER Double groupoids and crossed modules Cahiers de topologie et géométrie différentielle catégoriques, tome 17, no 4 (1976), p. 343-362 <http://www.numdam.org/item?id=CTGDC_1976__17_4_343_0> © Andrée C. Ehresmann et les auteurs, 1976, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE Vol. X VII -4 (1976) ET GEOMETRIE DIFFERENTIELLE DOUBLE GROUPOI DS AND CROSSED MODULES by Ronald BROWN and Christopher B. SPENCER* INTRODUCTION The notion of double category has occured often in the literature ( see for example (1 , 6 , 7 , 9 , 13 , 14] ). In this paper we study a more parti- cular algebraic object which we call a special double groupoid with special connection. The theory of these objects might be called «2-dimensional grou- poid theory ». The reason is that groupoid theory derives much of its tech- nique and motivation from the fundamental groupoid of a space, a device obtained by homotopies from paths on a space in such a way as to permit cancellations. The 2-dimensional animal corresponding to a groupoid should have features derived from operations on squares in a space. Thus it should have the algebraic analogue of the horizontal and vertical compositions of squares ; it should also permit cancellations. -
Symplectic Structures on Gauge Theory?
Commun. Math. Phys. 193, 47 – 67 (1998) Communications in Mathematical Physics c Springer-Verlag 1998 Symplectic Structures on Gauge Theory? Nai-Chung Conan Leung School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455, USA Received: 27 February 1996 / Accepted: 7 July 1997 [ ] Abstract: We study certain natural differential forms ∗ and their equivariant ex- tensions on the space of connections. These forms are defined usingG the family local index theorem. When the base manifold is symplectic, they define a family of sym- plectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations). Contents 1 Introduction ............................................... 47 2 Higher Index of the Universal Family ........................... 49 2.1 Dirac operators ............................................. 50 2.2 ∂-operator ................................................ 52 3 Moment Map and Stable Bundles .............................. 53 4 Space of Generalized Connections ............................. 58 5 Higer Level Chern–Simons Forms ............................. 60 [ ] 6 Equivariant Extensions of ∗ ................................ 63 1. Introduction In this paper, we study natural differential forms [2k] on the space of connections on a vector bundle E over X, A i [2k] FA (A)(B1,B2,...,B2 )= Tr[e 2π B1B2 B2 ] Aˆ(X). k ··· k sym ZX They are invariant closed differential forms on . They are introduced as the family local indexG for universal family of operators. A ? This research is supported by NSF grant number: DMS-9114456 48 N-C. C. Leung We use them to define higher Chern-Simons forms of E: [ ] ch(E; A0,...,A )=∗ L(A0,...,A ), l l ◦ l l and discuss their properties. -
Lecture 1: Basic Concepts, Problems, and Examples
LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introduction to the basic concepts and some of the fundamental problems in symplectic geometry/topology, where along the way various examples are also given for the purpose of illustration. We will often give statements without proofs, which means that their proof is either beyond the scope of this course or will be treated more systematically in later lectures. 1. Symplectic Manifolds Throughout we will assume that M is a C1-smooth manifold without boundary (unless specific mention is made to the contrary). Very often, M will also be closed (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifold M is a 2-form ! 2 Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. d! = 0). (Recall that a 2-form ! 2 Ω2(M) is said to be nondegenerate if for every point p 2 M, !(u; v) = 0 for all u 2 TpM implies v 2 TpM equals 0.) The pair (M; !) is called a symplectic manifold. Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ! is equivalent to the condition that M has an even dimension 2n and the top wedge product !n ≡ ! ^ ! · · · ^ ! is nowhere vanishing on M, i.e., !n is a volume form. In particular, M must be orientable, and is canonically oriented by !n. The nondegeneracy condition is also equivalent to the condition that M is almost complex, i.e., there exists an endomorphism J of TM such that J 2 = −Id. -
From Double Lie Groupoids to Local Lie 2-Groupoids
Smith ScholarWorks Mathematics and Statistics: Faculty Publications Mathematics and Statistics 12-1-2011 From Double Lie Groupoids to Local Lie 2-Groupoids Rajan Amit Mehta Pennsylvania State University, [email protected] Xiang Tang Washington University in St. Louis Follow this and additional works at: https://scholarworks.smith.edu/mth_facpubs Part of the Mathematics Commons Recommended Citation Mehta, Rajan Amit and Tang, Xiang, "From Double Lie Groupoids to Local Lie 2-Groupoids" (2011). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/mth_facpubs/91 This Article has been accepted for inclusion in Mathematics and Statistics: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected] FROM DOUBLE LIE GROUPOIDS TO LOCAL LIE 2-GROUPOIDS RAJAN AMIT MEHTA AND XIANG TANG Abstract. We apply the bar construction to the nerve of a double Lie groupoid to obtain a local Lie 2-groupoid. As an application, we recover Haefliger’s fun- damental groupoid from the fundamental double groupoid of a Lie groupoid. In the case of a symplectic double groupoid, we study the induced closed 2-form on the associated local Lie 2-groupoid, which leads us to propose a definition of a symplectic 2-groupoid. 1. Introduction In homological algebra, given a bisimplicial object A•,• in an abelian cate- gory, one naturally associates two chain complexes. One is the diagonal complex diag(A) := {Ap,p}, and the other is the total complex Tot(A) := { p+q=• Ap,q}. The (generalized) Eilenberg-Zilber theorem [DP61] (see, e.g. [Wei95, TheoremP 8.5.1]) states that diag(A) is quasi-isomorphic to Tot(A). -
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020 Lectures by R. Inanç˙ Baykur University of Massachusetts Amherst Disclaimer These notes were taken during lecture using the vimtex package of the editor neovim. Any errors are mine and not the instructor’s. In addition, my notes are picture-free (but will include commutative diagrams) and are a mix of my mathematical style (omit lengthy computations, use category theory) and that of the instructor. If you find any errors, please contact me at [email protected]. Contents Contents • 2 1 January 21 • 5 1.1 Course Description • 5 1.2 Organization • 5 1.2.1 Notational conventions•5 1.3 Basic Notions • 5 1.4 Symplectic Linear Algebra • 6 2 January 23 • 8 2.1 More Basic Linear Algebra • 8 2.2 Compatible Complex Structures and Inner Products • 9 3 January 28 • 10 3.1 A Big Theorem • 10 3.2 More Compatibility • 11 4 January 30 • 13 4.1 Homework Exercises • 13 4.2 Subspaces of Symplectic Vector Spaces • 14 5 February 4 • 16 5.1 Linear Algebra, Conclusion • 16 5.2 Symplectic Vector Bundles • 16 6 February 6 • 18 6.1 Proof of Theorem 5.11 • 18 6.2 Vector Bundles, Continued • 18 6.3 Compatible Triples on Manifolds • 19 7 February 11 • 21 7.1 Obtaining Compatible Triples • 21 7.2 Complex Structures • 21 8 February 13 • 23 2 3 8.1 Kähler Forms Continued • 23 8.2 Some Algebraic Geometry • 24 8.3 Stein Manifolds • 25 9 February 20 • 26 9.1 Stein Manifolds Continued • 26 9.2 Topological Properties of Kähler Manifolds • 26 9.3 Complex and Symplectic Structures on 4-Manifolds • 27 10 February 25 -
Hamiltonian and Symplectic Symmetries: an Introduction
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 3, July 2017, Pages 383–436 http://dx.doi.org/10.1090/bull/1572 Article electronically published on March 6, 2017 HAMILTONIAN AND SYMPLECTIC SYMMETRIES: AN INTRODUCTION ALVARO´ PELAYO In memory of Professor J.J. Duistermaat (1942–2010) Abstract. Classical mechanical systems are modeled by a symplectic mani- fold (M,ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called sym- plectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton’s equations. Since then a number of connections with combinatorics, finite- dimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent re- sults on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry. 1. Introduction Symplectic geometry is concerned with the study of a notion of signed area, rather than length, distance, or volume. It can be, as we will see, less intuitive than Euclidean or metric geometry and it is taking mathematicians many years to understand its intricacies (which is work in progress). The word “symplectic” goes back to the 1946 book [164] by Hermann Weyl (1885–1955) on classical groups. -
Crossed Modules, Double Group-Groupoids and Crossed Squares
Filomat 34:6 (2020), 1755–1769 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL2006755T University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat Crossed Modules, Double Group-Groupoids and Crossed Squares Sedat Temela, Tunc¸ar S¸ahanb, Osman Mucukc aDepartment of Mathematics, Recep Tayyip Erdogan University, Rize, Turkey bDepartment of Mathematics, Aksaray University, Aksaray, Turkey cDepartment of Mathematics, Erciyes University, Kayseri, Turkey Abstract. The purpose of this paper is to obtain the notion of crossed module over group-groupoids con- sidering split extensions and prove a categorical equivalence between these types of crossed modules and double group-groupoids. This equivalence enables us to produce various examples of double groupoids. We also prove that crossed modules over group-groupoids are equivalent to crossed squares. 1. Introduction In this paper we are interested in crossed modules of group-groupoids associated with the split exten- sions and producing new examples of double groupoids in which the sets of squares, edges and points are group-groupoids. The idea of crossed module over groups was initially introduced by Whitehead in [29, 30] during the investigation of the properties of second relative homotopy groups for topological spaces. The categori- cal equivalence between crossed modules over groups and group-groupoids which are widely called in literature 2-groups [3], -groupoids or group objects in the category of groupoids [9], was proved by Brown and Spencer in [9, TheoremG 1]. Following this equivalence normal and quotient objects in these two cate- gories have been recently compared and associated objects in the category of group-groupoids have been characterized in [22]. -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
Symplectic Topology, Geometry and Gauge Theory Lisa Jeffrey
Symplectic Topology, Geometry and Gauge Theory Lisa Jeffrey 1 Symplectic geometry has its roots in classical mechanics. A prototype for a symplectic manifold is the phase space which parametrizes the position q and momentum p of a classical particle. If the Hamiltonian (kinetic + potential en- ergy) is p2 H = + V (q) 2 then the motion of the particle is described by Hamilton’s equations dq ∂H = = p dt ∂p dp ∂H ∂V = − = − dt ∂q ∂q 2 In mathematical terms, a symplectic manifold is a manifold M endowed with a 2-form ω which is: • closed: dω = 0 (integral of ω over a 2-dimensional subman- ifold which is the boundary of a 3-manifold is 0) • nondegenerate (at any x ∈ M ω gives a map ∗ from the tangent space TxM to its dual Tx M; nondegeneracy means this map is invertible) If M = R2 is the phase space equipped with the Hamiltonian H then the symplectic form ω = dq ∧ dp transforms the 1-form ∂H ∂H dH = dp + dq ∂p ∂q to the vector field ∂H ∂H X = (− , ) H ∂q ∂p 3 The flow associated to XH is the flow satisfying Hamilton’s equations. Darboux’s theorem says that near any point of M there are local coordinates x1, . , xn, y1, . , yn such that Xn ω = dxi ∧ dyi. i=1 (Note that symplectic structures only exist on even-dimensional manifolds) Thus there are no local invariants that distin- guish between symplectic structures: at a local level all symplectic forms are identical. In contrast, Riemannian metrics g have local invariants, the curvature tensors R(g), such that if R(g1) =6 R(g2) then there is no smooth map that pulls back g1 to g2. -
Holonomic D-Modules and Positive Characteristic
Japan. J. Math. 4, 1–25 (2009) DOI: 10.1007/s11537-009-0852-x Holonomic -modules and positive characteristic Maxim Kontsevich Received: 8 December 2008 / Revised: 6 January 2009 / Accepted: 11 February 2009 Published online: ????? c The Mathematical Society of Japan and Springer 2009 Communicated by: Toshiyuki Kobayashi Abstract. We discuss a hypothetical correspondence between holonomic -modules on an al- gebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to posi- tive characteristic. Keywords and phrases: rings of differential operators and their modules, characteristic p methods Mathematics Subject Classification (2000): 13N10, 13A35 1. Quantization and -modules 1.1. Quantization Let us recall the heuristic dictionary of quantization. symplectic C∞-manifold (Y,ω) ←→ complex Hilbert space H complex-valued functions on Y ←→ operators in H Lagrangian submanifolds L ⊂ Y ←→ vectors v ∈ H In the basic example of the cotangent bundle Y = T ∗X the space H is L2(X), functions on Y which are polynomial along fibers of the projection Y = T ∗X → X correspond to differential operators, and with the Lagrangian manifold L ⊂ Y This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008. M. KONTSEVICH Institut des Hautes Etudes´ Scientifiques, 35 route de Chartres, Bures-sur-Yvette 91440, France (e-mail: [email protected]) 2 M. Kontsevich of the form L = graphdF for some function F ∈ C∞(Y) we associate (approxi- mately) vector exp(iF/) where → 0 is a small parameter (“Planck constant”). -
Symplectic Differential Geometry Definition 1.1. a Two Form Ω on A
1. Lecture 2: Symplectic differential geometry Definition 1.1. A two form ! on a manifold M is symplectic if and only if 1) 8x 2 M, !(x) is symplectic on TxM; 2) d! = 0 (! is closed). Examples: 1) (R2n; σ) is symplectic manifold. 2) If N is a manifold, then ∗ T N = f(q; p)jp linear form on TxMg is a symplectic manifold. Let q1; ··· ; qn be local coordinates on N and let p1; ··· ; pn be the dual coordinates. Then the symplectic form is defined by n X i ! = dp ^ dqi: i=1 One can check that ! does not depend on the choice of coordinates and is a symplectic form. We can also define a one form n X i λ = pdq = p dqi: i=1 It is well defined and dλ = !. 3) Projective algebraic manifolds(See also K¨ahlermanifolds) CP n has a canonical symplectic structure σ and is also a complex manifold. The complex structure J and the symplectic form σ are compatible. CP n has a hermitian metric h. For any z 2 CP n, h(z) n is a hermitian inner product on TzCP . h = g + iσ, where g is a Riemannian metric and σ(Jξ; ξ) = g(ξ; ξ). Claim: A complex submanifold M of CP n carries a natural sym- plectic structure. Indeed, consider σjM . It's obviously skew-symmetric and closed. We must prove that σjM is non-degenerate. This is true because if ξ 2 TxMf0g and Jξ 2 TxM, so !(x)(ξ; Jξ) 6= 0 Definition 1.2. A submanifold in symplectic manifold (M; !) is La- 1 grangian if and only if !jTxL = 0 for all x 2 M and dim L = 2 dim M.