Preface The present volume is intended to be a text for a graduate-level course. It discusses in some detail the groups that are popularly known as the clas- sical groups, as they were named by Hermann Weyl [74]. They are groups of matrices, or (perhaps more often) quotients of matrix groups by small (typically central) normal subgroups. The story begins, as Weyl suggested, with Her All-embracing Majesty, the General Linear Group GL(V) of all invertible linear transformations of a vector space V over a (commutative) field F. All further groups discussed are either subgroups of GL(V) or closely related quotient groups. Most of the classical groups are singled out within Her All-embracing Majesty for basically geometric reasons - they consist of invertible linear transformations that respect a bilinear form having some geometric signifi- cance, e.g. a quadratic form (hence preserving "distance"), or a symplectic form, etc. Accordingly, we develop the required geometric notions, albeit from an algebraic point of view, as the end results should apply to vector spaces over more-or-less arbitrary fields, finite or infinite. It is to be hoped that the end result is consonant with the title and intent of Emil Artin's deservedly famous book Geometric Algebra [3]. In particular, we do not em- ploy Lie-theoretic techniques, important as they are from many other points of view. The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years they have played a prominent role in the classification of the finite simple groups. IX
Previous Page Next Page