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The Classical Groups and K-Theory

Alexander J. Hahn, O.Timothy O'Meara (Autoren)

Buch | Hardcover

XV, 578 Seiten

1989 | 1989
Springer Berlin (Verlag)
978-3-540-17758-6 (ISBN)

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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).

Timothy O'Meara was born on January 1928. He was educated at the University of Cape Town and completed his doctoral work under Emil Artin at Princeton University in 1953. He has served on the faculties of the University of Otago, Princeton University and the University of Notre Dame. From 1978 to 1996 he was provost of the University of Notre Dame.

Notation and Conventions.- 1. General Linear Groups, Steinberg Groups, and K-Groups.- 2. Linear Groups over Division Rings.- 3. Isomorphism Theory for the Linear Groups.- 4. Linear Groups over General Classes of Rings.- 5. Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups.- 6. Unitary Groups over Division Rings.- 7. Clifford Algebras and Orthogonal Groups over Commutative Rings.- 8. Isomorphism Theory for the Unitary Groups.- 9. Unitary Groups over General Classes of Form Rings.- Concluding Remarks.- Index of Concepts.- Index of Symbols.

Erscheint lt. Verlag	10.8.1989
Reihe/Serie	Grundlehren der mathematischen Wissenschaften
Vorwort	J. Dieudonne
Zusatzinfo	XV, 578 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	989 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebra • Clifford Algebra • Commutative Ring • Gruppe (Mathematik) • K-Theorie • K-theory
ISBN-10	3-540-17758-2 / 3540177582
ISBN-13	978-3-540-17758-6 / 9783540177586
Zustand	Neuware