# The Book Of Involutions (colloquium Publications)

by Alexander Merkurjev /
1998 / English / PDF

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This monograph is an exposition of the theory of central simple
algebras with involution, in relation to linear algebraic groups.
It provides the algebra-theoretic foundations for much of the
recent work on linear algebraic groups over arbitrary fields.
Involutions are viewed as twisted forms of (hermitian) quadrics,
leading to new developments on the model of the algebraic theory of
quadratic forms. In addition to classical groups, phenomena related
to triality are also discussed, as well as groups of type $F_4$ or
$G_2$ arising from exceptional Jordan or composition algebras.
Several results and notions appear here for the first time, notably
the discriminant algebra of an algebra with unitary involution and
the algebra-theoretic counterpart to linear groups of type $D_4$.
This volume also contains a Bibliography and Index. Features:
original material not in print elsewhere a comprehensive discussion
of algebra-theoretic and group-theoretic aspects extensive notes
that give historical perspective and a survey on the literature
rational methods that allow possible generalization to more general
base rings

This monograph is an exposition of the theory of central simple
algebras with involution, in relation to linear algebraic groups.
It provides the algebra-theoretic foundations for much of the
recent work on linear algebraic groups over arbitrary fields.
Involutions are viewed as twisted forms of (hermitian) quadrics,
leading to new developments on the model of the algebraic theory of
quadratic forms. In addition to classical groups, phenomena related
to triality are also discussed, as well as groups of type $F_4$ or
$G_2$ arising from exceptional Jordan or composition algebras.
Several results and notions appear here for the first time, notably
the discriminant algebra of an algebra with unitary involution and
the algebra-theoretic counterpart to linear groups of type $D_4$.
This volume also contains a Bibliography and Index. Features:
original material not in print elsewhere a comprehensive discussion
of algebra-theoretic and group-theoretic aspects extensive notes
that give historical perspective and a survey on the literature
rational methods that allow possible generalization to more general
base rings