Representations of SL2(R)
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The Unitary Representations of the Poincaré Group in Any Spacetime
The unitary representations of the Poincar´e group in any spacetime dimension Xavier Bekaert a and Nicolas Boulanger b a Institut Denis Poisson, Unit´emixte de Recherche 7013, Universit´ede Tours, Universit´ed’Orl´eans, CNRS, Parc de Grandmont, 37200 Tours (France) [email protected] b Service de Physique de l’Univers, Champs et Gravitation Universit´ede Mons – UMONS, Place du Parc 20, 7000 Mons (Belgium) [email protected] An extensive group-theoretical treatment of linear relativistic field equa- tions on Minkowski spacetime of arbitrary dimension D > 2 is presented in these lecture notes. To start with, the one-to-one correspondence be- tween linear relativistic field equations and unitary representations of the isometry group is reviewed. In turn, the method of induced representa- tions reduces the problem of classifying the representations of the Poincar´e group ISO(D 1, 1) to the classification of the representations of the sta- − bility subgroups only. Therefore, an exhaustive treatment of the two most important classes of unitary irreducible representations, corresponding to massive and massless particles (the latter class decomposing in turn into the “helicity” and the “infinite-spin” representations) may be performed via the well-known representation theory of the orthogonal groups O(n) (with D 4 <n<D ). Finally, covariant field equations are given for each − unitary irreducible representation of the Poincar´egroup with non-negative arXiv:hep-th/0611263v2 13 Jun 2021 mass-squared. Tachyonic representations are also examined. All these steps are covered in many details and with examples. The present notes also include a self-contained review of the representation theory of the general linear and (in)homogeneous orthogonal groups in terms of Young diagrams. -
A NEW APPROACH to RANK ONE LINEAR ALGEBRAIC GROUPS An
A NEW APPROACH TO RANK ONE LINEAR ALGEBRAIC GROUPS DANIEL ALLCOCK Abstract. One can develop the basic structure theory of linear algebraic groups (the root system, Bruhat decomposition, etc.) in a way that bypasses several major steps of the standard development, including the self-normalizing property of Borel subgroups. An awkwardness of the theory of linear algebraic groups is that one must develop a lot of material before one can even characterize PGL2. Our goal here is to show how to develop the root system, etc., using only the completeness of the flag variety, its immediate consequences, and some facts about solvable groups. In particular, one can skip over the usual analysis of Cartan subgroups, the fact that G is the union of its Borel subgroups, the connectedness of torus centralizers, and the normalizer theorem (that Borel subgroups are self-normalizing). The main idea is a new approach to the structure of rank 1 groups; the key step is lemma 5. All algebraic geometry is over a fixed algebraically closed field. G always denotes a connected linear algebraic group with Lie algebra g, T a maximal torus, and B a Borel subgroup containing it. We assume the structure theory for connected solvable groups, and the completeness of the flag variety G/B and some of its consequences. Namely: that all Borel subgroups (resp. maximal tori) are conjugate; that G is nilpotent if one of its Borel subgroups is; that CG(T )0 lies in every Borel subgroup containing T ; and that NG(B) contains B of finite index and (therefore) is self-normalizing. -
A Note on Presentation of General Linear Groups Over a Finite Field
Southeast Asian Bulletin of Mathematics (2019) 43: 217–224 Southeast Asian Bulletin of Mathematics c SEAMS. 2019 A Note on Presentation of General Linear Groups over a Finite Field Swati Maheshwari and R. K. Sharma Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India Email: [email protected]; [email protected] Received 22 September 2016 Accepted 20 June 2018 Communicated by J.M.P. Balmaceda AMS Mathematics Subject Classification(2000): 20F05, 16U60, 20H25 Abstract. In this article we have given Lie regular generators of linear group GL(2, Fq), n where Fq is a finite field with q = p elements. Using these generators we have obtained presentations of the linear groups GL(2, F2n ) and GL(2, Fpn ) for each positive integer n. Keywords: Lie regular units; General linear group; Presentation of a group; Finite field. 1. Introduction Suppose F is a finite field and GL(n, F) is the general linear the group of n × n invertible matrices and SL(n, F) is special linear group of n × n matrices with determinant 1. We know that GL(n, F) can be written as a semidirect product, GL(n, F)= SL(n, F) oF∗, where F∗ denotes the multiplicative group of F. Let H and K be two groups having presentations H = hX | Ri and K = hY | Si, then a presentation of semidirect product of H and K is given by, −1 H oη K = hX, Y | R,S,xyx = η(y)(x) ∀x ∈ X,y ∈ Y i, where η : K → Aut(H) is a group homomorphism. Now we summarize some literature survey related to the presentation of groups. -
The General Linear Group
18.704 Gabe Cunningham 2/18/05 [email protected] The General Linear Group Definition: Let F be a field. Then the general linear group GLn(F ) is the group of invert- ible n × n matrices with entries in F under matrix multiplication. It is easy to see that GLn(F ) is, in fact, a group: matrix multiplication is associative; the identity element is In, the n × n matrix with 1’s along the main diagonal and 0’s everywhere else; and the matrices are invertible by choice. It’s not immediately clear whether GLn(F ) has infinitely many elements when F does. However, such is the case. Let a ∈ F , a 6= 0. −1 Then a · In is an invertible n × n matrix with inverse a · In. In fact, the set of all such × matrices forms a subgroup of GLn(F ) that is isomorphic to F = F \{0}. It is clear that if F is a finite field, then GLn(F ) has only finitely many elements. An interesting question to ask is how many elements it has. Before addressing that question fully, let’s look at some examples. ∼ × Example 1: Let n = 1. Then GLn(Fq) = Fq , which has q − 1 elements. a b Example 2: Let n = 2; let M = ( c d ). Then for M to be invertible, it is necessary and sufficient that ad 6= bc. If a, b, c, and d are all nonzero, then we can fix a, b, and c arbitrarily, and d can be anything but a−1bc. This gives us (q − 1)3(q − 2) matrices. -
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. -
Abelian Varieties with Complex Multiplication and Modular Functions, by Goro Shimura, Princeton Univ
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 36, Number 3, Pages 405{408 S 0273-0979(99)00784-3 Article electronically published on April 27, 1999 Abelian varieties with complex multiplication and modular functions, by Goro Shimura, Princeton Univ. Press, Princeton, NJ, 1998, xiv + 217 pp., $55.00, ISBN 0-691-01656-9 The subject that might be called “explicit class field theory” begins with Kro- necker’s Theorem: every abelian extension of the field of rational numbers Q is a subfield of a cyclotomic field Q(ζn), where ζn is a primitive nth root of 1. In other words, we get all abelian extensions of Q by adjoining all “special values” of e(x)=exp(2πix), i.e., with x Q. Hilbert’s twelfth problem, also called Kronecker’s Jugendtraum, is to do something2 similar for any number field K, i.e., to generate all abelian extensions of K by adjoining special values of suitable special functions. Nowadays we would add that the reciprocity law describing the Galois group of an abelian extension L/K in terms of ideals of K should also be given explicitly. After K = Q, the next case is that of an imaginary quadratic number field K, with the real torus R/Z replaced by an elliptic curve E with complex multiplication. (Kronecker knew what the result should be, although complete proofs were given only later, by Weber and Takagi.) For simplicity, let be the ring of integers in O K, and let A be an -ideal. Regarding A as a lattice in C, we get an elliptic curve O E = C/A with End(E)= ;Ehas complex multiplication, or CM,by .If j=j(A)isthej-invariant ofOE,thenK(j) is the Hilbert class field of K, i.e.,O the maximal abelian unramified extension of K. -
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. -
Maximality of Hyperspecial Compact Subgroups Avoiding Bruhat–Tits Theory Tome 67, No 1 (2017), P
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Marco MACULAN Maximality of hyperspecial compact subgroups avoiding Bruhat–Tits theory Tome 67, no 1 (2017), p. 1-21. <http://aif.cedram.org/item?id=AIF_2017__67_1_1_0> © Association des Annales de l’institut Fourier, 2017, Certains droits réservés. Cet article est mis à disposition selon les termes de la licence CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE. http://creativecommons.org/licenses/by-nd/3.0/fr/ L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 67, 1 (2017) 1-21 MAXIMALITY OF HYPERSPECIAL COMPACT SUBGROUPS AVOIDING BRUHAT–TITS THEORY by Marco MACULAN Abstract. — Let k be a complete non-archimedean field (non trivially valued). Given a reductive k-group G, we prove that hyperspecial subgroups of G(k) (i.e. those arising from reductive models of G) are maximal among bounded subgroups. The originality resides in the argument: it is inspired by the case of GLn and avoids all considerations on the Bruhat–Tits building of G. Résumé. — Soit k un corps non-archimédien complet et non trivialement va- lué. Étant donné un k-groupe réductif G, nous démontrons que les sous-groupes hyperspéciaux de G(k) (c’est-à-dire ceux qui proviennent des modèles réductifs de G) sont maximaux parmi les sous-groupes bornés. -
Normal Subgroups of the General Linear Groups Over Von Neumann Regular Rings L
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 96, Number 2, February 1986 NORMAL SUBGROUPS OF THE GENERAL LINEAR GROUPS OVER VON NEUMANN REGULAR RINGS L. N. VASERSTEIN1 ABSTRACT. Let A be a von Neumann regular ring or, more generally, let A be an associative ring with 1 whose reduction modulo its Jacobson radical is von Neumann regular. We obtain a complete description of all subgroups of GLn A, n > 3, which are normalized by elementary matrices. 1. Introduction. For any associative ring A with 1 and any natural number n, let GLn A be the group of invertible n by n matrices over A and EnA the subgroup generated by all elementary matrices x1'3, where 1 < i / j < n and x E A. In this paper we describe all subgroups of GLn A normalized by EnA for any von Neumann regular A, provided n > 3. Our description is standard (see Bass [1] and Vaserstein [14, 16]): a subgroup H of GL„ A is normalized by EnA if and only if H is of level B for an ideal B of A, i.e. E„(A, B) C H C Gn(A, B). Here Gn(A, B) is the inverse image of the center of GL„(,4/S) (when n > 2, this center consists of scalar invertible matrices over the center of the ring A/B) under the canonical homomorphism GL„ A —►GLn(A/B) and En(A, B) is the normal subgroup of EnA generated by all elementary matrices in Gn(A, B) (when n > 3, the group En(A, B) is generated by matrices of the form (—y)J'lx1'Jy:i''1 with x € B,y £ A,l < i ^ j < n, see [14]). -
Generalized Quaternions
GENERALIZED QUATERNIONS KEITH CONRAD 1. introduction The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomor- phism). The other one, D4, can be constructed as a semi-direct product: ∼ ∼ × ∼ D4 = Aff(Z=(4)) = Z=(4) o (Z=(4)) = Z=(4) o Z=(2); where the elements of Z=(2) act on Z=(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section2 and then jazz up the construction in Section3 to make an infinite family of similar groups with Q8 as the simplest member. In Section4 we will compare this family with the dihedral groups and see how it fits into a bigger picture. 2. The quaternion group from a semi-direct product The group Q8 is built out of its subgroups hii and hji with the overlapping condition i2 = j2 = −1 and the conjugacy relation jij−1 = −i = i−1. More generally, for odd a we have jaij−a = −i = i−1, while for even a we have jaij−a = i. We can combine these into the single formula a (2.1) jaij−a = i(−1) for all a 2 Z. These relations suggest the following way to construct the group Q8. Theorem 2.1. Let H = Z=(4) o Z=(4), where (a; b)(c; d) = (a + (−1)bc; b + d); ∼ The element (2; 2) in H has order 2, lies in the center, and H=h(2; 2)i = Q8. -
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). -
Some Finiteness Results for Groups of Automorphisms of Manifolds 3
SOME FINITENESS RESULTS FOR GROUPS OF AUTOMORPHISMS OF MANIFOLDS ALEXANDER KUPERS Abstract. We prove that in dimension =6 4, 5, 7 the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of Rn and various types of automorphisms of 2-connected manifolds. Contents 1. Introduction 1 2. Homologically and homotopically finite type spaces 4 3. Self-embeddings 11 4. The Weiss fiber sequence 19 5. Proofs of main results 29 References 38 1. Introduction Inspired by work of Weiss on Pontryagin classes of topological manifolds [Wei15], we use several recent advances in the study of high-dimensional manifolds to prove a structural result about diffeomorphism groups. We prove the classifying spaces of such groups are often “small” in one of the following two algebro-topological senses: Definition 1.1. Let X be a path-connected space. arXiv:1612.09475v3 [math.AT] 1 Sep 2019 · X is said to be of homologically finite type if for all Z[π1(X)]-modules M that are finitely generated as abelian groups, H∗(X; M) is finitely generated in each degree. · X is said to be of finite type if π1(X) is finite and πi(X) is finitely generated for i ≥ 2. Being of finite type implies being of homologically finite type, see Lemma 2.15. Date: September 4, 2019. Alexander Kupers was partially supported by a William R.