DOCSLIB.ORG

An Introduction to Lie Groups and Symplectic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Lie Groups and Symplectic Geometry An Introduction to Lie Groups and Symplectic Geometry A series of nine lectures on Lie groups and symplectic geometry delivered at the Regional Geometry Institute in Park City, Utah, 24 June–20 July 1991. by Robert L. Bryant Duke University Durham, NC [email protected] This is an unofficial version of the notes and was last modified on 23 July 2018. Introduction These are the lecture notes for a short course entitled “Introduction to Lie groups and symplectic geometry” that I gave at the 1991 Regional Geometry Institute at Park City, Utah starting on 24 June and ending on 11 July. The course really was designed to be an introduction, aimed at an audience of stu- dents who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for Lie groups and symplectic geometry. My purpose was not to provide an exhaustive treatment of either Lie groups, which would have been impossible even if I had had an entire year, or of symplectic manifolds, which has lately undergone something of a revolution. Instead, I tried to provide an introduction to what I regard as the basic concepts of the two subjects, with an emphasis on examples that drove the development of the theory. I deliberately tried to include a few topics that are not part of the mainstream subject, such as Lie’s reduction of order for differential equations and its relation with the notion of a solvable group on the one hand and integration of ODE by quadrature on the other. I also tried, in the later lectures to introduce the reader to some of the global methods that are now becoming so important in symplectic geometry. However, a full treatment of these topics in the space of nine lectures beginning at the elementary level was beyond my abilities. After the lectures were over, I contemplated reworking these notes into a comprehen- sive introduction to modern symplectic geometry and, after some soul-searching, finally decided against this. Thus, I have contented myself with making only minor modifications and corrections, with the hope that an interested person could read these notes in a few weeks and get some sense of what the subject was about. An essential feature of the course was the exercise sets. Each set begins with elemen- tary material and works up to more involved and delicate problems. My object was to provide a path to understanding of the material that could be entered at several different levels and so the exercises vary greatly in difficulty. Many of these exercise sets are obvi- ously too long for any person to do them during the three weeks the course, so I provided extensive hints to aid the student in completing the exercises after the course was over. I want to take this opportunity to thank the many people who made helpful sugges- tions for these notes both during and after the course. Particular thanks goes to Karen Uhlenbeck and Dan Freed, who invited me to give an introductory set of lectures at the RGI, and to my course assistant, Tom Ivey, who provided invaluable help and criticism in the early stages of the notes and tirelessly helped the students with the exercises. While the faults of the presentation are entirely my own, without the help, encouragement, and proofreading contributed by these folks and others, neither these notes nor the course would never have come to pass. I.1 2 Background Material and Basic Terminology. In these lectures, I assume that the reader is familiar with the basic notions of manifolds, vector fields, and differential forms. All manifolds will be assumed to be both second countable and Hausdorff. Also, unless I say otherwise, I generally assume that all maps and manifolds are C∞. Since it came up several times in the course of the course of the lectures, it is probably worth emphasizing the following point: A submanifold of a smooth manifold X is, by definition, a pair (S, f) where S is a smooth manifold and f: S X is a one-to-one immersion. In particular, f need not be an embedding. → The notation I use for smooth manifolds and mappings is fairly standard, but with a few slight variations: If f: X Y is a smooth mapping, then f ′: TX TY denotes the induced mapping → ′ → on tangent bundles, with f (x) denoting its restriction to TxX. (However, I follow tradition when X = R and let f ′(t) stand for f ′(t)(∂/∂t) for all t R. I trust that this abuse of notation will not cause confusion.) ∈ For any vector space V , I generally use Ap(V ) (instead of, say, Λp(V ∗)) to denote the space of alternating (or exterior) p-forms on V . For a smooth manifold M, I denote the space of smooth, alternating p-forms on M by p(M). The algebra of all (smooth) differential forms on M is denoted by ∗(M). A A I generally reserve the letter d for the exterior derivative d: p(M) p+1(M). A →A For any vector field X on M, I will denote left-hook with X (often called interior product with X) by the symbol X . This is the graded derivation of degree 1 of ∗(M) that satisfies X (df) = Xf for all smooth functions f on M. For example,− theA Cartan formula for the Lie derivative of differential forms is written in the form LX φ = X dφ + d(X φ). Jets. Occasionally, it will be convenient to use the language of jets in describing certain constructions. Jets provide a coordinate free way to talk about the Taylor expansion of some mapping up to a specified order. No detailed knowledge about these objects will be needed in these lectures, so the following comments should suffice: If f and g are two smooth maps from a manifold Xm to a manifold Y n, we say that f and g agree to order k at x X if, first, f(x) = g(x) = y Y and, second, when u: U Rm and v: V Rn are local∈ coordinate systems centered∈ on x and y respectively, the functions→ F = v →f u−1 and G = v g u−1 have the same Taylor series at 0 Rm up to and including order◦ k◦. Using the Chain◦ ◦ Rule, it is not hard to show that this conditi∈ on is independent of the choice of local coordinates u and v centered at x and y respectively. The notation f x,k g will mean that f and g agree to order k at x. This is easily ≡ k seen to define an equivalence relation. Denote the x,k-equivalence class of f by j (f)(x), and call it the k-jet of f at x. ≡ For example, knowing the 1-jet at x of a map f: X Y is equivalent to knowing both f(x) and the linear map f ′(x): T T Y . → x → f(x) I.2 3 The set of k-jets of maps from X to Y is usually denoted by J k(X,Y ). It is not hard to show that J k(X,Y ) can be given a unique smooth manifold structure in such a way that, for any smooth f: X Y , the obvious map jk(f): X J k(X,Y ) is also smooth. → → These jet spaces have various functorial properties that we shall not need at all. The main reason for introducing this notion is to give meaning to concise statements like “The critical points of f are determined by its 1-jet”, “The curvature at x of a Riemannian metric g is determined by its 2-jet at x”, or, from Lecture 8, “The integrability of an almost complex structure J: TX TX is determined by its 1-jet”. Should the reader wish to learn more about jets, I recommend→ the first two chapters of [GG]. Basic and Semi-Basic. Finally, I use the following terminology: If π: V X is a smooth submersion, a p-form φ p(V ) is said to be π-basic if it can be written→ in the form φ = π∗(ϕ) for some ϕ ∈ Ap(X) and π-semi-basic if, for any π-vertical*vector field X, we have X φ = 0. When∈ the A map π is clear from context, the terms “basic” or “semi-basic” are used. It is an elementary result that if the fibers of π are connected and φ is a p-form on V with the property that both φ and dφ are π-semi-basic, then φ is actually π-basic. At least in the early lectures, we will need very little in the way of major theorems, but we will make extensive use of the following results: The Implicit Function Theorem: If f: X Y is a smooth map of manifolds and y• Y is a regular value of f, then f −1(y) X →is a smooth embedded submanifold of X, with∈ ⊂ T f −1(y) = ker(f ′(x): T X T Y ) x x → y Existence and Uniqueness of Solutions of ODE: If X is a vector field on a smooth• manifold M, then there exists an open neighborhood U of 0 M in R M and a smooth mapping F : U M with the following properties: { } × × → i. F (0, m) = m for all m M. ∈ ii. For each m M, the slice U = t R (t, m) U is an open interval in R ∈ m { ∈ | ∈ } (containing 0) and the smooth mapping φm: Um M defined by φm(t) = F (t, m) is an integral curve of X.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Material on Algebraic and Lie Groups
    2 Lie groups and algebraic groups. 2.1 Basic Definitions. In this subsection we will introduce the class of groups to be studied. We first recall that a Lie group is a group that is also a differentiable manifold 1 and multiplication (x, y xy) and inverse (x x ) are C1 maps. An algebraic group is a group7! that is also an algebraic7! variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we first consider GL(n, C) as an affi ne algebraic group. Here Mn(C) denotes the space of n n matrices and GL(n, C) = g Mn(C) det(g) =) . Now Mn(C) is given the structure nf2 2 j 6 g of affi ne space C with the coordinates xij for X = [xij] . This implies that GL(n, C) is Z-open and as a variety is isomorphic with the affi ne variety 1 Mn(C) det . This implies that (GL(n, C)) = C[xij, det ]. f g O Lemma 1 If G is an algebraic group over an algebraically closed field, F , then every point in G is smooth. Proof. Let Lg : G G be given by Lgx = gx. Then Lg is an isomorphism ! 1 1 of G as an algebraic variety (Lg = Lg ). Since isomorphisms preserve the set of smooth points we see that if x G is smooth so is every element of Gx = G. 2 Proposition 2 If G is an algebraic group over an algebraically closed field F then the Z-connected components Proof.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Be the Integral Symplectic Group and S(G) Be the Set of All Positive Integers Which Can Occur As the Order of an Element in G
    FINITE ORDER ELEMENTS IN THE INTEGRAL SYMPLECTIC GROUP KUMAR BALASUBRAMANIAN, M. RAM MURTY, AND KARAM DEO SHANKHADHAR Abstract For g 2 N, let G = Sp(2g; Z) be the integral symplectic group and S(g) be the set of all positive integers which can occur as the order of an element in G. In this paper, we show that S(g) is a bounded subset of R for all positive integers g. We also study the growth of the functions f(g) = jS(g)j, and h(g) = maxfm 2 N j m 2 S(g)g and show that they have at least exponential growth. 1. Introduction Given a group G and a positive integer m 2 N, it is natural to ask if there exists k 2 G such that o(k) = m, where o(k) denotes the order of the element k. In this paper, we make some observations about the collection of positive integers which can occur as orders of elements in G = Sp(2g; Z). Before we proceed further we set up some notations and briefly mention the questions studied in this paper. Let G = Sp(2g; Z) be the group of all 2g × 2g matrices with integral entries satisfying > A JA = J 0 I where A> is the transpose of the matrix A and J = g g . −Ig 0g α1 αk Throughout we write m = p1 : : : pk , where pi is a prime and αi > 0 for all i 2 f1; 2; : : : ; kg. We also assume that the primes pi are such that pi < pi+1 for 1 ≤ i < k.
    [Show full text]
  • Matrix Lie Groups
    Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2).
    [Show full text]
  • LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
    LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1.
    [Show full text]
  • 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
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]
  • The History of the Abel Prize and the Honorary Abel Prize the History of the Abel Prize
    The History of the Abel Prize and the Honorary Abel Prize The History of the Abel Prize Arild Stubhaug On the bicentennial of Niels Henrik Abel’s birth in 2002, the Norwegian Govern- ment decided to establish a memorial fund of NOK 200 million. The chief purpose of the fund was to lay the financial groundwork for an annual international prize of NOK 6 million to one or more mathematicians for outstanding scientific work. The prize was awarded for the first time in 2003. That is the history in brief of the Abel Prize as we know it today. Behind this government decision to commemorate and honor the country’s great mathematician, however, lies a more than hundred year old wish and a short and intense period of activity. Volumes of Abel’s collected works were published in 1839 and 1881. The first was edited by Bernt Michael Holmboe (Abel’s teacher), the second by Sophus Lie and Ludvig Sylow. Both editions were paid for with public funds and published to honor the famous scientist. The first time that there was a discussion in a broader context about honoring Niels Henrik Abel’s memory, was at the meeting of Scan- dinavian natural scientists in Norway’s capital in 1886. These meetings of natural scientists, which were held alternately in each of the Scandinavian capitals (with the exception of the very first meeting in 1839, which took place in Gothenburg, Swe- den), were the most important fora for Scandinavian natural scientists. The meeting in 1886 in Oslo (called Christiania at the time) was the 13th in the series.
    [Show full text]