Classical Groups and Geometric Algebra

Larry C. Grove (Autor)

Buch | Softcover

169 Seiten

2002
American Mathematical Society (Verlag)
978-1-4704-7974-9 (ISBN)

Infos

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.

Autor

Larry C. Grove, University of Arizona, Tuscon, AZ.

Inhalt

Chapters
Chapter 0. Permutation actions
Chapter 1. The basic linear groups
Chapter 2. Bilinear forms
Chapter 3. Symplectic groups
Chapter 4. Symmetric forms and quadratic forms
Chapter 5. Orthogonal geometry (char $F/ne 2$)
Chapter 6. Orthogonal groups (char $F /ne 2$), I
Chapter 7. $O(V)$, $V$ Euclidean
Chapter 8. Clifford algebras (char $F /ne 2$)
Chapter 9. Orthogonal groups (char $F /ne 2$), II
Chapter 10. Hermitian forms and unitary spaces
Chapter 11. Unitary groups
Chapter 12. Orthogonal geometry (char $F = 2$)
Chapter 13. Clifford algebras (char $F = 2$)
Chapter 14. Orthogonal groups (char $F = 2$)
Chapter 15. Further developments

Produkt-Details

• Erscheint lt. Verlag: 30.12.2002
• Reihe/Serie: Graduate Studies in Mathematics
• Verlagsort: Providence
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• ISBN-10: 1-4704-7974-5 / 1470479745
• ISBN-13: 978-1-4704-7974-9 / 9781470479749
• Zustand: Neuware