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Group Theory – DRAFT 2019 February 26 Mathematical Methods in Physics - 231B { Group Theory { Eric D'Hoker Mani L. Bhaumik Institute for Theoretical Physics Department of Physics and Astronomy University of California, Los Angeles, CA 90095, USA [email protected] The purpose of this course is to present an introduction to standard and widely used methods of group theory in Physics, including Lie groups and Lie algebras, representation theory, tensors, spinors, structure theory of solvable and simple Lie algebras, homogeneous and symmetric spaces. Contents 1 Definition and examples of groups 5 1.1 A little history . .5 1.2 Groups . .6 1.3 Topological groups . .9 1.4 Lie Groups . 12 1.5 Vector spaces . 13 1.6 Lie Algebras . 13 1.7 Relating Lie groups and Lie algebras . 15 1.8 Tangent vectors, tangent space, and cotangent space . 16 2 Matrix Lie groups and Lie algebras 18 2.1 The general linear group GL(n)........................ 18 2.2 Closed subgroups of the general linear group . 21 2.3 The orthogonal groups SO(n) and Lie algebras so(n)............ 21 2.4 The symplectic group Sp(2n) and Lie algebra sp(2n)............ 23 2.5 The unitary group SU(n) and Lie algebra su(n)............... 25 2.6 Cartan subalgebras and subgroups and the center . 26 2.7 Summary of matrix Lie groups and Lie algebras . 26 2.8 Non-semi-simple matrix groups . 30 2.9 Invariant differential forms, metric, and measure . 31 2.10 Spontaneous symmetry breaking, order parameters . 37 2.11 Lie supergroups and Lie superalgebras . 38 3 Representations 41 3.1 Representations of groups . 41 3.2 Representations of Lie algebras . 43 3.3 Transformation of a vector space under a representation . 44 3.4 Direct sum of representations . 45 3.5 Schur's Lemma . 46 3.6 Unitary representations . 47 3.7 Tensor product of representations . 49 3.8 Characters of Representations . 50 4 Representations of SL(2; C), SU(2), and SO(2; 1) 53 4.1 Irreducible representations of the Lie group SU(2) . 53 4.2 Finite-dimensional representations of sl(2; C)................. 54 4.3 Infinite-dimensional representations of sl(2; C)................ 57 4.4 Harmonic oscillator representation of so(2; 1; R)............... 57 4.5 Unitary representations of so(2; 1; R)..................... 58 4.6 An example of the continuous series . 60 2 5 Tensor representations 62 5.1 Tensor product representations . 62 5.2 Symmetrization and anti-symmetrization . 63 5.3 Representations of SU(3)............................ 64 5.4 Representations of SU(n)........................... 66 6 Spinor representations 68 6.1 Spinor representations of SO(3; R)...................... 68 6.2 The Clifford Algebra . 69 6.3 Representations of the Clifford algebra . 70 6.4 Spinor representations of so(d)......................... 72 6.5 Reducibility and Weyl spinor representations . 73 6.6 Charge and complex conjugation . 74 6.7 Spinor representations of so(d − 1; 1; R).................... 75 7 Roots, weights, and representations 78 7.1 The Cartan-Killing form . 78 7.2 Weights . 79 7.3 Roots . 80 7.4 Raising and Lowering operators . 82 7.5 Finite-dimensional representations . 82 7.6 The example of SU(3) ............................. 84 7.7 The root lattice of An = sl(n +1)....................... 86 7.8 The root lattice of Dn = so(2n)........................ 87 7.9 The root lattice of Bn = so(2n +1)...................... 89 7.10 The root lattice of Cn = sp(2n +1)...................... 90 8 Representations of the classical Lie algebras 92 8.1 Weyl reflections and the Weyl group . 92 8.2 Finite-dimensional irreducible representations of sl(n + 1; C)........ 93 8.3 Spinor representation of so(2n + 1; C)..................... 94 8.4 Spinor representations of so(2n; C)...................... 94 9 The structure of semi-simple Lie algebras 96 9.1 Some properties of simple roots . 96 9.2 Classification of finite-dimensional simple Lie algebras . 97 10 Weyl's character formulas 102 10.1 Characters on the maximal torus . 103 10.2 Weyl's first formula . 104 3 Bibliography Standard texts for physicists • H. Georgi, Lie Algebras and Particle Physics, Benjamin/Cummings, 1982; • B.G. Wybourne, Classical Groups for Physicists, Wiley, 1974; • W. K. Tung, Group Theory in Physics, World Scientific; • P. Di Francesco, P. Mathieu, D. S´enechal, Conformal Field Theory, Springer 1997, Chapters 13 and 14. Classics and more mathematical • H. Weyl, Classical Groups, Princeton, 1946. • C. Chevalley, Theory of Lie Groups, Princeton, 1948; • E. Cartan, The Theory of Spinors, Hermann, Paris 1966, Dover; • S. Helgason, Differential Geometry, Lie Groups, Symmetric Spaces, Acad. Press 1978; • M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing (2005) (no group theory per se, but great for topology and differential geometry); Very useful but more mathematical • B.C. Hall, Lie Groups, Lie Algebras, and Representations, Springer Verlag Graduate Texts in Mathematics 222 (2015); • W. Fulton and J. Harris, Representation Theory, Springer Verlag, 1991; • D.P. Zelobenko, Compact Lie Groups and Their Representations, Tansl. Math. Monographs, Vol. 40, Am. Math. Soc., 1973; • N.J. Vilenkin, Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, Vol 22, Am. Math. Soc., 1968; • F.D. Murnaghan, The Theory of Group Representations, Johns Hopkins, 1938; Others • A.W. Knapp, Lie Groups beyond an Introduction, Birkh¨auser2004; • J. Humphreys, Introduction to Lie Algbras & Representation Theory, Springer, 1980; • N. Jacobson, Lie Algebras, Wiley, 1962; Dover, 1979; • I.R. Porteous, Clifford Algebras and the Classical Groups, Cambridge, 1995; 4 1 Definition and examples of groups In this section, we shall begin with a little history, and then present the basic definitions and some simple examples of groups, subgroups, topological groups, discrete groups, parametric groups, Lie groups, and Lie algebras. 1.1 A little history The mathematical concept of a group goes back to Joseph-Louis Lagrange (1736 - 1813) and Niels Abel (1802 - 1829), and was articulated by Evariste Galois (1811 - 1832) in the context of subgroups of the group of permutations of the roots of a polynomial. Lie groups and Lie algebras were introduced by Sophus Lie (1842 - 1899) around 1870. In physics the concept of a group arises almost invariably in the context of a symmetry or invariance of a physical quantity or of an equation which described a physical quantity. Greek antiquity appreciated the beauty of regular polygons and polyhedra. Galileo and Newton undoubtedly understood the notions of translation and rotation invariance, such as in Galilean relativity, and used them without invoking the notion of a group. Poincar´e(1854 - 1912) observed the invariance of Maxwell's equations (without sources) under Lorentz and Poincar´etransformation. It is fair to state that Einstein (1879 - 1955) pioneered elevating the symmetry of the equations to a physical principle in his 1905 theory of special relativity (Poincar´einvariance), but even more clearly in his 1915 theory of general relativity (invariance under general coordinate transformations). Since then, group theory has become a dominant organizing principle of modern physics: Herman Weyl (1885 - 1955) applied group theory to quantum mechanics, Lev Landau (1908 - 1968) based his theory of second order phase transitions on the group-theoretic symmetry properties of an order parameter; Murray Gell-Mann (1929 - present) used SU(3) group theory to predict the Ω particle; and the Standard Model of Particle Physics is built on the Yang-Mills theory for the gauge group SU(3)c ×SU(2)L ×U(1)Y , though some of this symmetry is spontaneously broken and not manifest at low energies. One of the biggest theoretical questions in modern theoretical and experimental physics is to find out what lies beyond the Standard Model. The most popular theoretical specu- lations are grand unification, which is based on extending SU(3)c × SU(2)L × U(1)Y to a larger unifying group, and supersymmetry which is based on extending the Poincar´egroup to a supergroup, with fermionic parameters. One of the most active areas of research over the past decade has been the conformal bootstrap program, based on exploiting the prop- erties of the conformal group to study quantum field theories that do not necessarily admit a Lagrangian or Hamiltonian description. 5 1.2 Groups A set G equipped with a binary operation ? is a group G? provided G and ? satisfy, 1. Closure: For any pair of elements g1; g2 2 G the operation obeys g1 ? g2 2 G; 2. Associativity: For any triplet of elements g1; g2; g3 2 G the operation obeys (g1 ? g2) ? g3 = g1 ? (g2 ? g3) = g1 ? g2 ? g3; 3. Identity: There exists an element e 2 G such that e ? g = g ? e = g for all g 2 G; 4. Inverse: For every g 2 G, there exists a g−1 2 G such that g ? g−1 = g−1 ? g = e. The group axioms imply that the identity element e is unique, and that for every element −1 g, the inverse g is unique. The direct product G1 × G2 of two groups G1 and G2 with respective operations ∗1 and ∗2 is a group with elements (g1; g2) under the operation ∗ defined by (g1; g2) ∗ (h1; h2) = (g1 ∗1 h1; g2 ∗2 h2) with g1; h1 2 G1 and g2; h2 2 G2. A group G? is commutative or Abelian if the product satisfies g1 ? g2 = g2 ? g1 for all g1; g2 2 G. Otherwise, the group is said to be non-commutative or non-Abelian. When no confusion is expected to arise, one often drops reference to the operation in denoting a group, so that G∗ is simply denoted G. 1.2.1 Subgroups, invariant subgroups, and simple groups A subgroup H of a group G is a subset H ⊂ G such that h1 ? h2 2 H for all h1; h2 2 H and h−1 2 H for all h 2 H.
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