Lie groups and lie algebras [electronic resource] : a physicist's perspective / Adam M. Bincer

Author: Bincer, Adam M. (Adam Marian)
Published: Oxford : Oxford University Press, 2013.
Physical Description: 1 online resource

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Contents

Machine generated contents note: ch. 1 Generalities -- Definitions of group, isomorphism, representation, vector space and algebra. Biographical notes on Galois, Abel and Jacobi are given -- ch. 2 Lie groups and Lie algebras -- Lie Groups, infinitesimal generators, structure constants, Cartan's metric tensor, simple and semisimple groups and algebras, compact and non-compact groups. Biographical notes on Euler, Lie and Cartan are given -- ch. 3 Rotations: SO(3) and SU(2) -- Rotations and reflections, connectivity, center, universal covering group -- ch. 4 Representations of SU(2) -- Irreducible representations, Casimir operators, addition of angular momenta, Clebsch-Gordan coefficients, the Wigner-Eckart theorem, multiplicity. Biographical notes on Casimir, Weyl, Clebsch, Gordan and Wigner are given -- ch. 5 The so(n) algebra and Clifford numbers -- Spin(n), spinors and semispinors, Schur's lemma. Biographical notes on Clifford and Schur are given -- ch. 6 Reality properties of spinors -- Conjugate, orthogonal and symplectic representations -- ch. 7 Clebsch-Gordan series for spinors -- Antisymmetric tensors, duality -- ch. 8 The center and outer automorphisms of Spin(n) -- Inversion, Z2, Z4 and Z2XZ2 centers. A biographical note on Dynkin is given -- ch. 9 Composition algebras -- Hurwitz's theorem, quaternions and octonions, non-associativity. Biographical notes on Hurwitz, Hamilton, Graves, Cayley and Frobenius are given -- ch. 10 The exceptional group G2 -- Automorphisms of octonions, quaternions and complex numbers. A biographical note on Racah is given -- ch. 11 Casimir operators for orthogonal groups -- The invariant & tensor, integrity basis. A biographical note on Pfaff is given -- ch. 12 Classical groups -- Orthogonal, symplectic and unitary groups; their dimensions and connectivity. Biographical notes on Lorentz, de Sitter, Liouville, Maxwell and Thomas are given -- ch. 13 Unitary groups -- Generators, tensors, Casimir operators; anomaly cancellation -- ch. 14 The symmetric group Sr and Young tableaux -- Permutations, the symmetric group, its representations and their dimension, Young patterns and tableaux. A biographical note on Young is given -- ch. 15 Reduction of SU(n) tensors -- Irreducible tensors and their dimensions, conjugate representations, reduction of the Kronecker product. A biographical note on Kronecker is given -- ch. 16 Cartan basis, simple roots and fundamental weights -- Cartan subalgebra, Weyl reflections -- ch. 17 Cartan classification of semisimple algebras -- The rank one A1 algebra, the rank two A2,B2=C2,D2=A1A1 and G2 algebras, the rank n An,Bn,Cn and Dn series of algebras, the F4 and the En, 6 [≤]n[≤]8, exceptional algebras -- ch. 18 Dynkin diagrams -- Rules for constructing Dynkin diagrams, simply laced algebras, folding -- ch. 19 The Lorentz group -- Lorentz Casimirs, principal and complementary series of unitary representations, finite-dimensional representations, the universal cover SL(2, C), polar decomposition. Biographical notes on Minkowski, Klein, Gordon, Dirac and Proca are given -- ch. 20 The Poincare and Liouville groups -- Semidirect product, Pauli-Lubanski four-vector, Poincare Casimirs, representations of the Poincare group, conformal group, Virasoro and Kac-Moody algebras. Biographical notes on Poincare, Pauli, Lubanski, Kac and Moody are given -- ch. 21 The Coulomb problem in n space dimensions -- Heisenberg algebra, Lenz-Runge n-vector, energy levels of the hydrogen atom. Biographical notes on Coulomb, Heisenberg, Lenz and Runge are given.

Summary

This text gives an introduction to group theory for physicists with a focus on lie groups and lie algebras.

Subject(s)

ISBN

9780191745492 (ebook)

Bibliography Note

Includes bibliographical references and index.

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