This book is a revised version of the first edition and is intended as a sequel and companion volume to the fourth edition of Linear Algebra (Graduate Texts in Mathematics 23).
As before, the terminology and basic results of Linear Algebra are frequently used without reference. In particular, the reader should be familiar with Chapters 1-5 and the first part of Chapter 6 of that book, although other sections are occasionally used.
In this new version of Multilinear Algebra, Chapters 1-5 remain essentially unchanged from the previous edition. Chapter 6 has been completely rewritten and split into three (Chapters 6, 7, and 8). Some of the proofs have been simplified and a substantial amount of new material has been added. This applies particularly to the study of characteristic coefficients and the Pffoffian.
The old Chapter 7 remains as it stood, except that it is now Chapter 9.
The old Chapter 8 has been suppressed and the material which it contained (multilinear functions) has been relocated at the end of Chapters 3, 5, and 9.
The last two chapters on Clifford algebras and their representations are completely new. In view of the growing importance of Clifford algebras
and the relatively few references available, it was felt that these chapters would be useful to both mathematicians and physicists.
In Chapter 10 Clifford algebras are introduced via universal properties and treated in a fashion analogous to exterior algebra. After the basic isomorphism theorems for these algebras (over an arbitrary inner product space) have been established the chapter proceeds to a discussion of finite-dimensional Clifford algebras. The treatment culminates in the complete classification of Clifford algebras over finite-dimensional complex and real inner product spaces.
The book concludes with Chapter 11 on representations of Clifford algebras. The twisted adjoint representation which leads to the definition of the spin-groups is an important example. A version of Wedderburn's theorem is the key to the classification of all representations of the Clifford algebra over an 8-dimensional real vector space with a negative definite inner product. The results are applied in the last section of this chapter to study orthogonal multiplications between Euclidean spaces and the existence of orthonormal frames on the sphere. In particular, it is shown that the (n -1)-sphere admits an orthonormal k-frame where k is the Radon-Hurwitz number corresponding to n. A deep theorem of F. Adams states that this result can not be improved.
The problems at the end of Chapter 11 include a basis-free definition of the Cayley algebra via the complex cross-product analogous to the definition of quaternions in Section 7.23 of the fourth edition of Linear Algebra.
Finally, the Cayley multiplication is used to obtain concrete forms of some of the isomorphisms in the table at the end of Chapter 10.
Author(s): Werner Greub
Series: Universitext
Edition: 2nd
Publisher: Springer
Year: 1978
Language: English
Pages: C, viii+294, B
Cover
Universitext
Multiinear Algebra, 2nd Edition
Copyright
© 1967 by Springer-Verlag Berlin Heidelberg
© 1978 by Springer-Verlag New York Inc.
ISBN-13:978-0-387-90284-5
e-ISBN-13:978-1-4613-9425-9
DO!: 10.1007/ 978-1-4613-9425-9
QA 184.G74 1978 512'.5
LCCN 78-949
Preface
Table of Contents
1 Tensor Products
Multilinear Mappings
1.1. Bilinear Mappings
1.2. Bilinear Mappings of Subspaces and Factor Spaces
1.3. Multilinear Mappings
PROBLEMS
The Tensor Product
1.4. The Universal Property
1.5. Elementary Properties
1.6. Uniqueness
PROBLEMS
Subspaces and Factor Spaces
1.9. Tensor Products of Subspaces
1.10. Tensor Product of Factor Spaces
Direct Decompositions
1.11. Tensor Product of Direct Sums
1.12. Direct Decompositions
1.13. Tensor Product of Basis Vectors
1.14. Application to Bilinear Mappings
1.15. Intersection of Tensor Products
PROBLEMS
Linear Maps
1.16. Tensor Product of Linear Maps
1.17. Example
1.18. Compositions
1.19. Image Space and Kernel
PROBLEMS
Tensor Product of Several Vector Spaces
1.20. The Universal Property
Dual Spaces
1.21. Bilinear Mappings
1.22. Bilinear Functions
1.23. Dual Mappings
1.24. Example
1.25. Inner Product Spaces
1.26. The Composition Algebra
PROBLEMS
Finite-Dimensional Vector Spaces
1.27.
1.28. The Isomorphism T
1.29 The Algebra of Linear Transformations
1.30. The Endomorphisms of A
PROBLEMS
2 Tensor Products of Vector Spaces with Additional Structure
Tensor Products of Algebras
2.1. The Structure Map
2.2. The Canonical Tensor Product of Algebras
2.3. Tensor Product of Homomorphisms
2.4. Antiderivations
Tensor Products of G-Graded Vector Spaces
2.5. Poincare Series
2.6. Tensor Products of Several G-Graded Vector Spaces
2.7. Dual G-Graded Spaces
2.8. Anticommutative Tensor Products of Graded Algebras
2.9. Homomorphisms and Antiderivations
Tensor Products of Differential Spaces
2.10. Tensor Products of Differential Spaces
2.11. Tensor Products of Dual Differential Spaces
2.12. Graded Differential Spaces
2.13. Dual Graded Differential Spaces
PROBLEM
Tensor Products of Differential Algebras
2.14. The Structure Map of the Homology Algebra
2.15. Tensor Products of Differential Algebras
2.16. The Algebra H(A) p H(B)
2.17. Graded Differential Algebras
3 Tensor Algebra
Tensors
3.1.
3.2. Tensor Algebra
3.3. The Universal Property of ®E
3.4. Universal Pairs
3.5. The Induced Homomorphism
3.6. The Derivation Induced by a Linear Transformation
3.7. Tensor Algebra Over a G-Graded Vector Space
PROBLEMS
Tensors Over a Pair of Dual Spaces
3.8.
3.9. The Induced Homomorphism
3.10. The Induced Derivation
Mixed Tensors
3.11.
3.12. The Mixed Tensor Algebra
3.13. Contraction
3.14. Tensorial Maps
PROBLEMS
Tensor Algebra Over an Inner Product Space
3.15. The Induced Inner Product
3.16. The Isomorphism -r®
3.17. The Metric Tensors
PROBLEMS
The Algebra of Multilinear Functions
3.18. The Algebra T(E)
3.20. The Isomorphism (x)E* - T"(E)
3.21. The Algebra T.(E)
3.22. The Duality Between T"(E) and T(E)
3.23. The Algebra T(E)
4 Skew-Symmetry and Symmetry in the Tensor Algebra
Skew-Symmetric Tensors
4.1. The Space Np(E)
4.2. The Alternator
4.3. Dual Spaces
4.4. The Skew-Symmetric Part of a Product
The Factor Algebra ®E/N(E)
4.5. The Ideal N(E)
4.6. The Algebra ®E/N(E)
4.7. Skew-Symmetric Tensors
4.8. The Induced Scalar Product
PROBLEM
Symmetric Tensors
4.9. The Space Mp(E)
4.10. The Symmetrizer
4.11. Dual Spaces
4.12. The Symmetric Part of a Product
PROBLEMS
The Factor Algebra ®E/M(E)
4.13. The Ideal M(E)
4.14. The Algebra QE/M(E)
4.15. Symmetric Tensors
4.16. The Induced Scalar Product
5 Exterior Algebra
Skew-Symmetric Mappings
5.1. Skew-Symmetric Mappings
PROBLEMS
Exterior Algebra
5.2. The Universal Property
5.3. Uniqueness and Existence
5.4. Exterior Algebra
5.5. The Universal Property of A E
5.6. Exterior Algebra Over Dual Spaces
5.7. Exterior Algebra Over a Vector Space of Finite Dimension
PROBLEMS
Homomorphisms, Derivations and Antiderivations
5.8. The Induced Homomorphism
5.9. Dual Mappings
5.10. The Induced Derivation
5.11. Antiderivations
5.12. a-Antiderivations
PROBLEMS
The Operator i(a)
5.13.
5.14. The Operator i(h)
PROBLEMS
Exterior Algebra Over a Direct Sum
5.15.
5.16. Direct Sums of Linear Maps
5.17. Derivations
5.18. Direct Sums of Dual Spaces
5.19. The Diagonal Mapping
5.20. Direct Sums of Several Vector Spaces
5.21. Exterior Algebra Over a Graded Vector Space
PROBLEMS
Ideals in AE
5.22. Graded Ideals
5.23. Direct Decompositions
5.24. Linear Maps
5.25. Invertible Elements, Maximum and Minimum Ideals
5.26. The Annihilator
5.27.
PROBLEMS
Ideals and Duality
5.28.
PROBLEMS
The Algebra of Skew-Symmetric Functions
5.29. Skew-Symmetric Functions
5.30. The Algebra A (E)
5.31. Homomorphisms and Derivations
5.32. The Operator iA(h)
5.33. The Isomorphism A E* 4 T' (E)
5.34. The Algebra A,(E)
6 Mixed Exterior Algebra
The Algebra \Lambda (E*, E)
6.1. Skew-Symmetric Maps of Type (p, q)
6.2. The Algebra A (E*, E)
6.3. The Box Product of Linear Transformations
The composition Product
6.4
6.5
Poincare Duality
6.6. The Isomorphism TE
6.7. The Unit Tensors
6.8. The Poincare Isomorphism
6.9. Naturality
6.10. The Isomorphism DE
6.11. Naturality
6.12. The Intersection Product
6.13. The Duals of the Basis Elements
6.14. The External Product
6.15. Euclidean Spaces
PROBLEMS
Applications to Linear Transformations
6.16. The Isomorphism T
The Skew Tensor Product of n E* and A E
6.17. The Algebra A E* p n E
6.18. The Inner Product in E* © E
7 Applications to Linear Transformations
The Isomorphism DL
7.1. Definition
7.2. The Determinant
7.3. The Adjoint Tensor
7.4. The Classical Adjoint Transformation
Characteristic Coefficients
7.5. Definition
7.6. The Linear Transformations Ap(p)
7.7. The Trace Coefficients
7.8. Application to the Elementary Symmetric Functions
7.9. Complex Vector Spaces
PROBLEMS
8 Skew and Skew-Hermitian Transformations
The Characteristic Coefficients of a Skew Linear Transformation
8.1. Definition
8.2. The Isomorphisms \Phi_E and \Psi_E
8.3. The Isomorphism i
The Pfaffian of a Skew Linear Transformation
8.4. The Pfaffian
8.5. Direct Sums
8.6. Euclidean Spaces
Skew-Hermitian Transformations
8.7. The Isomorphisms \delta and \tau
9 Symmetric Tensor Algebra
Symmetric Tensor Algebra
9.1. Symmetric Mappings
9.2. The Universal Property
9.3. Symmetric Algebra
9.4. Symmetric Algebras Over Dual Spaces
9.5. Homomorphisms and Derivations
9.6. The Operator i(a)
9.7. Zero Divisors
9.8. Symmetric Algebra Over a Direct Sum
9.9. Symmetric Tensor Algebras Over a Graded Space
9.10. Symmetric Algebra Over a Vector Space of Finite Dimension
9.11. Poincare Series
9.12. Homogeneous Functions
PROBLEMS
Polynomial Algebras
9.13. Polynomial Algebras
9.14.
PROBLEMS
The Algebra of Symmetric Functions
9.15. Symmetric Functions
9.16. The Operator is(h)
9.17. The Algebra S,(E)
9.18. Homogeneous Functions and Homogeneous Polynomials
10 Clifford Algebras
Basic Properties
10.1. The Universal Property
10.2. Examples
10.3. Uniqueness and Existence
10.4. The Injectivity of iE
10.5. Homomorphisms
10.6. The Z2-Gradation of CE
10.7. Direct Decompositions
10.8. The Involution SE
PROBLEM
Clifford Algebras Over a Finite-Dimensional Space
10.9
10.10. The Canonical Element eo
10.11. Center and Anticenter
10.12. The Algebra C _ E
10.13. The Canonical Tensor Product of Clifford Algebras
10.14. The Direct Sum of Dual Spaces
PROBLEMS
Clifford Algebras over a Complex Vector Space
10.15. Clifford Algebras Over Complex Vector Spaces
10.16. Complexification of Real Clifford Algebras
Clifford Algebras Over a Real Vector Space
10.17
10.18. Inner Product Spaces With Signature Zero
10.19. Clifford Algebras of Low Dimensions
10.20
10.21. The Algebras Cn(+) and Cn(-)
10.22. The Algebras C(p, q)
PROBLEMS
11 Representations of Clifford Algebras
Basic Concepts
11.1. Representations of an Algebra
11.2. Representations of a Clifford Algebra
11.3. Orthogonal Representations
The Twisted Adjoint Representation
11.4
11.5. The Clifford Group
11.6. The Map \lambda E
11.7. The Homomorphism \Phi_E: \Gamma_E --> O(E)
11.8. The Group Pin
11.9. The Group Spin
PROBLEMS
The Spin Representation
11.10
11.11. The Spin Representation
11.12. The Hermitian Inner Product in A E 1
11.13. The Half Spin Representations
The Wedderburn Theorems
11.14. Invariant Linear Maps
11.15. The Isomorphism R
11.16
11.17. The Isomorphism BR
11.18
11.19
Representations of Ck( - )
11.20. The Radon-Hurwitz Number
11.21
11.22. Orthogonal Multiplications
11.23. Orthogonal Systems of Skew Transformations
11.24. Orthogonal Multiplications and Representations of Ck
11.25. Orthonormal k-Frames on S"-1
PROBLEMS
Index
Back Cover
