Lie Algebras of Finite and Affine Type

Описание

Preface xv
spaces with respect to a Cartan subalgebra. The well-known example of the
special linear Lie algebra is used to illustrate the general ideas. In Chapter 5
the Weyl group is introduced and shown to be a Coxeter group. This leads on
to the definition of the Cartan matrix and the Dynkin diagram. The possible
Dynkin diagrams and Cartan matrices are classified in Chapter 6, and in
Chapter 7 the existence and uniqueness of a semisimple Lie algebra with a
given Cartan matrix are proved. In Chapter 8 the finite dimensional simple
Lie algebras are discussed individually and their root systems determined.
Chapters 9 to 13 are concerned with the representation theory of finite
dimensional semisimple Lie algebras. We begin in Chapter 9 with the introduction of the universal enveloping algebra, of free Lie algebras and of Lie
algebras defined by generators and relations. The finite dimensional irreducible modules for semisimple Lie algebras are obtained in Chapter 10 as
quotients of infinite dimensional Verma modules with dominant integral highest weight. In Chapter 11 the enveloping algebra is studied in more detail. Its
centre is shown to be isomorphic to the algebra of polynomial functions on a
Cartan subalgebra invariant under the Weyl group, and to the algebra of polynomial functions on the Lie algebra invariant under the adjoint group. This
algebra is shown to be isomorphic to a polynomial algebra. The properties of
the Casimir element of the centre of the enveloping algebra are also discussed.
These are important in subsequent applications to representation theory. Characters of modules are introduced in Chapter 12, and Weyl’s character formula
for the irreducible modules is proved. The fundamental irreducible modules
for the finite dimensional simple Lie algebras are discussed individually in
Chapter 13. Their discussion involves exterior powers of modules, Clifford
algebras and spin modules, and contraction maps.
This concludes the development of the structure and representation theory
of the finite dimensional Lie algebras. This development has concentrated
particularly on the properties necessary to obtain the classification of the
simple Lie algebras and their finite dimensional irreducible modules. Among
the significant results omitted from our account are Ado’s theorem on the
existence of a faithful finite dimensional module, the radical splitting theorem
of Levi, the theorem of Malcev and Harish-Chandra on the conjugacy of
complements to the radical, and the cohomology theory of Lie algebras.
The theory of Kac–Moody algebras is introduced in Chapter 14, where the
Kac–Moody algebra associated to a generalised Cartan matrix is defined. In
fact there are two slightly different definitions of a Kac–Moody algebra which
have been used. There is a definition in terms of generators and relations
which appears the more natural, but there is a different definition, given by
Kac in his book, which is more convenient when one wishes to show that a

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