Portal de Periódicos da CAPES

An Introduction to Lie Groups and Lie Algebras, with Applications

1966; Society for Industrial and Applied Mathematics; Volume: 8; Issue: 1 Linguagem: Inglês

10.1137/1008002

1095-7200

Johan G. F. Belinfante, Bernard Kolman, Harvey A. Smith,

Quantum chaos and dynamical systems

Previous article Next article An Introduction to Lie Groups and Lie Algebras, with ApplicationsJ. G. Belinfante, B. Kolman, and H. A. SmithJ. G. Belinfante, B. Kolman, and H. A. Smithhttps://doi.org/10.1137/1008002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] George Allen Baker, Jr., Degeneracy of the n-dimensional, isotropic, harmonic oscillator, Phys. Rev. (2), 103 (1956), 1119–1120 10.1103/PhysRev.103.1119 MR0081170 0071.42505 CrossrefISIGoogle Scholar[2A] V. Bargmann and , M. Moshinsky, Group theory of harmonic oscillators. I. The collective modes, Nuclear Phys., 18 (1960), 697–712 10.1016/0029-5582(60)90438-7 MR0121199 0096.23902 CrossrefISIGoogle Scholar[2B] V. Bargmann and , M. Moshinsky, Group theory of harmonic oscillators. II. The integrals of motion for the quadrupole-quadrupole interaction, Nuclear Phys., 23 (1961), 177–199 10.1016/0029-5582(61)90253-X MR0122476 0094.43904 CrossrefISIGoogle Scholar[3] R. E. Behrends, , J. Dreitlein, , C. Fronsdal and , W. Lee, Simple groups and strong interaction symmetries, Rev. Modern Phys., 34 (1962), 1–40 10.1103/RevModPhys.34.1 MR0156617 0108.22604 CrossrefGoogle Scholar[4] L. C. Biedenharn, Wigner coefficients for the $R\sb{4}$ group and some applications, J. Mathematical Phys., 2 (1961), 433–441 10.1063/1.1703728 MR0146268 0098.12802 CrossrefISIGoogle Scholar[5] F. Bruhat, Lectures on Lie Groups and Representations of Locally Compact Groups, Lecture notes, Tata Institute of Fundamental Research, Bombay, 1958 Google Scholar[6A] E. Cartan, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France, 41 (1913), 53–96 MR1504700 CrossrefGoogle Scholar[6B] Elie Cartan, Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3), 31 (1914), 263–355 MR1509178 CrossrefGoogle Scholar[7] Kuo-tsai Chen, Decompostion of differential equations, Math. Ann., 146 (1962), 263–278 10.1007/BF01470955 MR0140621 0166.02802 CrossrefISIGoogle Scholar[8] Claude Chevalley, Theory of Lie groups. I, Princeton University Press, Princeton, N. J., 1946 1957xi+217 pp. (Third printing 1957) MR0082628 0063.00842 CrossrefGoogle Scholar[9] A. Cohen, An Introduction to the Lie Theory of One-Parameter Groups, Stechert, New York, 1931 Google Scholar[10] P. M. Cohn, Lie groups, Cambridge Tracts in Mathematics and Mathematical Physics, no. 46, Cambridge University Press, New York, N.Y., 1957vii+164 MR0103940 0084.03201 Google Scholar[11] E. B. Dynkin, The structure of semisimple Lie algebras, Amer. Math. Soc. Transl. 17, (1950), , reprinted as 9 (1962), pp. 328–469 Google Scholar[12] A. R. Edmonds, Angular momentum in quantum mechanics, Investigations in Physics, Vol. 4, Princeton University Press, Princeton, N.J., 1957viii+146 MR0095700 0079.42204 CrossrefGoogle Scholar[13A] J. P. Elliott, Collective motion in the nuclear shell model. I. Classification schemes for states of mixed configurations, Proc. Roy. Soc. London. Ser. A., 245 (1958), 128–145 MR0092620 0082.43901 CrossrefISIGoogle Scholar[13B] J. P. Elliott, Collective motion in the nuclear shell model. II. The introduction of intrinsic wave-functions, Proc. Roy. Soc. London. Ser. A, 245 (1958), 562–581 MR0094178 0082.43901 CrossrefISIGoogle Scholar[14A] F. Gantmacher, Canonical representation of automorphisms of a complex semisimple Lie group, Mat. Sb., 5 (1939), 101–146 Google Scholar[14B] Felix Gantmacher, On the classification of real simple Lie groups, Rec. Math. [Mat. Sbornik] N.S., 5 (47) (1939), 217–250 MR0002141 0022.31503 Google Scholar[15] Murray Gell-Mann and , Yuval Ne'eman, The eightfold way, W. A. Benjamin, Inc., New York-Amsterdam, 1964xi+317 MR0180214 Google Scholar[16] Murray Gell-Mann and , Yuval Ne'eman, Current-generated algebras, Ann. Physics, 30 (1964), 360–369 10.1016/0003-4916(64)90122-8 MR0178842 CrossrefISIGoogle Scholar[17] Roger Godement, A theory of spherical functions. I, Trans. Amer. Math. Soc., 73 (1952), 496–556 MR0052444 0049.20103 CrossrefISIGoogle Scholar[18] F. Gursey, Group theoretical concepts and methods in elementary particle physics, Lectures of the Istanbul Summer School of Theoretical Physics, July 16-August 4, Vol. 1962, Gordon and Breach Science Publishers, New York, 1964vii+425 MR0170636 Google Scholar[19] Sigurur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York, 1962xiv+486 MR0145455 0111.18101 Google Scholar[20] K. Helmers, Symplectic invariants and Flowers' classification of shell model states, Nuclear Phys., 23 (1961), 594–611 10.1016/0029-5582(61)90285-1 MR0128410 CrossrefISIGoogle Scholar[21] J. C. Herz, Contribution to the algebraic theory of partial differential equations, Ann. Sci. Ecole Norm. Sup. (3), 71 (1954), 321–362 CrossrefGoogle Scholar[22] W. C. Hoffman, Pattern recognition and the method of isoclines, Boeing Scientific Report, Boeing Scientific Research Laboratories, Seattle, 1964 Google Scholar[23] K. H. Hofmann and , Paul Mostert, Splitting in topological groups, Mem. Amer. Math. Soc. No., 43 (1963), 75– MR0151544 0163.02705 Google Scholar[24] L. Infeld and , T. E. Hull, The factorization method, Rev. Modern Physics, 23 (1951), 21–68 10.1103/RevModPhys.23.21 MR0043308 0043.38602 CrossrefGoogle Scholar[25] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962ix+331 MR0143793 0121.27504 Google Scholar[26] J. M. Jauch and , E. L. Hill, On the problem of degeneracy in quantum mechanics, Phys. Rev., 57 (1940), 641–645 10.1103/PhysRev.57.641 MR0001669 0023.28503 CrossrefGoogle Scholar[27] H. Joos, Lectures on Lie algebras for physicists, Acts Phys. Austriaca, 1 (1964), 1–18 Google Scholar[28] Michiji Konuma, , Kazuhisa Shima and , Morihiro Wada, Simple Lie algebras of rank 3 and symmetries of elementary particles in the strong interaction, Progr. Theoret. Phys. Suppl. No., 28 (1963), 1–128 MR0183414 0129.43401 CrossrefISIGoogle Scholar[29] Wilhelm Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), 649–673 MR0067873 0056.34102 CrossrefISIGoogle Scholar[30] W. Mayer and , T. Y. Thomas, Foundations of the theory of Lie groups, Ann. of Math. (2), 36 (1935), 770–822 MR1503252 0012.05502 CrossrefGoogle Scholar[31] Willard Miller, Jr., On Lie algebras and some special functions of mathematical physics, Mem. Amer. Math. Soc. No., 50 (1964), iv+43 MR0173031 0132.29602 Google Scholar[32] M. Moshinsky, The harmonic oscillator and supermultiplet theory, Nuclear Phys., 31 (1962), 384–405 10.1016/0029-5582(62)90758-7 CrossrefISIGoogle Scholar[33] F. J. Murray, Perturbation theory and Lie algebras, J. Mathematical Phys., 3 (1962), 451–468 10.1063/1.1724245 MR0146308 0149.46601 CrossrefISIGoogle Scholar[34] M. A. Naimark, Linear representations of the Lorentz group, Translated by Ann Swinfen and O. J. Marstrand; translation edited by H. K. Farahat. A Pergamon Press Book, The Macmillan Co., New York, 1964xiv+450 MR0170977 0116.32904 Google Scholar[35] James Wei and , Edward Norman, Lie algebraic solution of linear differential equations, J. Mathematical Phys., 4 (1963), 575–581 10.1063/1.1703993 MR0149053 0133.34202 CrossrefISIGoogle Scholar[36] W. Pauli, Jr., On the hydrogen spectrum from the standpoint of the new quantum mechanics, Z. Physik, 36 (1926), 336–363 CrossrefGoogle Scholar[37] L. Pontryagin, Topological Groups, Moscow, 1954 Google Scholar[38] G. Racah, Group Theory and Spectroscopy, Lecture notes, Institute for Advanced Studies, Princeton, 1951 Google Scholar[39] Peter J. Redmond, Generalization of the Runge-Lenz vector in the presence of an electric field, Phys. Rev. (2), 133 (1964), B1352–B1353 10.1103/PhysRev.133.B1352 MR0182190 CrossrefISIGoogle Scholar[40] M. E. Rose, Elementary Theory of Angular Momentum, John Wiley, New York, 1957 0079.20102 CrossrefGoogle Scholar[41] Hans Samelson, Topology of Lie groups, Bull. Amer. Math. Soc., 58 (1952), 2–37 MR0045129 0047.16701 CrossrefISIGoogle Scholar[42] N. Ya. Vilenkin, Bessel functions and representations of the group of Euclidean motions, Uspehi Mat. Nauk (N.S.), 11 (1956), 69–112 MR0086268 Google Scholar[43] N. Ya. Vilenkin, On the theory of associated spherical functions, Dokl. Akad. Nauk SSSR (N.S.), 111 (1956), 742–744 MR0086267 0094.24705 Google Scholar[44] B. L. van der Waerden, The classification of simple Lie groups, Math. Z., 37 (1933), 446–462 10.1007/BF01474586 MR1545406 0007.29201 CrossrefGoogle Scholar[45] J. Wei and , E. Norman, On global representations of the solutions of linear differential equations as a product of exponentials, Proc. Amer. Math. Soc., 15 (1964), 327–334 MR0160009 0119.07202 CrossrefGoogle Scholar[46] Louis Weisner, Group-theoretic origin of certain generating functions, Pacific J. Math., 5 (1955), 1033–1039 MR0086905 0067.29401 CrossrefGoogle Scholar[47] G. H. Weiss and , A. A. Maradudin, The Baker-Hausdorff formula and a problem in crystal physics, J. Mathematical Phys., 3 (1962), 771–777 10.1063/1.1724280 MR0147248 0108.44804 CrossrefISIGoogle Scholar[48] H. Weyl, Harmonics on homogeneous manifolds, Ann. of Math. (2), 35 (1934), 486–499 MR1503175 0010.01201 CrossrefGoogle Scholar[49] Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939xii+302 MR0000255 0020.20601 Google Scholar[50] Eyvind H. Wichmann, Note on the algebraic aspect of the integration of a system of ordinary linear differential equations, J. Mathematical Phys., 2 (1961), 876–880 10.1063/1.1724235 MR0136831 0106.29003 CrossrefISIGoogle Scholar[51] E. P. Wigner, The Application of Group Theory to the Special Functions of Mathematical Physics, Lecture notes, Princeton, 1955 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Lie algebras of Lie groups, Kac-Moody groups, supergroups, and some specialized topics in finite- and infinite-dimensional Lie algebrasIntroduction to Finite and Infinite Dimensional Lie (Super)algebras | 1 Jan 2016 Cross Ref BibliographyIntroduction to Finite and Infinite Dimensional Lie (Super)algebras | 1 Jan 2016 Cross Ref Appendix☆☆“To view the full reference list for the book, click here”Introduction to Finite and Infinite Dimensional Lie (Super)algebras | 1 Jan 2016 Cross Ref Representation Theory for Risk on Markowitz-Tversky-Kahneman TopologySSRN Electronic Journal | 1 Jan 2012 Cross Ref Accurate Estimation of ICA Weight Matrix by Implicit Constraint Imposition Using Lie GroupIEEE Transactions on Neural Networks, Vol. 20, No. 10 | 1 Oct 2009 Cross Ref Geometrical methods for non-negative ICA: Manifolds, Lie groups and toral subalgebrasNeurocomputing, Vol. 67 | 1 Aug 2005 Cross Ref Liquid Sloshing Dynamics11 August 2009 Cross Ref Canonical forms for systems of two second-order ordinary differential equationsJournal of Physics A: Mathematical and General, Vol. 34, No. 13 | 26 March 2001 Cross Ref Some open problems in matrix theory arising in linear systems and controlLinear Algebra and its Applications, Vol. 162-164 | 1 Feb 1992 Cross Ref A Novel Architecture for Real Time Pick Up of 3D Motion and 3D Layout Information from The Flow of The Optic ArrayReal-Time Object Measurement and Classification | 1 Jan 1988 Cross Ref The equivalence of uniquely divisible semigroups and uniquely representable semigroups on the two-cellSemigroup Forum, Vol. 4, No. 1 | 1 Dec 1972 Cross Ref Computer Approaches to the Representations of Lie AlgebrasJournal of the ACM, Vol. 19, No. 4 | 1 Oct 1972 Cross Ref Generation of the Weyl group on a computerJournal of Computational Physics, Vol. 7, No. 2 | 1 Apr 1971 Cross Ref Fourth Picture in Quantum MechanicsThe Journal of Chemical Physics, Vol. 53, No. 4 | 15 Aug 1970 Cross Ref An Introduction to Lie Groups and Lie Algebras, with Applications. III: Computational Methods and Applications of Representation TheoryJ. G. Belinfante and B. KolmanSIAM Review, Vol. 11, No. 4 | 18 July 2006AbstractPDF (3082 KB)An Introduction to Lie Groups and Lie Algebras, with Applications. II: The Basic Methods and Results of Representation TheoryJ. G. Belinfante, B. Kolman, and H. A. SmithSIAM Review, Vol. 10, No. 2 | 18 July 2006AbstractPDF (4649 KB) Volume 8, Issue 1| 1966SIAM Review1-143 History Submitted:05 April 1965Published online:18 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1008002Article page range:pp. 11-46ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics