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Arxiv:Hep-Th/0510191V1 22 Oct 2005
Maximal Subgroups of the Coxeter Group W (H4) and Quaternions Mehmet Koca,∗ Muataz Al-Barwani,† and Shadia Al-Farsi‡ Department of Physics, College of Science, Sultan Qaboos University, PO Box 36, Al-Khod 123, Muscat, Sultanate of Oman Ramazan Ko¸c§ Department of Physics, Faculty of Engineering University of Gaziantep, 27310 Gaziantep, Turkey (Dated: September 8, 2018) The largest finite subgroup of O(4) is the noncrystallographic Coxeter group W (H4) of order 14400. Its derived subgroup is the largest finite subgroup W (H4)/Z2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W (H2) × W (H2)]×Z4 and W (H3)×Z2 possess noncrystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3)×SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W (H4) with sets of quaternion pairs acting on the quaternionic root systems. PACS numbers: 02.20.Bb INTRODUCTION The noncrystallographic Coxeter group W (H4) of order 14400 generates some interests [1] for its relevance to the quasicrystallographic structures in condensed matter physics [2, 3] as well as its unique relation with the E8 gauge symmetry associated with the heterotic superstring theory [4]. The Coxeter group W (H4) [5] is the maximal finite subgroup of O(4), the finite subgroups of which have been classified by du Val [6] and by Conway and Smith [7]. -
Invariant Differential Operators 1. Derivatives of Group Actions
(October 28, 2010) Invariant differential operators Paul Garrett [email protected] http:=/www.math.umn.edu/~garrett/ • Derivatives of group actions: Lie algebras • Laplacians and Casimir operators • Descending to G=K • Example computation: SL2(R) • Enveloping algebras and adjoint functors • Appendix: brackets • Appendix: proof of Poincar´e-Birkhoff-Witt We want an intrinsic approach to existence of differential operators invariant under group actions. n n The translation-invariant operators @=@xi on R , and the rotation-invariant Laplacian on R are deceptively- easily proven invariant, as these examples provide few clues about more complicated situations. For example, we expect rotation-invariant Laplacians (second-order operators) on spheres, and we do not want to write a formula in generalized spherical coordinates and verify invariance computationally. Nor do we want to be constrained to imbedding spheres in Euclidean spaces and using the ambient geometry, even though this succeeds for spheres themselves. Another basic example is the operator @2 @2 y2 + @x2 @y2 on the complex upper half-plane H, provably invariant under the linear fractional action of SL2(R), but it is oppressive to verify this directly. Worse, the goal is not merely to verify an expression presented as a deus ex machina, but, rather to systematically generate suitable expressions. An important part of this intention is understanding reasons for the existence of invariant operators, and expressions in coordinates should be a foregone conclusion. (No prior acquaintance with Lie groups or Lie algebras is assumed.) 1. Derivatives of group actions: Lie algebras For example, as usual let > SOn(R) = fk 2 GLn(R): k k = 1n; det k = 1g act on functions f on the sphere Sn−1 ⊂ Rn, by (k · f)(m) = f(mk) with m × k ! mk being right matrix multiplication of the row vector m 2 Rn. -
324 COHOMOLOGY of DIHEDRAL GROUPS of ORDER 2P Let D Be A
324 PHYLLIS GRAHAM [March 2. P. M. Cohn, Unique factorization in noncommutative power series rings, Proc. Cambridge Philos. Soc 58 (1962), 452-464. 3. Y. Kawada, Cohomology of group extensions, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1963), 417-431. 4. J.-P. Serre, Cohomologie Galoisienne, Lecture Notes in Mathematics No. 5, Springer, Berlin, 1964, 5. , Corps locaux et isogênies, Séminaire Bourbaki, 1958/1959, Exposé 185 6. , Corps locaux, Hermann, Paris, 1962. UNIVERSITY OF MICHIGAN COHOMOLOGY OF DIHEDRAL GROUPS OF ORDER 2p BY PHYLLIS GRAHAM Communicated by G. Whaples, November 29, 1965 Let D be a dihedral group of order 2p, where p is an odd prime. D is generated by the elements a and ]3 with the relations ap~fi2 = 1 and (ial3 = or1. Let A be the subgroup of D generated by a, and let p-1 Aoy Ai, • • • , Ap-\ be the subgroups generated by j8, aft, • * • , a /3, respectively. Let M be any D-module. Then the cohomology groups n n H (A0, M) and H (Aif M), i=l, 2, • • • , p — 1 are isomorphic for 1 l every integer n, so the eight groups H~ (Df Af), H°(D, M), H (D, jfef), 2 1 Q # (A Af), ff" ^. M), H (A, M), H-Wo, M), and H»(A0, M) deter mine all cohomology groups of M with respect to D and to all of its subgroups. We have found what values this array takes on as M runs through all finitely generated -D-modules. All possibilities for the first four members of this array are deter mined as a special case of the results of Yang [4]. -
Dihedral Groups Ii
DIHEDRAL GROUPS II KEITH CONRAD We will characterize dihedral groups in terms of generators and relations, and describe the subgroups of Dn, including the normal subgroups. We will also introduce an infinite group that resembles the dihedral groups and has all of them as quotient groups. 1. Abstract characterization of Dn −1 −1 The group Dn has two generators r and s with orders n and 2 such that srs = r . We will show every group with a pair of generators having properties similar to r and s admits a homomorphism onto it from Dn, and is isomorphic to Dn if it has the same size as Dn. Theorem 1.1. Let G be generated by elements x and y where xn = 1 for some n ≥ 3, 2 −1 −1 y = 1, and yxy = x . There is a surjective homomorphism Dn ! G, and if G has order 2n then this homomorphism is an isomorphism. The hypotheses xn = 1 and y2 = 1 do not mean x has order n and y has order 2, but only that their orders divide n and divide 2. For instance, the trivial group has the form hx; yi where xn = 1, y2 = 1, and yxy−1 = x−1 (take x and y to be the identity). Proof. The equation yxy−1 = x−1 implies yxjy−1 = x−j for all j 2 Z (raise both sides to the jth power). Since y2 = 1, we have for all k 2 Z k ykxjy−k = x(−1) j by considering even and odd k separately. Thus k (1.1) ykxj = x(−1) jyk: This shows each product of x's and y's can have all the x's brought to the left and all the y's brought to the right. -
Minimal Dark Matter Models with Radiative Neutrino Masses
Master’s thesis Minimal dark matter models with radiative neutrino masses From Lagrangians to observables Simon May 1st June 2018 Advisors: Prof. Dr. Michael Klasen, Dr. Karol Kovařík Institut für Theoretische Physik Westfälische Wilhelms-Universität Münster Contents 1. Introduction 5 2. Experimental and observational evidence 7 2.1. Dark matter . 7 2.2. Neutrino oscillations . 14 3. Gauge theories and the Standard Model of particle physics 19 3.1. Mathematical background . 19 3.1.1. Group and representation theory . 19 3.1.2. Tensors . 27 3.2. Representations of the Lorentz group . 31 3.2.1. Scalars: The (0, 0) representation . 35 1 1 3.2.2. Weyl spinors: The ( 2 , 0) and (0, 2 ) representations . 36 1 1 3.2.3. Dirac spinors: The ( 2 , 0) ⊕ (0, 2 ) representation . 38 3.2.4. Majorana spinors . 40 1 1 3.2.5. Lorentz vectors: The ( 2 , 2 ) representation . 41 3.2.6. Field representations . 42 3.3. Two-component Weyl spinor formalism and van der Waerden notation 44 3.3.1. Definition . 44 3.3.2. Correspondence to the Dirac bispinor formalism . 47 3.4. The Standard Model . 49 3.4.1. Definition of the theory . 52 3.4.2. The Lagrangian . 54 4. Component notation for representations of SU(2) 57 4.1. SU(2) doublets . 59 4.1.1. Basic conventions and transformation of doublets . 60 4.1.2. Dual doublets and scalar product . 61 4.1.3. Transformation of dual doublets . 62 4.1.4. Adjoint doublets . 63 4.1.5. The question of transpose and conjugate doublets . -
Lecture 7 - Complete Reducibility of Representations of Semisimple Algebras
Lecture 7 - Complete Reducibility of Representations of Semisimple Algebras September 27, 2012 1 New modules from old A few preliminaries are necessary before jumping into the representation theory of semisim- ple algebras. First a word on creating new g-modules from old. Any Lie algebra g has an action on a 1-dimensional vector space (or F itself), given by the trivial action. Second, any action on spaces V and W can be extended to an action on V ⊗ W by forcing the Leibnitz rule: for any basis vector v ⊗ w 2 V ⊗ W we define x:(v ⊗ w) = x:v ⊗ w + v ⊗ x:w (1) One easily checks that x:y:(v ⊗ w) − y:x:(v ⊗ w) = [x; y]:(v ⊗ w). Assuming g has an action on V , it has an action on its dual V ∗ (recall V ∗ is the vector space of linear functionals V ! F), given by (v:f)(x) = −f(x:v) (2) for any functional f : V ! F in V ∗. This is in fact a version of the \forcing the Leibnitz rule." That is, recalling that we defined x:(f(v)) = 0, we define x:f 2 V ∗ implicitly by x: (f(v)) = (x:f)(v) + f(x:v): (3) For any vector spaces V , W , we have an isomorphism Hom(V; W ) ≈ V ∗ ⊗ W; (4) so Hom(V; W ) is a g-module whenever V and W are. This can be defined using the above rules for duals and tensor products, or, equivalently, by again forcing the Leibnitz rule: for F 2 Hom(V; W ), we define x:F 2 Hom(V; W ) implicitly by x:(F (v)) = (x:F )(v) + F (x:v): (5) 1 2 Schur's lemma and Casimir elements Theorem 2.1 (Schur's Lemma) If g has an irreducible representation on gl(V ) and if f 2 End(V ) commutes with every x 2 g, then f is multiplication by a constant. -
An Identity Crisis for the Casimir Operator
An Identity Crisis for the Casimir Operator Thomas R. Love Department of Mathematics and Department of Physics California State University, Dominguez Hills Carson, CA, 90747 [email protected] April 16, 2006 Abstract 2 P ij The Casimir operator of a Lie algebra L is C = g XiXj and the action of the Casimir operator is usually taken to be C2Y = P ij g XiXjY , with ordinary matrix multiplication. With this defini- tion, the eigenvalues of the Casimir operator depend upon the repre- sentation showing that the action of the Casimir operator is not well defined. We prove that the action of the Casimir operator should 2 P ij be interpreted as C Y = g [Xi, [Xj,Y ]]. This intrinsic definition does not depend upon the representation. Similar results hold for the higher order Casimir operators. We construct higher order Casimir operators which do not exist in the standard theory including a new type of Casimir operator which defines a complex structure and third order intrinsic Casimir operators for so(3) and so(3, 1). These opera- tors are not multiples of the identity. The standard theory of Casimir operators predicts neither the correct operators nor the correct num- ber of invariant operators. The quantum theory of angular momentum and spin, Wigner’s classification of elementary particles as represen- tations of the Poincar´eGroup and quark theory are based on faulty mathematics. The “no-go theorems” are shown to be invalid. PACS 02.20S 1 1 Introduction Lie groups and Lie algebras play a fundamental role in classical mechan- ics, electrodynamics, quantum mechanics, relativity, and elementary particle physics. -
A Quantum Group in the Yang-Baxter Algebra
A Quantum Group in the Yang-Baxter Algebra Alexandros Aerakis December 8, 2013 Abstract In these notes we mainly present theory on abstract algebra and how it emerges in the Yang-Baxter equation. First we review what an associative algebra is and then introduce further structures such as coalgebra, bialgebra and Hopf algebra. Then we discuss the con- struction of an universal enveloping algebra and how by deforming U[sl(2)] we obtain the quantum group Uq[sl(2)]. Finally, we discover that the latter is actually the Braid limit of the Yang-Baxter alge- bra and we use our algebraic knowledge to obtain the elements of its representation. Contents 1 Algebras 1 1.1 Bialgebra and Hopf algebra . 1 1.2 Universal enveloping algebra . 4 1.3 The quantum Uq[sl(2)] . 7 2 The Yang-Baxter algebra and the quantum Uq[sl(2)] 10 1 Algebras 1.1 Bialgebra and Hopf algebra To start with, the most familiar algebraic structure is surely that of an asso- ciative algebra. Definition 1 An associative algebra A over a field C is a linear vector space V equipped with 1 • Multiplication m : A ⊗ A ! A which is { bilinear { associative m(1 ⊗ m) = m(m ⊗ 1) that pictorially corresponds to the commutative diagram A ⊗ A ⊗ A −−−!1⊗m A ⊗ A ? ? ? ?m ym⊗1 y A ⊗ A −−−!m A • Unit η : C !A which satisfies the axiom m(η ⊗ 1) = m(1 ⊗ η) = 1: =∼ =∼ A ⊗ C A C ⊗ A ^ m 1⊗η > η⊗1 < A ⊗ A By reversing the arrows we get another structure which is called coalgebra. -
MAT313 Fall 2017 Practice Midterm II the Actual Midterm Will Consist of 5-6 Problems
MAT313 Fall 2017 Practice Midterm II The actual midterm will consist of 5-6 problems. It will be based on subsections 127-234 from sections 6-11. This is a preliminary version. I might add few more problems to make sure that all important concepts and methods are covered. Problem 1. 2 Let P be a set of lines through 0 in R . The group SL(2; R) acts on X by linear transfor- mations. Let H be the stabilizer of a line defined by the equation y = 0. (1) Describe the set of matrices H. (2) Describe the orbits of H in X. How many orbits are there? (3) Identify X with the set of cosets of SL(2; R). Solution. (1) Denote the set of lines by P = fLm;ng, where Lm;n is equal to f(x; y)jmx+ny = 0g. a b Note that Lkm;kn = Lm;n if k 6= 0. Let g = c d , ad − bc = 1 be an element of SL(2; R). Then gLm;n = f(x; y)jm(ax + by) + n(cx + dy) = 0g = Lma+nc;mb+nd. a b By definition the line f(x; y)jy = 0g is equal to L0;1. Then c d L0;1 = Lc;d. The line Lc;d coincides with L0;1 if c = 0. Then the stabilizer Stab(L0;1) coincides with a b H = f 0 d g ⊂ SL(2; R). (2) We already know one trivial H-orbit O = fL0;1g. The complement PnfL0;1g is equal to fLm:njm 6= 0g = fL1:n=mjm 6= 0g = fL1:tg. -
Notes on Geometry of Surfaces
Notes on Geometry of Surfaces Contents Chapter 1. Fundamental groups of Graphs 5 1. Review of group theory 5 2. Free groups and Ping-Pong Lemma 8 3. Subgroups of free groups 15 4. Fundamental groups of graphs 18 5. J. Stalling's Folding and separability of subgroups 21 6. More about covering spaces of graphs 23 Bibliography 25 3 CHAPTER 1 Fundamental groups of Graphs 1. Review of group theory 1.1. Group and generating set. Definition 1.1. A group (G; ·) is a set G endowed with an operation · : G × G ! G; (a; b) ! a · b such that the following holds. (1) 8a; b; c 2 G; a · (b · c) = (a · b) · c; (2) 91 2 G: 8a 2 G, a · 1 = 1 · a. (3) 8a 2 G, 9a−1 2 G: a · a−1 = a−1 · a = 1. In the sequel, we usually omit · in a · b if the operation is clear or understood. By the associative law, it makes no ambiguity to write abc instead of a · (b · c) or (a · b) · c. Examples 1.2. (1)( Zn; +) for any integer n ≥ 1 (2) General Linear groups with matrix multiplication: GL(n; R). (3) Given a (possibly infinite) set X, the permutation group Sym(X) is the set of all bijections on X, endowed with mapping composition. 2 2n −1 −1 (4) Dihedral group D2n = hr; sjs = r = 1; srs = r i. This group can be visualized as the symmetry group of a regular (2n)-polygon: s is the reflection about the axe connecting middle points of the two opposite sides, and r is the rotation about the center of the 2n-polygon with an angle π=2n. -
Arxiv:2009.00393V2 [Hep-Th] 26 Jan 2021 Supersymmetric Localisation and the Conformal Bootstrap
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 007, 38 pages Harmonic Analysis in d-Dimensional Superconformal Field Theory Ilija BURIC´ DESY, Notkestraße 85, D-22607 Hamburg, Germany E-mail: [email protected] Received September 02, 2020, in final form January 15, 2021; Published online January 25, 2021 https://doi.org/10.3842/SIGMA.2021.007 Abstract. Superconformal blocks and crossing symmetry equations are among central in- gredients in any superconformal field theory. We review the approach to these objects rooted in harmonic analysis on the superconformal group that was put forward in [J. High Energy Phys. 2020 (2020), no. 1, 159, 40 pages, arXiv:1904.04852] and [J. High Energy Phys. 2020 (2020), no. 10, 147, 44 pages, arXiv:2005.13547]. After lifting conformal four-point functions to functions on the superconformal group, we explain how to obtain compact expressions for crossing constraints and Casimir equations. The later allow to write superconformal blocks as finite sums of spinning bosonic blocks. Key words: conformal blocks; crossing equations; Calogero{Sutherland models 2020 Mathematics Subject Classification: 81R05; 81R12 1 Introduction Conformal field theories (CFTs) are a class of quantum field theories that are interesting for several reasons. On the one hand, they describe the critical behaviour of statistical mechanics systems such as the Ising model. Indeed, the identification of two-dimensional statistical systems with CFT minimal models, first suggested in [2], was a celebrated early achievement in the field. For similar reasons, conformal theories classify universality classes of quantum field theories in the Wilsonian renormalisation group paradigm. On the other hand, CFTs also play a role in the description of physical systems that do not posses scale invariance, through certain \dualities". -
Quantum Mechanics 2 Tutorial 9: the Lorentz Group
Quantum Mechanics 2 Tutorial 9: The Lorentz Group Yehonatan Viernik December 27, 2020 1 Remarks on Lie Theory and Representations 1.1 Casimir elements and the Cartan subalgebra Let us first give loose definitions, and then discuss their implications. A Cartan subalgebra for semi-simple1 Lie algebras is an abelian subalgebra with the nice property that op- erators in the adjoint representation of the Cartan can be diagonalized simultaneously. Often it is claimed to be the largest commuting subalgebra, but this is actually slightly wrong. A correct version of this statement is that the Cartan is the maximal subalgebra that is both commuting and consisting of simultaneously diagonalizable operators in the adjoint. Next, a Casimir element is something that is constructed from the algebra, but is not actually part of the algebra. Its main property is that it commutes with all the generators of the algebra. The most commonly used one is the quadratic Casimir, which is typically just the sum of squares of the generators of the algebra 2 2 2 2 (e.g. L = Lx + Ly + Lz is a quadratic Casimir of so(3)). We now need to say what we mean by \square" 2 and \commutes", because the Lie algebra itself doesn't have these notions. For example, Lx is meaningless in terms of elements of so(3). As a matrix, once a representation is specified, it of course gains meaning, because matrices are endowed with multiplication. But the Lie algebra only has the Lie bracket and linear combinations to work with. So instead, the Casimir is living inside a \larger" algebra called the universal enveloping algebra, where the Lie bracket is realized as an explicit commutator.