DOCSLIB.ORG

Lie Group, Lie Algebra, Exponential Mapping, Linearization Killing Form, Cartan's Criteria

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lie Group, Lie Algebra, Exponential Mapping, Linearization Killing Form, Cartan's Criteria American Journal of Computational and Applied Mathematics 2016, 6(5): 182-186 DOI: 10.5923/j.ajcam.20160605.02 On Extracting Properties of Lie Groups from Their Lie Algebras M-Alamin A. H. Ahmed1,2 1Department of Mathematics, Faculty of Science and Arts – Khulais, University of Jeddah, Saudi Arabia 2Department of Mathematics, Faculty of Education, Alzaeim Alazhari University (AAU), Khartoum, Sudan Abstract The study of Lie groups and Lie algebras is very useful, for its wider applications in various scientific fields. In this paper, we introduce a thorough study of properties of Lie groups via their lie algebras, this is because by using linearization of a Lie group or other methods, we can obtain its Lie algebra, and using the exponential mapping again, we can convey properties and operations from the Lie algebra to its Lie group. These relations helpin extracting most of the properties of Lie groups by using their Lie algebras. We explain this extraction of properties, which helps in introducing more properties of Lie groups and leads to some important results and useful applications. Also we gave some properties due to Cartan’s first and second criteria. Keywords Lie group, Lie algebra, Exponential mapping, Linearization Killing form, Cartan’s criteria 1. Introduction 2. Lie Group The Properties and applications of Lie groups and their 2.1. Definition (Lie group) Lie algebras are too great to overview in a small paper. Lie groups were greatly studied by Marius Sophus Lie (1842 – A Lie group G is an abstract group and a smooth 1899), who intended to solve or at least to simplify ordinary n-dimensional manifold so that multiplication differential equations, and since then the research continues G×→ G G:,( a b) → ab and inverse G→→ Ga: a−1 to disclose their properties and applications in various are smooth. scientific fields [1, 2, 5, 6]. Lie groups have algebraic properties derived from their group axioms, and geometric 2.2. Examples properties due to identification of their group operations with points in a topological space, namely every Lie group is a a) The general linear group GL( n,)= differentiable manifold [3, 4, 7]. Almost all of the Lie groups ∈=≠nn× nn× encountered in many applications are matrix groups, and this {AA :det 0} is an open set of , it simplifies their study. This simplification can be done by is a manifold with the two operations, matrix linearizing the Lie group in the neighborhood of its identity, multiplication and matrix inverse which are C∞ which results in what is called a Lie algebra [1]. The Lie mappings. So it is a Lie group. algebra retains most of, but not all of the properties of the original Lie group. Also most of the Lie group properties can b) The complex linear group GL( n,) , the group of be recovered by the inverse of the linearization operation, nonsingular nn× matrices is also a Lie group. namely, the exponential mapping. A Lie algebra is a linear c) We can take the two subsets of the general linear vector space, and this makes it easy to study it by using tools of linear algebra rather than studying the original Lie group group: the special linear group SL( n,) which is with its complicated structure. In this manner we can extract the group of matrices of determinant 1 and the other is properties of a Lie group from its Lie algebra, and this opens the orthogonal group On( ) consisting of all the door to many scientific applications, for instance in T mathematics and physics. matrices A satisfying AA= I, these two subsets are also Lie groups. * Corresponding author: [email protected] (M-Alamin A. H. Ahmed) Matrix groups are very important examples of Lie groups Published online at http://journal.sapub.org/ajcam because of their various scientific applications. There are Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved many examples of Lie groups different from matrix groups. American Journal of Computational and Applied Mathematics 2016, 6(5): 182-186 183 3. Lie Algebra linearization immediately as we see in the following example: 3.1. Definition (Lie Algebra) 4.2. Example A Lie algebra over a field K ( or ) is a vector space V over K with a skew – symmetric k- bilinear form (the Lie To linearize the Lie group bracket) [ ,:] VV×→ V, which satisfies the Jacobi identity: SL(2,)= {M∈= GL(2,) ,det M 1} X,,[ YZ] ++= Y ,,[ ZX] Z ,[ XY ,] 0 (3.1) We parametrize the group by: 1+ab For all XYZ,, ∈ V →= (4.1) (abc,,) M( abc ,,) (1+bc) The tangent space TGe at the identity of a Lie group c (1+ a) canonically has the structure of Lie algebra. This Lie algebra encodes in it much information about the Lie group. So we then we let (abc,,) to become infinitesimals: have to define the Lie algebra structure on TGe and this can abc,,→→δδδ abc ,, M δδδ abc ,, be shown if we identified this structure with the Lie algebra ( ) ( ) ( ) for instance for some classical groups. 1+δδab (4.2) = (1+δδbc) 3.2. Example δ c (1+δ a) × Since GL( n,) is an open subset of nn (the vector so the basis vectors in the Lie algebra X, X and X are space of all nn× real matrices), the tangent space to ab c × the coefficients of the first order infinitesimals: GL( n,) at the identity I is nn itself. 1+δδab Also the tangent space TI ( SL( n,)) is the subspace of (δδδa,, b c)→+ I2 δ aXabc + δ bX + δ cX =(4.3) δδca1− nn× × consisting of all nn matrices of trace 0. We get the basis vectors from (4-1) as follows: And for the group of orthogonal matrices On( ) , the tangent space at the identity is consisting of skew – ∂M 10 symmetric matrices. These are some examples of Lie X a = = , ∂a (0,0,0) 01− algebras of some groups of matrices considered as tangent spaces at the identity. ∂ 01 M (4.4) Xb = = , ∂b (0,0,0) 00 4. Linearization of a Lie Group ∂M 00 X = = The Lie group may have a complicated structure which c ∂c (0,0,0) 10 make it difficult to study it through some mathematical tools, instead of this we linearize the group in the neighborhood of These are the basis vectors for the Lie algebra sl (2,) its identity, that is by getting a vector space at the identity, this vector space is its Lie algebra. Therefore, Linearization in our example. of Lie group makes it possible to study it by using tools of Linear algebra. Moreover most of the Lie group properties can be recovered by the inverse of the linearization operation, 5. Exponential Mapping that is by the exponential mapping. The following diagram Suppose V is a finite dimensional vector space equipped expresses how to go from the Lie group to its Lie algebra and with a complete norm . over the field or . If vice versa by using these two operations: End( V ) denotes the algebra of linear automorphisms on , Linearization V Lie group ←→ Lie alg ebra n Exponentiation and GL( V ) the general linear group. Then for V = , End( V ) is identified with M n () ( nn× matrices ) and 4.1. How To Linearize a Lie Group? GL( V)= GL( n,) . In order to linearize a Lie group to get its Lie algebra we need an explicit parametrization of the underlying manifold 5.1. Definition (Matrix Group) and also we need an available expression for the group composition law. For matrix groups the group composition A matrix group is a closed subgroup of GL( V ) (the law is matrix multiplication and every matrix group is a general linear group), where V is a finite dimensional closed subset of the general linear group, so we can do vector space. 184 M-Alamin A. H. Ahmed: On Extracting Properties of Lie Groups from Their Lie Algebras 5.2. Definition (Exponential map) 5.10. Theorem (The Lie Algebra of a Matrix Group) The exponential map on End( V ) is defined by For a matrix group G⊆ GL( V ) , the set ∞ n =∈∈∈ A g{ A EndV:exp( tA) G for all t R} is a Lie algebra, exp A =, A ∈ End V (5.1) ( ) ∑ ( ) called the Lie algebra of G . n =0 n! Sketch of the Proof: If G is a Lie group with Lie algebra g , then the The theorem can be proved by showing that: exponential map is defined as: exp(t[ A , B])= exp([ tA , B]) ∈∈ G for all t . Thus the exp:g→= GwhereexpXφφ 1 with' 0 = X , ( ) XX( ) ( ) commutator [AB, ]∈ g. ∀∈Xg 5.11. Computing Lie Algebras 5.3. Definition (One – Parameter Subgroup) For matrix groups we can use some methods to find their A one – parameter subgroup of a Lie group G is a Lie algebras. For instance we can use the following continuous homomorphism: α :,( +→) G proposition for this purpose. 5.4. Example 5.12. Proposition For the Lie groups GL( n,,) and GL( n ) , we have Let β (.,.) be a continuous bilinear form on the vector (ts, ) exp( A)= eA , e = etA e sA ,.. ieφ ( t)= etA is a one parameter space V and set G=∈{ h GL( V) :, ∀∈ x y V , ' A ββ= group. Furthermore φφ(0)= A and henceexp( A)=( I) = e . (hx,hy) ( x , y)} , then the Lie algebra is g=∈{ A EndV:, ∀∈ x y V ,ββ( Ax , y) +( x , Ay) = 0} 5.5. Proposition [2] Proof For V a finite dimensional normed vector space and ∈ β = A∈ End( V ) , the map t→exp( tA) is a one – parameter Let Ag, then (exp(tA) x ,exp( tA) y) −1 β x,, y for all x y∈ V .
Load more

Recommended publications
  • Contents 1 Root Systems
    Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations.
    [Show full text]
  • Introduction to Lie Algebras and Representation Theory (Following Humphreys)
    INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY (FOLLOWING HUMPHREYS) Ulrich Thiel Abstract. These are my (personal and incomplete!) notes on the excellent book Introduction to Lie Algebras and Representation Theory by Jim Humphreys [1]. Almost everything is basically copied word for word from the book, so please do not share this document. What is the point of writing these notes then? Well, first of all, they are my personal notes and I can write whatever I want. On the more serious side, despite what I just said, I actually tried to spell out most things in my own words, and when I felt I need to give more details (sometimes I really felt it is necessary), I have added more details. My actual motivation is that eventually I want to see what I can shuffle around and add/remove to reflect my personal thoughts on the subject and teach it more effectively in the future. To be honest, I do not want you to read these notes at all. The book is excellent already, and when you come across something you do not understand, it is much better to struggle with it and try to figure it out by yourself, than coming here and see what I have to say about it (if I have to say anything additionally). Also, these notes are not complete yet, will contain mistakes, and I will not keep it up to date with the current course schedule. It is just my notebook for the future. Document compiled on June 6, 2020 Ulrich Thiel, University of Kaiserslautern.
    [Show full text]
  • Finite Order Automorphisms on Real Simple Lie Algebras
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 364, Number 7, July 2012, Pages 3715–3749 S 0002-9947(2012)05604-2 Article electronically published on February 15, 2012 FINITE ORDER AUTOMORPHISMS ON REAL SIMPLE LIE ALGEBRAS MENG-KIAT CHUAH Abstract. We add extra data to the affine Dynkin diagrams to classify all the finite order automorphisms on real simple Lie algebras. As applications, we study the extensions of automorphisms on the maximal compact subalgebras and also study the fixed point sets of automorphisms. 1. Introduction Let g be a complex simple Lie algebra with Dynkin diagram D. A diagram automorphism on D of order r leads to the affine Dynkin diagram Dr. By assigning integers to the vertices, Kac [6] uses Dr to represent finite order g-automorphisms. This paper extends Kac’s result. It uses Dr to represent finite order automorphisms on the real forms g0 ⊂ g. As applications, it studies the extensions of finite order automorphisms on the maximal compact subalgebras of g0 and studies their fixed point sets. The special case of g0-involutions (i.e. automorphisms of order 2) is discussed in [2]. In general, Gothic letters denote complex Lie algebras or spaces, and adding the subscript 0 denotes real forms. The finite order automorphisms on compact or complex simple Lie algebras have been classified by Kac, so we ignore these cases. Thus we always assume that g0 is a noncompact real form of the complex simple Lie algebra g (this automatically rules out complex g0, since in this case g has two simple factors). 1 2 2 2 3 The affine Dynkin diagrams are D for all D, as well as An,Dn, E6 and D4.By 3 r Definition 1.1 below, we ignore D4.
    [Show full text]
  • Killing Forms, W-Invariants, and the Tensor Product Map
    Killing Forms, W-Invariants, and the Tensor Product Map Cameron Ruether Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of Master of Science in Mathematics1 Department of Mathematics and Statistics Faculty of Science University of Ottawa c Cameron Ruether, Ottawa, Canada, 2017 1The M.Sc. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics Abstract Associated to a split, semisimple linear algebraic group G is a group of invariant quadratic forms, which we denote Q(G). Namely, Q(G) is the group of quadratic forms in characters of a maximal torus which are fixed with respect to the action of the Weyl group of G. We compute Q(G) for various examples of products of the special linear, special orthogonal, and symplectic groups as well as for quotients of those examples by central subgroups. Homomorphisms between these linear algebraic groups induce homomorphisms between their groups of invariant quadratic forms. Since the linear algebraic groups are semisimple, Q(G) is isomorphic to Zn for some n, and so the induced maps can be described by a set of integers called Rost multipliers. We consider various cases of the Kronecker tensor product map between copies of the special linear, special orthogonal, and symplectic groups. We compute the Rost multipliers of the induced map in these examples, ultimately concluding that the Rost multipliers depend only on the dimensions of the underlying vector spaces. ii Dedications To root systems - You made it all worth Weyl.
    [Show full text]
  • Math 222 Notes for Apr. 1
    Math 222 notes for Apr. 1 Alison Miller 1 Real lie algebras and compact Lie groups, continued Last time we showed that if G is a compact Lie group with Z(G) finite and g = Lie(G), then the Killing form of g is negative definite, hence g is semisimple. Now we’ll do a converse. Proposition 1.1. Let g be a real lie algebra such that Bg is negative definite. Then there is a compact Lie group G with Lie algebra Lie(G) =∼ g. Proof. Pick any connected Lie group G0 with Lie algebra g. Since g is semisimple, the center Z(g) = 0, and so the center Z(G0) of G0 must be discrete in G0. Let G = G0/Z(G)0; then Lie(G) =∼ Lie(G0) =∼ g. We’ll show that G is compact. Now G0 has an adjoint action Ad : G0 GL(g), with kernel ker Ad = Z(G). Hence 0 ∼ G = Im(Ad). However, the Killing form Bg on g is Ad-invariant, so Im(Ad) is contained in the subgroup of GL(g) preserving the Killing! form. Since Bg is negative definite, this subgroup is isomorphic to O(n) (where n = dim(g). Since O(n) is compact, its closed subgroup Im(Ad) is also compact, and so G is compact. Note: I found a gap in this argument while writing it up; it assumes that Im(Ad) is closed in GL(g). This is in fact true, because Im(Ad) = Aut(g) is the group of automorphisms of the Lie algebra g, which is closed in GL(g); but proving this fact takes a bit of work.
    [Show full text]
  • 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.
    [Show full text]
  • Lie Groups and Lie Algebras
    version 1.6d — 24.11.2008 Lie Groups and Lie Algebras Lecture notes for 7CCMMS01/CMMS01/CM424Z Ingo Runkel j j Contents 1 Preamble 3 2 Symmetry in Physics 4 2.1 Definitionofagroup ......................... 4 2.2 RotationsandtheEuclideangroup . 6 2.3 Lorentzand Poincar´etransformations . 8 2.4 (*)Symmetriesinquantummechanics . 9 2.5 (*)Angularmomentuminquantummechanics . 11 3 MatrixLieGroupsandtheirLieAlgebras 12 3.1 MatrixLiegroups .......................... 12 3.2 Theexponentialmap......................... 14 3.3 TheLiealgebraofamatrixLiegroup . 15 3.4 A little zoo of matrix Lie groups and their Lie algebras . 17 3.5 Examples: SO(3) and SU(2) .................... 19 3.6 Example: Lorentz group and Poincar´egroup . 21 3.7 Final comments: Baker-Campbell-Hausdorff formula . 23 4 Lie algebras 24 4.1 RepresentationsofLiealgebras . 24 4.2 Irreducible representations of sl(2, C)................ 27 4.3 Directsumsandtensorproducts . 29 4.4 Ideals ................................. 32 4.5 TheKillingform ........................... 34 5 Classification of finite-dimensional, semi-simple, complex Lie algebras 37 5.1 Workinginabasis .......................... 37 5.2 Cartansubalgebras. 38 5.3 Cartan-Weylbasis .......................... 42 5.4 Examples: sl(2, C) and sl(n, C)................... 47 5.5 TheWeylgroup............................ 49 5.6 SimpleLiealgebrasofrank1and2. 50 5.7 Dynkindiagrams ........................... 53 5.8 ComplexificationofrealLiealgebras . 56 6 Epilogue 62 A Appendix: Collected exercises 63 2 Exercise 0.1 : I certainly did not manage to remove all errors from this script. So the first exercise is to find all errors and tell them to me. 1 Preamble The topic of this course is Lie groups and Lie algebras, and their representations. As a preamble, let us have a quick look at the definitions. These can then again be forgotten, for they will be restated further on in the course.
    [Show full text]
  • Knot Homology Groups from Instantons
    Knot homology groups from instantons P. B. Kronheimer and T. S. Mrowka Harvard University, Cambridge MA 02138 Massachusetts Institute of Technology, Cambridge MA 02139 1 Introduction 1.1 An observation of some coincidences For a knot or link K in S 3, the Khovanov homology Kh.K/ is a bigraded abelian group whose construction can be described in entirely combinatorial terms [15]. If we forget the bigrading, then as abelian groups we have, for example, Kh.unknot/ Z2 D and Kh.trefoil/ Z4 Z=2: D ˚ The second equality holds for both the right- and left-handed trefoils, though the bigrading would distinguish these two cases. The present paper was motivated in large part by the observation that the 4 group Z Z=2 arises in a different context. Pick a basepoint y0 in the com- ˚ plement of the knot or link, and consider the space of all homomorphisms 3 1 S K; y0 SU.2/ W n ! satisfying the additional constraint that  i 0à .m/ is conjugate to (1) 0 i for every m in the conjugacy class of a meridian of the link. (There is one such conjugacy class for each component of K, once the components are oriented. The orientation does not matter here, because the above element of SU.2/ is conjugate to its inverse.) Let us write 3 R.K/ Hom 1 S K; y0 ; SU.2/ n 2 for the set of these homomorphisms. Note that we are not defining R.K/ as a set of equivalence classes of such homomorphisms under the action of conju- gation by SU.2/.
    [Show full text]
  • Basics of Lie Theory (Proseminar in Theoretical Physics)
    Basics of April 4, 2013 Lie Theory Andreas Wieser Page 1 Basics of Lie Theory (Proseminar in Theoretical Physics) Andreas Wieser ETH Z¨urich April 4, 2013 Tutor: Dr. Peter Vrana Abstract The goal of this report is to give an insight into the theory of Lie groups and Lie algebras. After an introduction (Matrix Lie groups), the first topic is Lie groups focusing on the construction of the associated Lie algebra and on basic notions. The second part is on Lie algebras, where especially semisimple and simple Lie algebras are of interest. The Cartan decomposition is stated and elementary properties are proven. The final part is on the classification of finite-dimensional, simple Lie algebras. Basics of April 4, 2013 Lie Theory Andreas Wieser Page 2 Contents 1 An Intoductory Example 3 1.1 The Matrix Group SO(3) . .3 1.2 Other Matrix Groups . .4 2 Lie Groups 5 2.1 Definition and Examples . .5 2.2 Construction of the Lie-Bracket . .6 2.3 Associated Lie algebra . .7 2.3.1 Examples of complex Lie algebras . .7 2.4 Representations of Lie Groups and Lie Algebras . .8 3 Lie Algebras 9 3.1 Basic Notions . .9 3.2 Representation Theory of sl2C ..................... 11 3.3 Cartan Decomposition . 13 3.4 The Killing Form . 16 3.5 Properties of the Cartan Decomposition . 17 3.6 The Weyl Group . 20 4 Classification of Simple Lie Algebras 21 4.1 Ordering of the Roots . 21 4.2 Angles between the Roots . 22 4.3 Examples of Root Systems . 23 4.4 Further Properties of the Root System .
    [Show full text]
  • COMPLEX LIE ALGEBRAS Contents 1. Lie Algebras 1 2. the Killing
    COMPLEX LIE ALGEBRAS ADAM KAYE Abstract. We prove that every Lie algebra can be decomposed into a solvable Lie algebra and a semisimple Lie algebra. Then we show that every complex semisimple Lie algebra is a direct sum of simple Lie algebras. Finally, we give a complete classification of simple complex Lie algebras. Contents 1. Lie algebras 1 2. The Killing Form and Cartan's Criterion 3 3. Root Space Decomposition 4 4. Classifying Root Systems 5 5. Classifying Dynkin Diagrams 7 Acknowledgements 11 References 11 1. Lie algebras Definition 1.1. A Lie algebra L over a field F is a finite dimensional vector space over F with a bracket operation [−; −]: L × L ! L with the following properties for all x; y; z 2 L, a 2 F : i) bilinearity: [ax; y] = a[x; y] = [x; ay], [x + y; z] = [x; z] + [y; z], [x; y + z] = [x; y] + [x; z] ii) anti-symmetry: [x; x] = 0 iii) the Jacobi identity: [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0 Lemma 1.1. For all x; y 2 L, [x; y] = −[y; x] and [x; 0] = 0. Example 1.2. Let F be a field and let gl(n; F ) be the algebra of n by n matrices over F . If we define [−; −]: gl(n; F ) × gl(n; F ) ! gl(n; F ) by [x; y] 7! xy − yx then this makes gl(n; F ) a Lie algebra over F . Given any vector space V over F we may similarly define the Lie algebra gl(V ) of endomorphisms of V .
    [Show full text]
  • Diffeomorphism Group of a Closed Surface*)
    LAPP-TH-254/89 . June :;/•. .u Diffeomorphism Group of a Closed Surface*) L. Frappât, E. Ragoucy, P. Sorba, F. Thuillier LAPP, Chemin de Bellevue, BP 110, F-74941 ANNECY-LE-VIEUX cedex and H. H0gâsen Universitetet i Oslo, P.O. 1048 Blindern, N-0316 OSLO 3 Abstract We extend the notion of loop algebra ^(S1) to algebras C(M), M being a closed surface with dim M > 1. We point out and discuss the correspondence between the central extensions of such generalized Kac-Moody algebras and the volume preserving diffeomorphism algebra on M. *) Work supported in part by D JÎJE.T. under contract n0 88/1334/DRET/DS/SR. L.Â.P.P. Bf III) l-74')4l \N\I ( VU-Wri \ ( I.!>1 \ • 11-1 ITl K)M 50.2.'.32 4^ • « 1 i.l.l \ 1»^ 1«« I- • HIIXOIMI 5O27')4')S 1. Introduction Quite recently, it has been shown that the symmetries of a closed bosonic membrane W M form the area preserving subgroup SDiff(M) of the diffeomorphism Diff(M) Pl. Explicit realizations of the corresponding Lie algebra when M is either the torus T^ = S1XS1 or the two dimensional sphere S2 have been given. Some efforts have recently been brought to a better understanding of such a symmetry and of its properties : among themselves a research of the Virasoro algebra from SDiff(T2) 131, the determination of the central extensions of SDiff(M) t2»4] and the study of the possible connections between this algebra and the SU(°°) group (51. It is on the eventual existence of (generalized) Kac-Moody algebras behind the already determined symmetry of a membrane theory that we would like to think in this letter.
    [Show full text]
  • 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...............
    [Show full text]