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Books > Science & Mathematics > Mathematics > Algebra > Groups & group theory
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The Classical Groups and K-Theory (Paperback, Softcover reprint of hardcover 1st ed. 1989)
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The Classical Groups and K-Theory (Paperback, Softcover reprint of hardcover 1st ed. 1989)
Series: Grundlehren der mathematischen Wissenschaften, 291
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It is a great satisfaction for a mathematician to witness the
growth and expansion of a theory in which he has taken some part
during its early years. When H. Weyl coined the words "classical
groups", foremost in his mind were their connections with invariant
theory, which his famous book helped to revive. Although his
approach in that book was deliberately algebraic, his interest in
these groups directly derived from his pioneering study of the
special case in which the scalars are real or complex numbers,
where for the first time he injected Topology into Lie theory. But
ever since the definition of Lie groups, the analogy between simple
classical groups over finite fields and simple classical groups
over IR or C had been observed, even if the concept of "simplicity"
was not quite the same in both cases. With the discovery of the
exceptional simple complex Lie algebras by Killing and E. Cartan,
it was natural to look for corresponding groups over finite fields,
and already around 1900 this was done by Dickson for the
exceptional Lie algebras G and E * However, a deep reason for this
2 6 parallelism was missing, and it is only Chevalley who, in 1955
and 1961, discovered that to each complex simple Lie algebra
corresponds, by a uniform process, a group scheme (fj over the ring
Z of integers, from which, for any field K, could be derived a
group (fj(K).
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