Classical groups and geometric algebra

Larry C. Grove0821820192, 9780821820193

“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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Table of contents :
Contents……Page 6
Preface……Page 8
0. Permutation Actions……Page 10
1. The Basic Linear Groups……Page 14
2. Bilinear Forms……Page 22
3. Symplectic Groups……Page 29
4. Symmetric Forms and Quadratic Forms……Page 39
5. Orthogonal Geometry (char F neq 2)……Page 47
6. Orthogonal Groups (char F neq 2), I……Page 52
7. O(V), V Euclidean……Page 66
8. Clifford Algebras (char F neq 2)……Page 71
9. Orthogonal Groups (char F neq 2), II……Page 80
10. Hermitian Forms and Unitary Spaces……Page 90
11. Unitary Groups……Page 98
12. Orthogonal Geometry (char F = 2)……Page 117
13. Clifford Algebras (char F = 2)……Page 123
14. Orthogonal Groups (char F = 2)……Page 130
15. Further Developments……Page 154
Bibliography……Page 163
List of Notation……Page 166
Index……Page 168