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Lecture Notes Group Theory Course held by Notes written by Prof. Kenichi Konishi Francesco Paolo Maiale Department of Mathematics Pisa University February 17, 2018 Disclaimer These are the notes I have written during the Group Theory course, held by Professor Kenichi Konishi in the first semester of the academic year 2017/2018. These include all the topics that were discussed during the lectures, but I took the liberty to extend the Mathematical side (e.g., the homotopy group chapter). To report any mistakes, misprints, or if you have any question, feel free to send me an email to francescopaolo (dot) maiale (at) gmail (dot) com. Acknowledgments I would like to thank the user Gonzalo Medina, from StackExchange, for the code (avail- able here) of the margin notes frame. Contents I Representation Theory7 1 Group Theory8 1.1 Set Theory.....................................8 1.2 Elementary Definitions and Basic Examples................... 10 1.2.1 Main Examples in Physics......................... 12 1.2.2 Back to Group Theory........................... 17 1.3 Finite Groups.................................... 21 2 Lie Groups and Lie Algebras 25 2.1 Definitions and Main Properties.......................... 25 2.1.1 Local Behavior: Lie Algebras....................... 26 2.1.2 Adjoint Representation.......................... 27 2.1.3 Examples.................................. 28 2.2 Lie Algebra..................................... 30 2.2.1 SubAlgebras................................ 31 2.3 Killing Form.................................... 32 2.3.1 Examples.................................. 33 2.4 Casimir Operator.................................. 34 3 The Fundamental Group π1(M) 36 3.1 Topological Spaces................................. 36 3.2 Homotopy...................................... 37 3.2.1 Path-Components............................. 38 3.2.2 Homotopy.................................. 39 3.3 The Fundamental Group.............................. 40 3.3.1 Path Homotopy............................... 40 3.3.2 The Fundamental Group.......................... 41 3.3.3 Examples.................................. 42 3.4 Covering Space................................... 43 3.5 Application: SU(2; C) =6∼ SO(3; R) ........................ 44 3.6 Monodromy Group................................. 45 3.6.1 Examples.................................. 46 3.7 Higher-Order Homotopy Groups......................... 47 3.7.1 Definitions and Basic Properties..................... 48 3.7.2 Equivalent Definitions and Base Point.................. 48 3.7.3 Higher-Order Homotopy Groups in Physics............... 49 4 Haar Measures 50 4.1 Invariant Measure on a Topological Group.................... 50 4.1.1 Examples.................................. 53 5 Representation Theory 55 5.1 Introduction..................................... 55 5.2 Irreducible Representations............................ 56 5.2.1 Schur Lemmas............................... 57 5.2.2 Representations of Finite and Compact Groups............. 58 II Fundamental Groups in Physics 61 6 Special Unitary Group SU(2; C) 62 6.1 Finite Irreducible Representations........................ 63 6.1.1 Introduction................................ 63 6.1.2 Admissible Dimensions of Finite Representations............ 65 6.2 Fundamental Representation of SU(2; C) .................... 67 6.2.1 Elements in the Fundamental Representation.............. 68 6.2.2 Pseudo-Real (Fundamental) Representation............... 69 6.2.3 Simpleness of SU(2; C) .......................... 71 6.3 Elements in Irreducible Representations..................... 71 6.4 Tensor Product of Representations........................ 72 6.4.1 Clebsh-Gordan Decomposition...................... 73 6.5 Comparison: SO(3; R) and SU(2; C) ....................... 74 7 Special Unitary Group SU(3; C) 76 7.1 Finite Irreducible Representations........................ 77 7.1.1 Construction via Weight Diagrams.................... 77 7.1.2 Generators of the Special Unitary Group SU(2; C) ........... 78 7.2 Quark Model.................................... 81 7.2.1 Young Tableaux.............................. 82 7.2.2 Adjoint Representation.......................... 87 7.2.3 Multiplicity................................. 87 7.2.4 Baryons, Resonances and Colors Model................. 88 7.3 Fundamental Representation in SU(2; C) × U(1; C) ............... 90 8 Special Orthogonal Group SO(4; R) 91 8.1 Representations of SO(4; R) ............................ 91 9 Euclidean Groups En 94 9.1 Two-Dimensional Euclidean Group E2 ...................... 94 9.1.1 Irreducible Representation of E2 ..................... 98 10 Lorentz Group 102 10.1 Introduction..................................... 102 10.2 Irreducible Representations of T+: Part I.................... 103 10.2.1 Generators................................. 103 10.2.2 Lie Algebra of T+ ............................. 105 10.3 Spinorial Representations............................. 106 10.4 Irreducible Representations of T+: Part II.................... 110 10.4.1 Generators of T+ .............................. 110 10.4.2 Finite Irreducible Representations of T+ ................. 111 10.5 Chirality....................................... 112 10.5.1 Standard Model.............................. 112 11 Poincaré Group 114 11.1 Introduction..................................... 114 11.2 Irreducible Representations of P(1; 3) ...................... 115 11.2.1 Generators................................. 115 11.3 Little Group..................................... 117 III Applications in Quantum Mechanics 121 12 Roots and Weights 122 12.1 Introduction..................................... 122 12.2 Weight and Roots in SU(3; C) .......................... 125 12.3 Simple Roots.................................... 129 12.3.1 Angle Between Simple Roots....................... 131 12.3.2 Simple Roots Lie Algebra g ...................... 132 12.4 Dynkin Diagram.................................. 133 12.5 Cartan Matrices.................................. 135 12.5.1 The construction of the Lie algebra su(3; C) ............... 135 12.5.2 Weyl Reflection Group........................... 136 13 Quantum Physics Applications 137 13.1 3-Dimensional Harmonic Oscillator........................ 137 13.2 Hydrogen Atom................................... 139 13.3 Wigner-Eckart Theorem in SU(2; C) - SO(3; R) ................. 141 Part I Representation Theory 7 Chapter 1 Group Theory In this first chapter, the principal goal is to present (briefly) the fundamental notions of group theory we will be using in this course, and to give a sketchy idea of the groups we will study in the second half of the course. 1.1 Set Theory In this section, we briefly recall some of the basic definitions of set theory we will be using throughout this course, and we will also introduce the notation. Definition 1.1 (Set Operations). Let M and N be sets. (a) The set M is a subset of N, and we denote it by M ⊆ N, if and only if a 2 M =) a 2 N: (b) The set M [ N is the union of M and N, that is, a 2 M [ N () a 2 M or a 2 N: (c) The set M \ N is the intersection of M and N, that is, a 2 M \ N () a 2 M and a 2 N: (d) The set N n M is the difference between N and M, that is, a 2 N n M () a 2 N and a2 = M: (e) The set M × N is the Cartesian product of M and N, that is, (a; b) 2 M × N () a 2 M and b 2 N: Furthermore, we denote by M ⊗n the Cartesian product M × · · · × M. n 9 1.1. SET THEORY Definition 1.2 (Mapping). Let M and N be sets. (a) A mapping f : M −! N is injective (=into) if and only if f(b) 6= f(a) for every a 6= b 2 M. (b) A mapping f : M −! N is surjective (=onto) if and only if for every b 2 N there exists a 2 M such that f(a) = b. (c) A mapping f : M −! N is bijective if and only if f is both injective and surjective. U U M N M N Figure 1.1: Left: M [ N. Right: M \ N. U U N M M N Figure 1.2: Left: M is a subset of N. Right: N n M. Definition 1.3 (Equivalence Relation). Let M be a set. An equivalence relation ∼ is a subset of the Cartesian product M ⊗2 satisfying the following properties: (i) Reflexive. For every a 2 M, it turns out that a ∼ a. (ii) Symmetric. For every couple (a; b) 2 M ⊗2, it turns out that a ∼ b () b ∼ a. (iii) Transitive. For every triple (a; b; c) 2 M ⊗3 satisfying a ∼ b and b ∼ c, it turns out that a ∼ c. Moreover, given a set M and an equivalence relation ∼, we denote by [a] the equivalence class of a 2 M, that is, [a] := fb 2 M : b ∼ ag : CHAPTER 1. GROUP THEORY 10 Example 1.1. Let R be the set of all real numbers. Then, a ∼ b () a − b 2 R is an equivalence relation. In fact, the reflexive property is obvious (a − a = 0 2 Z for all a 2 R), while the symmetric property follows from the fact that a − b 2 Z =) −(a − b) 2 Z =) b − a 2 Z: The unique nontrivial property is the transitiveness, but a simple algebraic trick shows that a − c = a ± b − c = (a − b) + (b − c) 2 Z; and this is enough to infer that ∼ is an equivalence relation. Caution! Remark 1.1. If M is a set and ∼ an equivalence relation on M, then it is always possible1 The set of repre- to write M as the disjoint union of the equivalence classes, that is, sentatives R is not unique! G M = [a] a2R where R ⊂ M denotes a maximal collection of elements a 2 M that are not equivalent, i.e., a 6= b 2 R =) a 6∼ b: 1.2 Elementary Definitions and Basic Examples In this section, we introduce the basic definitions of group theory, and we briefly explain some of the leading examples we will be dealing with in this course. We also present the notion of representation, which will be used in the next chapter to introduce Lie groups and Lie algebras. Definition 1.4 (Group). A group is a set, G, together with a mapping · : G × G −! G, called group product, satisfying the following properties: 1) Closure. For every g1; g2 2 G, the product g1 · g2 also belongs to G. 2) Associativity. For every g1; g2; g3 2 G, it turns out that g1 · (g2 · g3) = (g1 · g2) · g3: 3) Left Identity. There exists e 2 G such that e · g = g for every g 2 G. 4) Left Inverse. For every g 2 G there exists an element, denoted by g−1, such that g−1 · g = e: Lemma 1.5.
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  • 324 COHOMOLOGY of DIHEDRAL GROUPS of ORDER 2P Let D Be A

    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].