Ebook: The Classical Groups and K-Theory

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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The book gives a comprehensive account of the basic algebraic properties of the classical groups over rings. Much of the theory appears in book form for the first time, and most proofs are given in detail. The book also includes a revised and expanded version of Dieudonn?'s classical theory over division rings. The authors analyse congruence subgroups, normal subgroups and quotient groups, they describe their isomorphisms and investigate connections with linear and hermitian K-theory. A first insight is offered through the simplest case of the general linear group. All the other classical groups, notably the symplectic, unitary and orthogonal groups, are dealt with uniformly as isometry groups of generalized quadratic modules. New results on the unitary Steinberg groups, the associated K2-groups and the unitary symbols in these groups lead to simplified presentation theorems for the classical groups. Related material such as the K-theory exact sequences of Bass and Sharpe and the Merkurjev-Suslin theorem is outlined. From the foreword by J. Dieudonn?: "All mathematicians interested in classical groups should be grateful to these two outstanding investigators for having brought together old and new results (many of them their own) into a superbly organized whole. I am confident that their book will remain for a long time the standard reference in the theory."

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The book gives a comprehensive account of the basic algebraic properties of the classical groups over rings. Much of the theory appears in book form for the first time, and most proofs are given in detail. The book also includes a revised and expanded version of Dieudonn?'s classical theory over division rings. The authors analyse congruence subgroups, normal subgroups and quotient groups, they describe their isomorphisms and investigate connections with linear and hermitian K-theory. A first insight is offered through the simplest case of the general linear group. All the other classical groups, notably the symplectic, unitary and orthogonal groups, are dealt with uniformly as isometry groups of generalized quadratic modules. New results on the unitary Steinberg groups, the associated K2-groups and the unitary symbols in these groups lead to simplified presentation theorems for the classical groups. Related material such as the K-theory exact sequences of Bass and Sharpe and the Merkurjev-Suslin theorem is outlined. From the foreword by J. Dieudonn?: "All mathematicians interested in classical groups should be grateful to these two outstanding investigators for having brought together old and new results (many of them their own) into a superbly organized whole. I am confident that their book will remain for a long time the standard reference in the theory."
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-2
Notation and Conventions....Pages 3-4
General Linear Groups, Steinberg Groups, and K-Groups....Pages 5-67
Linear Groups over Division Rings....Pages 68-95
Isomorphism Theory for the Linear Groups....Pages 96-138
Linear Groups over General Classes of Rings....Pages 139-182
Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups....Pages 183-291
Unitary Groups over Division Rings....Pages 292-380
Clifford Algebras and Orthogonal Groups over Commutative Rings....Pages 381-440
Isomorphism Theory for the Unitary Groups....Pages 441-507
Unitary Groups over General Classes of Form Rings....Pages 508-542
Concluding Remarks....Pages 543-544
Back Matter....Pages 545-578

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The book gives a comprehensive account of the basic algebraic properties of the classical groups over rings. Much of the theory appears in book form for the first time, and most proofs are given in detail. The book also includes a revised and expanded version of Dieudonn?'s classical theory over division rings. The authors analyse congruence subgroups, normal subgroups and quotient groups, they describe their isomorphisms and investigate connections with linear and hermitian K-theory. A first insight is offered through the simplest case of the general linear group. All the other classical groups, notably the symplectic, unitary and orthogonal groups, are dealt with uniformly as isometry groups of generalized quadratic modules. New results on the unitary Steinberg groups, the associated K2-groups and the unitary symbols in these groups lead to simplified presentation theorems for the classical groups. Related material such as the K-theory exact sequences of Bass and Sharpe and the Merkurjev-Suslin theorem is outlined. From the foreword by J. Dieudonn?: "All mathematicians interested in classical groups should be grateful to these two outstanding investigators for having brought together old and new results (many of them their own) into a superbly organized whole. I am confident that their book will remain for a long time the standard reference in the theory."
Content:
Front Matter....Pages i-xv
Introduction....Pages 1-2
Notation and Conventions....Pages 3-4
General Linear Groups, Steinberg Groups, and K-Groups....Pages 5-67
Linear Groups over Division Rings....Pages 68-95
Isomorphism Theory for the Linear Groups....Pages 96-138
Linear Groups over General Classes of Rings....Pages 139-182
Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups....Pages 183-291
Unitary Groups over Division Rings....Pages 292-380
Clifford Algebras and Orthogonal Groups over Commutative Rings....Pages 381-440
Isomorphism Theory for the Unitary Groups....Pages 441-507
Unitary Groups over General Classes of Form Rings....Pages 508-542
Concluding Remarks....Pages 543-544
Back Matter....Pages 545-578
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