**Graduate Studies in Mathematics**

Volume: 39;
2002;
169 pp;
Hardcover

MSC: Primary 20; 11;
Secondary 51

Print ISBN: 978-0-8218-2019-3

Product Code: GSM/39

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-2091-8
Product Code: GSM/39.E**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

# Classical Groups and Geometric Algebra

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*Larry C. Grove*

“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.

#### Readership

Graduate students and research mathematicians interested in algebraic geometry, group theory, and generalizations.

#### Reviews & Endorsements

Textbook for an in-depth course … provides a nice discussion of various further topics in the study of classical groups and Chevalley groups. … the text would be great for a class or for students learning the material on their own. The topics are covered in a clean tight fashion with appropriate examples given where possible.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Classical Groups and Geometric Algebra

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Chapter 0. Permutation Actions 112 free
- Chapter 1. The Basic Linear Groups 516
- Chapter 2. Bilinear Forms 1324
- Chapter 3. Symplectic Groups 2132
- Chapter 4. Symmetric Forms and Quadratic Forms 3142
- Chapter 5. Orthogonal Geometry (char F ≠ 2) 3950
- Chapter 6. Orthogonal Groups (char F ≠ 2), I 4556
- Chapter 7. O(V), V Euclidean 5970
- Chapter 8. Clifford Algebras (char F ≠ 2) 6576
- Chapter 9. Orthogonal Groups (char F ≠ 2), II 7586
- Chapter 10. Hermitian Forms and Unitary Spaces 8596
- Chapter 11. Unitary Groups 93104
- Chapter 12. Orthogonal Geometry (char F = 2) 113124
- Chapter 13. Clifford Algebras (char F = 2) 119130
- Chapter 14. Orthogonal Groups (char F = 2) 127138
- Chapter 15. Further Developments 151162
- Bibliography 161172
- List of Notation 165176
- Index 167178
- Back Cover Back Cover1181