444 X A Character Study
was established by Lie, but we owe to Killing (1888–1890) the remarkable classifica-
tion of simple complex Lie algebras. Some gaps and inaccuracies in Killing’s pioneer-
ing work were filled and corrected in the thesis of Cartan (1894). The classification of
simple real Lie algebras is due to Cartan (1914). The representation theory of semisim-
ple Lie algebras and compact semisimple Lie groups is the creation of Cartan (1913)
and Weyl (1925–7). The introduction of groups generated by reflections is due to Weyl.
For the theory of Lie groups, see Chevalley [7], Warner [44], Varadarajan [43],
Helgason [22] and Barut and Raczka [2]. Thelast reference also has information
on representations of noncompact Lie groups and applications to quantum theory.
The purely algebraic theory of Lie algebras is discussed by Jacobson [28] and
Humphreys [25]. Niederle [38] gives a survey of the applications of the exceptional
Lie algebras and Lie superalgebras in particle physics. Groups generated by reflections
are treated by Humphreys [26], Bourbaki [5] and Kac [30], while Cohen [8] gives a
useful overview.
The character theory of locally compact abelian groups, whose roots lie in
Dirichlet’s theorem on primes in arithmetic progressions, has given something back to
number theory in the adelic approach to algebraic number fields; see the thesis of Tate,
reproduced (pp. 305–347) in Cassels and Fr ̈ohlich [6], Lang [31] and Weil [47]. For a
broad historical perspective and future plans, see Mackey [35] and Langlands [32].
10 SelectedReferences
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(1996), 1717–1735.
[2] A.O. Barut and R. Raczka,Theory of group representations and applications, 2nd ed.,
Polish Scientific Publishers, Warsaw, 1986.
[3] P.T. Bateman, A theorem of Ingham implying that Dirichlet’sL-functions have no zeros
with real part one,Enseign. Math. 43 (1997), 281–284.
[4] J.L. Birman,Theory of crystal space groups and lattice dynamics, Springer-Verlag, Berlin,
1984.
[5] N. Bourbaki,Groupes et alg`ebres de Lie:Chapitres 4,5 et 6, Masson, Paris, 1981.
[6] J.W.S. Cassels and A. Fr ̈ohlich (ed.),Algebraic number theory, Academic Press, London,
1967.
[7] C. Chevalley,Theory of Lie groups I, Princeton University Press, Princeton, 1946.
[Reprinted, 1999]
[8] A.M. Cohen, Coxeter groups and three related topics,Generators and relations in groups
and geometries(ed. A. Barlottiet al.), pp. 235–278, Kluwer, Dordrecht, 1991.
[9] J.F. Cornwell,Group theory in physics, 3 vols., Academic Press, London, 1984–1989.
[10] C.W. Curtis,Pioneers of representation theory:Frobenius, Burnside, Schur, and Brauer,
American Mathematical Society, Providence, R.I., 1999.
[11] C.W. Curtis and I. Reiner,Methods of representation theory, 2 vols., Wiley, New York,
1990.
[12] H. Davenport,Multiplicative number theory, 3rd ed. revised by H.L. Montgomery,
Springer-Verlag, New York, 2000.
[13] L.E. Dickson,History of the theory of numbers, 3 vols., reprinted Chelsea, New York,
1966.
[14] H. Dym and H.P. McKean,Fourier series and integrals, Academic Press, Orlando, FL,
1972.