American Mathematical Society

Book Reviews
Book reviews do not contain an abstract. You may download the entire review from the links below.

Erratum: Bull. Amer. Math. Soc. 46 (2009), 175-175.

Full text review in PDF
This review is available free of charge

Lie groups. An approach through invariants and representations by Claudio Procesi: Bull. Amer. Math. Soc. 45 (2008), 661-674; Additional book information: Springer, New York, 2007, xxii + 596 pp., ISBN 978-0-387-26040-2, US$59.95$

References

David Hilbert, Hilbert’s invariant theory papers, Lie Groups: History, Frontiers and Applications, VIII, Math Sci Press, Brookline, Mass., 1978. Translated from the German by Michael Ackerman; With comments by Robert Hermann. MR 512034
Michael Atiyah and Wilfried Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62. MR 463358, DOI 10.1007/BF01389783
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), no. 1, 61–110. MR 496157, DOI 10.1016/0003-4916(78)90224-5
Richard E. Block and Robert Lee Wilson, Classification of the restricted simple Lie algebras, J. Algebra 114 (1988), no. 1, 115–259. MR 931904, DOI 10.1016/0021-8693(88)90216-5
Armand Borel, Groupes linéaires algébriques, Ann. of Math. (2) 64 (1956), 20–82 (French). MR 93006, DOI 10.2307/1969949
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
Armand Borel and Jacques Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150 (French). MR 207712
Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. MR 89473, DOI 10.2307/1969996
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1975 edition. MR 979493
F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923
C. Carmeli, G. Cassinelli, A. Toigo, and V. S. Varadarajan, Unitary representations of super Lie groups and applications to the classification and multiplet structure of super particles, Comm. Math. Phys. 263 (2006), no. 1, 217–258. MR 2207328, DOI 10.1007/s00220-005-1452-0
Élie Cartan, Œuvres complètes. Partie I, 2nd ed., Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1984 (French). Groups de Lie. [Lie groups]. MR 753096
H. Casimir and B. L. van der Waerden, Algebraischer Beweis der vollständigen Reduzibilität der Darstellungen halbeinfacher Liescher Gruppen, Math. Ann. 111 (1935), no. 1, 1–12 (German). MR 1512971, DOI 10.1007/BF01472196
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
Shiing-Shen Chern and Claude Chevalley, Obituary: Elie Cartan and his mathematical work, Bull. Amer. Math. Soc. 58 (1952), 217–250. MR 46972, DOI 10.1090/S0002-9904-1952-09588-4
Claude Chevalley, Sur la classification des algèbres de Lie simples et de leurs représentations, C. R. Acad. Sci. Paris 227 (1948), 1136–1138 (French). MR 27270
Claude Chevalley, Theory of Lie groups. I, Princeton Mathematical Series, vol. 8, Princeton University Press, Princeton, NJ, 1999. Fifteenth printing; Princeton Landmarks in Mathematics. MR 1736269
Claude Chevalley, Classification des groupes algébriques semi-simples, Springer-Verlag, Berlin, 2005 (French). Collected works. Vol. 3; Edited and with a preface by P. Cartier; With the collaboration of Cartier, A. Grothendieck and M. Lazard. MR 2124841
C. Chevalley, Sur certains groupes simples, Tohoku Math. J. (2) 7 (1955), 14–66 (French). MR 73602, DOI 10.2748/tmj/1178245104
Claude Chevalley, The algebraic theory of spinors and Clifford algebras, Springer-Verlag, Berlin, 1997. Collected works. Vol. 2; Edited and with a foreword by Pierre Cartier and Catherine Chevalley; With a postface by J.-P. Bourguignon. MR 1636473
P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195 (French). MR 1106898
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
Pierre Deligne and John W. Morgan, Notes on supersymmetry (following Joseph Bernstein), Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 41–97. MR 1701597
Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958. With an introduction by W. Magnus. MR 0104735
V. G. Drinfel′d, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR 934283
E. B. Dynkin, The structure of semi-simple algebras, Amer. Math. Soc. Translation 1950 (1950), no. 17, 143. MR 0035761
Thomas J. Enright, On the fundamental series of a real semisimple Lie algebra: their irreducibility, resolutions and multiplicity formulae, Ann. of Math. (2) 110 (1979), no. 1, 1–82. MR 541329, DOI 10.2307/1971244
Thomas J. Enright and V. S. Varadarajan, On an infinitesimal characterization of the discrete series, Ann. of Math. (2) 102 (1975), no. 1, 1–15. MR 476921, DOI 10.2307/1970970
Thomas J. Enright and Nolan R. Wallach, The fundamental series of representations of a real semisimple Lie algebra, Acta Math. 140 (1978), no. 1-2, 1–32. MR 476814, DOI 10.1007/BF02392301
Hans Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York-London, 1969. MR 0260926
Edward Formanek and Claudio Procesi, Mumford’s conjecture for the general linear group, Advances in Math. 19 (1976), no. 3, 292–305. MR 404279, DOI 10.1016/0001-8708(76)90026-8
Roe Goodman and Nolan R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR 1606831
W. J. Haboush, Reductive groups are geometrically reductive, Ann. of Math. (2) 102 (1975), no. 1, 67–83. MR 382294, DOI 10.2307/1970974
Harish-Chandra, On some applications of the universal enveloping algebra of a semisimple Lie algebra, Trans. Amer. Math. Soc. 70 (1951), 28–96. MR 44515, DOI 10.1090/S0002-9947-1951-0044515-0

[H2]

Harish-Chandra, Collected papers, Edited by V. S. Varadarajan, Springer-Verlag, New York, 1984, Vol. I., 566 pp; Vol. II., 539 pp; Vol. III., 670 pp; Vol. IV., 461 pp.

Harish-Chandra, Eisenstein series over finite fields, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968) Springer, New York, 1970, pp. 76–88. MR 0457579

Thomas Hawkins, Emergence of the theory of Lie groups, Sources and Studies in the History of Mathematics and Physical Sciences, Springer-Verlag, New York, 2000. An essay in the history of mathematics 1869–1926. MR 1771134, DOI 10.1007/978-1-4612-1202-7

Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112

Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034

Kenkichi Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. MR 29911, DOI 10.2307/1969548

Nathan Jacobson, Lie algebras, Dover Publications, Inc., New York, 1979. Republication of the 1962 original. MR 559927

Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234

V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, DOI 10.1016/0001-8708(77)90017-2

Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2

Wilhelm Killing, Die Zusammensetzung der stetigen endlichen Transformations-gruppen, Math. Ann. 34 (1889), no. 1, 57–122 (German). MR 1510568, DOI 10.1007/BF01446792

Anthony W. Knapp, Lie groups beyond an introduction, 2nd ed., Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1920389

Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130

Michel Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603 (French). MR 209286

[L]

Séminaire ``Sophus Lie'', 1, 1954-1955 Theorie des algebres de Lie. Topologie des groupes de Lie; Séminaire ``Sophus Lie'', 2, 1955-1956, Hyperalgèbre et groupes de Lie formels.

George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098

Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104

Robert V. Moody and Arturo Pianzola, Lie algebras with triangular decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1323858

J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Philos. Soc. 45 (1949), 99–124. MR 29330

David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602

David Mumford, Hilbert’s fourteenth problem–the finite generation of subrings such as rings of invariants, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Amer. Math. Soc.., Providence, R.I., 1976, pp. 431–444. MR 0435076

David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7

Jean-Pierre Serre, Complex semisimple Lie algebras, Springer-Verlag, New York, 1987. Translated from the French by G. A. Jones. MR 914496, DOI 10.1007/978-1-4757-3910-7

Jean-Pierre Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992. 1964 lectures given at Harvard University. MR 1176100

Wilfried Schmid, On a conjecture of Langlands, Ann. of Math. (2) 93 (1971), 1–42. MR 286942, DOI 10.2307/1970751

C. S. Seshadri, Theory of moduli, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R. I., 1975, pp. 263–304. MR 0396565

T. A. Springer, Linear algebraic groups, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR 1642713, DOI 10.1007/978-0-8176-4840-4

T. A. Springer, Caractères de groupes de Chevalley finis, Séminaire Bourbaki, 25ème année (1972/1973), Lecture Notes in Math., Vol. 383, Springer, Berlin, 1974, pp. Exp. No. 429, pp. 210–233 (French). MR 0463276

Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335

Robert Steinberg, Nagata’s example, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., vol. 9, Cambridge Univ. Press, Cambridge, 1997, pp. 375–384. MR 1635692

Robert Steinberg, The isomorphism and isogeny theorems for reductive algebraic groups, J. Algebra 216 (1999), no. 1, 366–383. MR 1694546, DOI 10.1006/jabr.1998.7776

V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6

V. S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Cambridge Studies in Advanced Mathematics, vol. 16, Cambridge University Press, Cambridge, 1999. Corrected reprint of the 1989 original. MR 1725738

V. S. Varadarajan, Geometry of quantum theory, 2nd ed., Springer-Verlag, New York, 1985. MR 805158

V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, Courant Lecture Notes in Mathematics, vol. 11, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2004. MR 2069561, DOI 10.1090/cln/011

V. S. Varadarajan, Harmonic analysis on real reductive groups, Lecture Notes in Mathematics, Vol. 576, Springer-Verlag, Berlin-New York, 1977. MR 0473111

David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, Mass., 1981. MR 632407

Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683

Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1992. MR 1170566

William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117

Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158

Hermann Weyl, Gesammelte Abhandlungen. Bände I, II, III, IV, Springer-Verlag, Berlin-New York, 1968 (German). Herausgegeben von K. Chandrasekharan. MR 0230597

[W3]

Weyl, Hermann, The Theory of Groups and Quantum Mechanics, Translated by H. P. Robertson, Dover, 1949, 422 pages. Translated from the original 1928 german edition of Gruppentheorie und Quantenmechanik.

S. L. Woronowicz, Tannaka-Kreĭn duality for compact matrix pseudogroups. Twisted $\textrm {SU}(N)$ groups, Invent. Math. 93 (1988), no. 1, 35–76. MR 943923, DOI 10.1007/BF01393687

Gregg Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. of Math. (2) 106 (1977), no. 2, 295–308. MR 457636, DOI 10.2307/1971097

Reviewer information

Reviewer: V. S. Varadarajan
Affiliation: University of California at Los Angeles
Email: vsv@math.ucla.edu

Additional Information

Journal: Bull. Amer. Math. Soc. 45 (2008), 661-674
DOI: https://doi.org/10.1090/S0273-0979-08-01201-9
Published electronically: July 1, 2008

Book Reviews Book reviews do not contain an abstract. You may download the entire review from the links below.

Book Reviews
Book reviews do not contain an abstract. You may download the entire review from the links below.