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From Arthur Cayley Via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and Superstrings to Cantorian Space–Time
Available online at www.sciencedirect.com Chaos, Solitons and Fractals 37 (2008) 1279–1288 www.elsevier.com/locate/chaos From Arthur Cayley via Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan, Emmy Noether and superstrings to Cantorian space–time L. Marek-Crnjac Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, P.O. Box 2964, SI-1001 Ljubljana, Slovenia Abstract In this work we present a historical overview of mathematical discoveries which lead to fundamental developments in super string theory, super gravity and finally to E-infinity Cantorian space–time theory. Cantorian space–time is a hierarchical fractal-like semi manifold with formally infinity many dimensions but a finite expectation number for these dimensions. The idea of hierarchy and self-similarity in science was first entertain by Right in the 18th century, later on the idea was repeated by Swedenborg and Charlier. Interestingly, the work of Mohamed El Naschie and his two contra parts Ord and Nottale was done independently without any knowledge of the above starting from non- linear dynamics and fractals. Ó 2008 Published by Elsevier Ltd. 1. Introduction Many of the profound mathematical discovery and dare I say also inventions which were made by the mathemati- cians Arthur Cayley, Felix Klein, Sophus Lie, Wilhelm Killing, Elie Cartan and Emmy Noether [1] are extremely important for high energy particles in general [2] as well as in the development of E-infinity, Cantorian space–time the- ory [3,4]. The present paper is dedicated to the historical background of this subject. 2. Arthur Cayley – beginner of the group theory in the modern way Arthur Cayley was a great British mathematician. -
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). -
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 -
Oldandnewonthe Exceptionalgroupg2 Ilka Agricola
OldandNewonthe ExceptionalGroupG2 Ilka Agricola n a talk delivered in Leipzig (Germany) on product, the Lie bracket [ , ]; as a purely algebraic June 11, 1900, Friedrich Engel gave the object it is more accessible than the original Lie first public account of his newly discovered group G. If G happens to be a group of matrices, its description of the smallest exceptional Lie Lie algebra g is easily realized by matrices too, and group G2, and he wrote in the corresponding the Lie bracket coincides with the usual commuta- Inote to the Royal Saxonian Academy of Sciences: tor of matrices. In Killing’s and Lie’s time, no clear Moreover, we hereby obtain a direct defi- distinction was made between the Lie group and nition of our 14-dimensional simple group its Lie algebra. For his classification, Killing chose [G2] which is as elegant as one can wish for. a maximal set h of linearly independent, pairwise 1 [En00, p. 73] commuting elements of g and constructed base Indeed, Engel’s definition of G2 as the isotropy vectors Xα of g (indexed over a finite subset R of group of a generic 3-form in 7 dimensions is at elements α ∈ h∗, the roots) on which all elements the basis of a rich geometry that exists only on of h act diagonally through [ , ]: 7-dimensional manifolds, whose full beauty has been unveiled in the last thirty years. [H,Xα] = α(H)Xα for all H ∈ h. This article is devoted to a detailed historical In order to avoid problems when doing so he chose and mathematical account of G ’s first years, in 2 the complex numbers C as the ground field. -
Chapter 18 Metrics, Connections, and Curvature on Lie Groups
Chapter 18 Metrics, Connections, and Curvature on Lie Groups 18.1 Left (resp. Right) Invariant Metrics Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all the Riemannian metrics on a Lie groups, those for which the left translations (or the right translations) are isometries are of particular interest because they take the group structure of G into account. This chapter makes extensive use of results from a beau- tiful paper of Milnor [38]. 831 832 CHAPTER 18. METRICS, CONNECTIONS, AND CURVATURE ON LIE GROUPS Definition 18.1. Ametric , on a Lie group G is called left-invariant (resp. right-invarianth i )i↵ u, v = (dL ) u, (dL ) v h ib h a b a b iab (resp. u, v = (dR ) u, (dR ) v ), h ib h a b a b iba for all a, b G and all u, v T G. 2 2 b ARiemannianmetricthatisbothleftandright-invariant is called a bi-invariant metric. In the sequel, the identity element of the Lie group, G, will be denoted by e or 1. 18.1. LEFT (RESP. RIGHT) INVARIANT METRICS 833 Proposition 18.1. There is a bijective correspondence between left-invariant (resp. right invariant) metrics on a Lie group G, and inner products on the Lie al- gebra g of G. If , be an inner product on g,andset h i u, v g = (dL 1)gu, (dL 1)gv , h i h g− g− i for all u, v TgG and all g G.Itisfairlyeasytocheck that the above2 induces a left-invariant2 metric on G. -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics. -
Mathematisches Forschungsinstitut Oberwolfach the History Of
Mathematisches Forschungsinstitut Oberwolfach Report No. 51/2004 The History of Differential Equations, 1670–1950 Organised by Thomas Archibald (Wolfville) Craig Fraser (Toronto) Ivor Grattan-Guinness (Middlesex) October 31st – November 6th, 2004 Mathematics Subject Classification (2000): 01A45, 01A50, 01A55, 01A60, 34-03, 35-03. Introduction by the Organisers Differential equations have been a major branch of pure and applied mathematics since their inauguration in the mid 17th century. While their history has been well studied, it remains a vital field of on-going investigation, with the emergence of new connections with other parts of mathematics, fertile interplay with applied subjects, interesting reformulation of basic problems and theory in various periods, new vistas in the 20th century, and so on. In this meeting we considered some of the principal parts of this story, from the launch with Newton and Leibniz up to around 1950. ‘Differential equations’ began with Leibniz, the Bernoulli brothers and others from the 1680s, not long after Newton’s ‘fluxional equations’ in the 1670s. Appli- cations were made largely to geometry and mechanics; isoperimetrical problems were exercises in optimisation. Most 18th-century developments consolidated the Leibnizian tradition, extend- ing its multi-variate form, thus leading to partial differential equations. General- isation of isoperimetrical problems led to the calculus of variations. New figures appeared, especially Euler, Daniel Bernoulli, Lagrange and Laplace. Development of the general theory of solutions included singular ones, functional solutions and those by infinite series. Many applications were made to mechanics, especially to astronomy and continuous media. In the 19th century: general theory was enriched by development of the un- derstanding of general and particular solutions, and of existence theorems. -
LIE, MARIUS SOPHUS (B
ndsbv4_L 9/25/07 1:15 PM Page 307 Licklider Lie Technical Documents.” American Documentation 17 (1966): reliable history book on IPTO based on the historical 186–189. documents of the agency that include normally inaccessible With Robert W. Taylor and Evan Herbert. “Computer as a records. Communication Device.” Science & Technology 76 (1968): O’Neill, Judy E. “The Role of ARPA in the Development of the 21–31. ARPANET, 1961–1972.” IEEE Annals of the History of With A. Vezza. “Applications of Information Technology.” Computing 17, no. 4 (1995): 76–81. Proceedings of the IEEE 66 (1978): 1330–1346. Waldrop, M. Mitchell. The Dream Machine: J. C. R. Licklider and the Revolution That Made Computing Personal. New York: OTHER SOURCES Viking, 2001. Biographical history book for general readers Aspray, William, and Arthur L. Norberg. “J. C. R. Licklider.” based on documents and interviews. Author’s interviews with OH 150. 28 October 1988, Cambridge, MA; Charles Louise Licklider and others result in a lively description of Babbage Institute, University of Minnesota, Minneapolis. A Licklider’s personality. scholarly oral history record. “Bolt Beranek and Newman: The First 40 Years.” Special issues, Chigusa Ishikawa Kita IEEE Annals of the History of Computing 27, no. 2 (2005); 28, no. 1 (2006). Some of these papers refer to Licklider’s contribution in the establishment of BBN’s role in the history of computing and networking: Leo Beranek, “BBN’s Earliest Days: Founding a Culture of Engineering Creativity” LIE, MARIUS SOPHUS (b. Nordfjordeide, (27, no. 2: 6–14); John A. Swets, “ABC’s of BBN: From Norway, 17 December 1842; d. -
E8, the MOST EXCEPTIONAL GROUP Contents 1. Introduction 1 2. What Is
E8, THE MOST EXCEPTIONAL GROUP SKIP GARIBALDI Abstract. The five exceptional simple Lie algebras over the complex number are included one within the other as g2 ⊂ f4 ⊂ e6 ⊂ e7 ⊂ e8. The biggest one, e8, is in many ways the most mysterious. This article surveys what is known about it including many recent results, focusing on the point of view of Lie algebras and algebraic groups over fields. Contents 1. Introduction1 2. What is E8?3 3. E8 as an automorphism group5 4. Constructing the Lie algebra via gradings7 5. E8 over the real numbers9 6. E8 over an arbitrary field 12 7. Tits' construction 13 8. Cohomological invariants; the Rost invariant 15 9. The kernel of the Rost invariant; Semenov's invariant 17 10. Witt invariants 18 11. Connection with division algebras 19 12. Other recent results on E8 20 About the author 21 Acknowledgments 21 References 22 1. Introduction The Lie algebra e8 or Lie group E8 was first sighted by a human being sometime in summer or early fall of 1887, by Wilhelm Killing as part of his program to classify the semisimple finite-dimensional Lie algebras over the complex numbers [94, pp. 162{163]. Since then, it has been a source of fascination for mathematicians and others in its role as the largest of the exceptional Lie algebras. (It appears, for example, as part of the fictional Beard-Einstein Conflation in the prize-winning novel Solar [120].) Killing's classification is now considered the core of a typical graduate course on Lie algebras, and the paper containing the key ideas, [104], has Date: Version of May 16, 2016. -
View This Volume's Front and Back Matter
Titles in This Series Volume 8 Kare n Hunger Parshall and David £. Rowe The emergenc e o f th e America n mathematica l researc h community , 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moore 1994 7 Hen k J. M. Bos Lectures in the history of mathematic s 1993 6 Smilk a Zdravkovska and Peter L. Duren, Editors Golden years of Moscow mathematic s 1993 5 Georg e W. Mackey The scop e an d histor y o f commutativ e an d noncommutativ e harmoni c analysis 1992 4 Charle s W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Pete r L. Duren, editor, et al. A century of mathematics in America, part III 1989 2 Pete r L. Duren, editor, et al. A century of mathematics in America, part II 1989 1 Pete r L. Duren, editor, et al. A century of mathematics in America, part I 1988 This page intentionally left blank https://doi.org/10.1090/hmath/008 History of Mathematics Volume 8 The Emergence o f the American Mathematical Research Community, 1876-1900: J . J. Sylvester, Felix Klein, and E. H. Moor e Karen Hunger Parshall David E. Rowe American Mathematical Societ y London Mathematical Societ y 1991 Mathematics Subject Classification. Primary 01A55 , 01A72, 01A73; Secondary 01A60 , 01A74, 01A80. Photographs o n th e cove r ar e (clockwis e fro m right ) th e Gottinge n Mathematisch e Ges - selschafft, Feli x Klein, J. J. Sylvester, and E. H. Moore. -
Sophus Lie: a Real Giant in Mathematics by Lizhen Ji*
Sophus Lie: A Real Giant in Mathematics by Lizhen Ji* Abstract. This article presents a brief introduction other. If we treat discrete or finite groups as spe- to the life and work of Sophus Lie, in particular cial (or degenerate, zero-dimensional) Lie groups, his early fruitful interaction and later conflict with then almost every subject in mathematics uses Lie Felix Klein in connection with the Erlangen program, groups. As H. Poincaré told Lie [25] in October 1882, his productive writing collaboration with Friedrich “all of mathematics is a matter of groups.” It is Engel, and the publication and editing of his collected clear that the importance of groups comes from works. their actions. For a list of topics of group actions, see [17]. Lie theory was the creation of Sophus Lie, and Lie Contents is most famous for it. But Lie’s work is broader than this. What else did Lie achieve besides his work in the 1 Introduction ..................... 66 Lie theory? This might not be so well known. The dif- 2 Some General Comments on Lie and His Impact 67 ferential geometer S. S. Chern wrote in 1992 that “Lie 3 A Glimpse of Lie’s Early Academic Life .... 68 was a great mathematician even without Lie groups” 4 A Mature Lie and His Collaboration with Engel 69 [7]. What did and can Chern mean? We will attempt 5 Lie’s Breakdown and a Final Major Result . 71 to give a summary of some major contributions of 6 An Overview of Lie’s Major Works ....... 72 Lie in §6. -
Einstein and Conformally Einstein Bi-Invariant Semi-Riemannian Metrics
Einstein and conformally Einstein bi-invariant semi-Riemannian metrics Kelli L. Francis-Staite Thesis submitted in partial fulfilment of the requirements for the degree of Master of Philosophy in Pure Mathematics at The University of Adelaide School of Mathematical Sciences September 2015 ii Signed Statement I certify that this work contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. In addition, I certify that no part of this work will, in the future, be used in a submission for any other degree or diploma in any university or other tertiary institution without the prior approval of the University of Adelaide and where applicable, any partner institution responsible for the joint-award of this degree. I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopying, subject to the provisions of the Copyright Act 1968. I also give permission for the digital version of my thesis to be made available on the web, via the University's digital research repository, the Library catalogue and also through web search engines, unless permission has been granted by the University to restrict access for a period of time. SIGNED: ........................ DATE: .......................... iii iv Contents Signed Statement iii Abstract vii Dedication ix Acknowledgements xi Introduction xiii 1 Semi-Riemannian metrics and their curvature1 1.1 Connections and curvature...........................2 1.2 Ricci and scalar curvature............................7 1.3 Differential operators and more on curvature.................9 1.4 The Schouten, Weyl, Cotton and Bach tensors...............