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Classical Groups and Geometric Algebra
Larry C. Grove, University of Arizona, Tuscon, AZ
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2002; 169 pp; hardcover
Volume: 39
ISBN-10: 0-8218-2019-2
ISBN-13: 978-0-8218-2019-3
List Price: US$44 Member Price: US$35.20
Order Code: GSM/39

"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.

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

Reviews

"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

• Permutation actions
• The basic linear groups
• Bilinear forms
• Symplectic groups
• Symmetric forms and quadratic forms
• Orthogonal geometry (char $$F\not= 2$$)
• Orthogonal groups (char $$F \not= 2$$), I
• $$O(V)$$, $$V$$ Euclidean
• Clifford algebras (char $$F \not = 2$$)
• Orthogonal groups (char $$F \not = 2$$), II
• Hermitian forms and unitary spaces
• Unitary groups
• Orthogonal geometry (char $$F = 2$$)
• Clifford algebras (char $$F = 2$$)
• Orthogonal groups (char $$F = 2$$)
• Further developments
• Bibliography
• List of notation
• Index