THE GROUP of DIFFEOMORPHISMS of a NON COMPACT MANIFOLD IS NOT REGULAR Jean-Pierre Magnot
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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. -
Modules and Lie Semialgebras Over Semirings with a Negation Map 3
MODULES AND LIE SEMIALGEBRAS OVER SEMIRINGS WITH A NEGATION MAP GUY BLACHAR Abstract. In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. We define a notion of lifting a module with a negation map, similarly to the tropicalization process, and use it to prove several theorems about semirings with a negation map which possess a lift. In the context of Lie semialgebras over semirings with a negation map, we first give basic definitions, and provide parallel constructions to the classical Lie algebras. We prove an ELT version of Cartan’s criterion for semisimplicity, and provide a counterexample for the naive version of the PBW Theorem. Contents Page 0. Introduction 2 0.1. Semirings with a Negation Map 2 0.2. Modules Over Semirings with a Negation Map 3 0.3. Supertropical Algebras 4 0.4. Exploded Layered Tropical Algebras 4 1. Modules over Semirings with a Negation Map 5 1.1. The Surpassing Relation for Modules 6 1.2. Basic Definitions for Modules 7 1.3. -morphisms 9 1.4. Lifting a Module Over a Semiring with a Negation Map 10 1.5. Linearly Independent Sets 13 1.6. d-bases and s-bases 14 1.7. Free Modules over Semirings with a Negation Map 18 2. -
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). -
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. -
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 -
CLIFFORD ALGEBRAS Property, Then There Is a Unique Isomorphism (V ) (V ) Intertwining the Two Inclusions of V
CHAPTER 2 Clifford algebras 1. Exterior algebras 1.1. Definition. For any vector space V over a field K, let T (V ) = k k k Z T (V ) be the tensor algebra, with T (V ) = V V the k-fold tensor∈ product. The quotient of T (V ) by the two-sided⊗···⊗ ideal (V ) generated byL all v w + w v is the exterior algebra, denoted (V ).I The product in (V ) is usually⊗ denoted⊗ α α , although we will frequently∧ omit the wedge ∧ 1 ∧ 2 sign and just write α1α2. Since (V ) is a graded ideal, the exterior algebra inherits a grading I (V )= k(V ) ∧ ∧ k Z M∈ where k(V ) is the image of T k(V ) under the quotient map. Clearly, 0(V )∧ = K and 1(V ) = V so that we can think of V as a subspace of ∧(V ). We may thus∧ think of (V ) as the associative algebra linearly gener- ated∧ by V , subject to the relations∧ vw + wv = 0. We will write φ = k if φ k(V ). The exterior algebra is commutative | | ∈∧ (in the graded sense). That is, for φ k1 (V ) and φ k2 (V ), 1 ∈∧ 2 ∈∧ [φ , φ ] := φ φ + ( 1)k1k2 φ φ = 0. 1 2 1 2 − 2 1 k If V has finite dimension, with basis e1,...,en, the space (V ) has basis ∧ e = e e I i1 · · · ik for all ordered subsets I = i1,...,ik of 1,...,n . (If k = 0, we put { } k { n } e = 1.) In particular, we see that dim (V )= k , and ∅ ∧ n n dim (V )= = 2n. -
Mixed States from Diffeomorphism Anomalies Arxiv:1109.5290V1
SU-4252-919 IMSc/2011/9/10 Quantum Gravity: Mixed States from Diffeomorphism Anomalies A. P. Balachandran∗ Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA and International Institute of Physics (IIP-UFRN) Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil Amilcar R. de Queirozy Instituto de Fisica, Universidade de Brasilia, Caixa Postal 04455, 70919-970, Brasilia, DF, Brazil October 30, 2018 Abstract In a previous paper, we discussed simple examples like particle on a circle and molecules to argue that mixed states can arise from anoma- lous symmetries. This idea was applied to the breakdown (anomaly) of color SU(3) in the presence of non-abelian monopoles. Such mixed states create entropy as well. arXiv:1109.5290v1 [hep-th] 24 Sep 2011 In this article, we extend these ideas to the topological geons of Friedman and Sorkin in quantum gravity. The \large diffeos” or map- ping class groups can become anomalous in their quantum theory as we show. One way to eliminate these anomalies is to use mixed states, thereby creating entropy. These ideas may have something to do with black hole entropy as we speculate. ∗[email protected] [email protected] 1 1 Introduction Diffeomorphisms of space-time play the role of gauge transformations in grav- itational theories. Just as gauge invariance is basic in gauge theories, so too is diffeomorphism (diffeo) invariance in gravity theories. Diffeos can become anomalous on quantization of gravity models. If that happens, these models cannot serve as descriptions of quantum gravitating systems. There have been several investigations of diffeo anomalies in models of quantum gravity with matter in the past. -
Area-Preserving Diffeomorphisms of the Torus Whose Rotation Sets Have
Area-preserving diffeomorphisms of the torus whose rotation sets have non-empty interior Salvador Addas-Zanata Instituto de Matem´atica e Estat´ıstica Universidade de S˜ao Paulo Rua do Mat˜ao 1010, Cidade Universit´aria, 05508-090 S˜ao Paulo, SP, Brazil Abstract ǫ In this paper we consider C1+ area-preserving diffeomorphisms of the torus f, either homotopic to the identity or to Dehn twists. We sup- e pose that f has a lift f to the plane such that its rotation set has in- terior and prove, among other things that if zero is an interior point of e e 2 the rotation set, then there exists a hyperbolic f-periodic point Q∈ IR such that W u(Qe) intersects W s(Qe +(a,b)) for all integers (a,b), which u e implies that W (Q) is invariant under integer translations. Moreover, u e s e e u e W (Q) = W (Q) and f restricted to W (Q) is invariant and topologi- u e cally mixing. Each connected component of the complement of W (Q) is a disk with diameter uniformly bounded from above. If f is transitive, u e 2 e then W (Q) =IR and f is topologically mixing in the whole plane. Key words: pseudo-Anosov maps, Pesin theory, periodic disks e-mail: [email protected] 2010 Mathematics Subject Classification: 37E30, 37E45, 37C25, 37C29, 37D25 The author is partially supported by CNPq, grant: 304803/06-5 1 Introduction and main results One of the most well understood chapters of dynamics of surface homeomor- phisms is the case of the torus. -
Right-Angled Artin Groups in the C Diffeomorphism Group of the Real Line
Right-angled Artin groups in the C∞ diffeomorphism group of the real line HYUNGRYUL BAIK, SANG-HYUN KIM, AND THOMAS KOBERDA Abstract. We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG ∞ group, embeds into the C diffeomorphism group of the real line. 1. Introduction The right-angled Artin group on a finite simplicial graph Γ is the following group presentation: A(Γ) = hV (Γ) | [vi, vj] = 1 if and only if {vi, vj}∈ E(Γ)i. Here, V (Γ) and E(Γ) denote the vertex set and the edge set of Γ, respec- tively. ∞ For a smooth oriented manifold X, we let Diff+ (X) denote the group of orientation preserving C∞ diffeomorphisms on X. A group G is said to embed into another group H if there is an injective group homormophism G → H. Our main result is the following. ∞ R Theorem 1. Every right-angled Artin group embeds into Diff+ ( ). Recall that a finitely generated group G is a limit group (or a fully resid- ually free group) if for each finite set F ⊂ G, there exists a homomorphism φF from G to a nonabelian free group such that φF is an injection when restricted to F . The class of limit groups fits into a larger class of groups, which we call residually RAAG groups. A group G is in this class if for each arXiv:1404.5559v4 [math.GT] 16 Feb 2015 1 6= g ∈ G, there exists a graph Γg and a homomorphism φg : G → A(Γg) such that φg(g) 6= 1. -
Introuduction to Representation Theory of Lie Algebras
Introduction to Representation Theory of Lie Algebras Tom Gannon May 2020 These are the notes for the summer 2020 mini course on the representation theory of Lie algebras. We'll first define Lie groups, and then discuss why the study of representations of simply connected Lie groups reduces to studying representations of their Lie algebras (obtained as the tangent spaces of the groups at the identity). We'll then discuss a very important class of Lie algebras, called semisimple Lie algebras, and we'll examine the repre- sentation theory of two of the most basic Lie algebras: sl2 and sl3. Using these examples, we will develop the vocabulary needed to classify representations of all semisimple Lie algebras! Please email me any typos you find in these notes! Thanks to Arun Debray, Joakim Færgeman, and Aaron Mazel-Gee for doing this. Also{thanks to Saad Slaoui and Max Riestenberg for agreeing to be teaching assistants for this course and for many, many helpful edits. Contents 1 From Lie Groups to Lie Algebras 2 1.1 Lie Groups and Their Representations . .2 1.2 Exercises . .4 1.3 Bonus Exercises . .4 2 Examples and Semisimple Lie Algebras 5 2.1 The Bracket Structure on Lie Algebras . .5 2.2 Ideals and Simplicity of Lie Algebras . .6 2.3 Exercises . .7 2.4 Bonus Exercises . .7 3 Representation Theory of sl2 8 3.1 Diagonalizability . .8 3.2 Classification of the Irreducible Representations of sl2 .............8 3.3 Bonus Exercises . 10 4 Representation Theory of sl3 11 4.1 The Generalization of Eigenvalues . -
Differential Topology: Exercise Sheet 3
Prof. Dr. M. Wolf WS 2018/19 M. Heinze Sheet 3 Differential Topology: Exercise Sheet 3 Exercises (for Nov. 21th and 22th) 3.1 Smooth maps Prove the following: (a) A map f : M ! N between smooth manifolds (M; A); (N; B) is smooth if and only if for all x 2 M there are pairs (U; φ) 2 A; (V; ) 2 B such that x 2 U, f(U) ⊂ V and ◦ f ◦ φ−1 : φ(U) ! (V ) is smooth. (b) Compositions of smooth maps between subsets of smooth manifolds are smooth. Solution: (a) As the charts of a smooth structure cover the manifold the above property is certainly necessary for smoothness of f : M ! N. To show that it is also sufficient, consider two arbitrary charts U;~ φ~ 2 A and V;~ ~ 2 B. We need to prove that the map ~ ◦ f ◦ φ~−1 : φ~(U~) ! ~(V~ ) is smooth. Therefore consider an arbitrary y 2 φ~(U~) such that there is some x 2 M such that x = φ~−1(y). By assumption there are charts (U; φ) 2 A; (V; ) 2 B such that x 2 U, −1 f(U) ⊂ V and ◦ f ◦ φ : φ(U) ! (V ) is smooth. Now we choose U0;V0 such ~ ~ that x 2 U0 ⊂ U \ U and f(x) 2 V0 ⊂ V \ V and therefore we can write ~ ~−1 ~ −1 −1 ~−1 ◦ f ◦ φ j = ◦ ◦ ◦ f ◦ φ ◦ φ ◦ φ : (1) φ~(U0) As a composition of smooth maps, we showed that ~ ◦ f ◦ φ~−1j is smooth. As φ~(U0) x 2 M was chosen arbitrarily we proved that ~ ◦ f ◦ φ~−1 is a smooth map. -
Matrix Lie Groups and Their Lie Algebras
Matrix Lie groups and their Lie algebras Alen Alexanderian∗ Abstract We discuss matrix Lie groups and their corresponding Lie algebras. Some common examples are provided for purpose of illustration. 1 Introduction The goal of these brief note is to provide a quick introduction to matrix Lie groups which are a special class of abstract Lie groups. Study of matrix Lie groups is a fruitful endeavor which allows one an entry to theory of Lie groups without requiring knowl- edge of differential topology. After all, most interesting Lie groups turn out to be matrix groups anyway. An abstract Lie group is defined to be a group which is also a smooth manifold, where the group operations of multiplication and inversion are also smooth. We provide a much simple definition for a matrix Lie group in Section 4. Showing that a matrix Lie group is in fact a Lie group is discussed in standard texts such as [2]. We also discuss Lie algebras [1], and the computation of the Lie algebra of a Lie group in Section 5. We will compute the Lie algebras of several well known Lie groups in that section for the purpose of illustration. 2 Notation Let V be a vector space. We denote by gl(V) the space of all linear transformations on V. If V is a finite-dimensional vector space we may put an arbitrary basis on V and identify elements of gl(V) with their matrix representation. The following define various classes of matrices on Rn: ∗The University of Texas at Austin, USA. E-mail: [email protected] Last revised: July 12, 2013 Matrix Lie groups gl(n) : the space of n