# Classical Groups and Geometric Algebra

American Mathematical Soc. - Mathematics - 169 pages
''Classical groups'', named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups. Thus the story begins, as Weyl suggested, with ''Her All-embracing Majesty'', the general linear group \$GL n(V)\$ of all invertible linear transformations of a vector space \$V\$ over a field \$F\$. All further groups discussed are either subgroups of \$GL n(V)\$ or closely related quotient groups. Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops 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. 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. This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles. It is intended for graduate students who have completed standard courses in linear algebra and abstract algebra. The author, L. C. Grove, is a well-known expert who has published extensively in the subject area.

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### Contents

 Permutation Actions 1 The Basic Linear Groups 5 Bilinear Forms 13 Symplectic Groups 21 Symmetric Forms and Quadratic Forms 31 Orthogonal Geometry char F 2 39 Orthogonal Groups char F 2 I 45 OV V Euclidean 59
 Orthogonal Groups char F 2 II 75 Hermitian Forms and Unitary Spaces 85 Unitary Groups 93 Orthogonal Geometry char F 2 113 Clifford Algebras char F 2 119 Orthogonal Groups char F 2 127 Further Developments 151 List of Notation 165

 Clifford Algebras char F 2 65