Multilinear Algebra 1st Edition by Douglas Geoffrey Northcott ISBN 978-0521262699 0521262690
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Product details:
ISBN 10: 0521262690
ISBN 13: 978-0521262699
Author: Douglas Geoffrey Northcott
Multilinear algebra has important applications in many different areas of mathematics but is usually learned in a rather haphazard fashion. The aim of this book is to provide a readable and systematic account of multilinear algebra at a level suitable for graduate students. Professor Northcott gives a thorough treatment of topics such as tensor, exterior, Grassmann, Hopf and co-algebras and ends each chapter with a section entitled ‘Comments and Exercises’. The comments contain convenient summaries and discussion of the content whilst the exercises provide an opportunity to test understanding and add extra material. Complete solutions are provided for those exercises that are particularly important or used later in the book. The volume as a whole is based on advanced lectures given by the author at the University of Sheffield.
Table of contents:
1 Multilinear mappings
General remarks
1.1 Multilinear mappings
1.2 The tensor notation
1.3 Tensor powers of a module
1.4 Alternating multilinear mappings
1.5 Symmetric multilinear mappings
1.6 Comments and exercises
1.7 Solutions to selected exercises
2 Some properties of tensor products
General remarks
2.1 Basic isomorphisms
2.2 Tensor products of homomorphisms
2.3 Tensor products and direct sums
2.4 Additional structure
2.5 Covariant extension
2.6 Comments and exercises
2.7 Solutions to selected exercises
3 Associative algebras
General remarks
3.1 Basic definitions
3.2 Tensor products of algebras
3.3 Graded algebras
3.4 A modified graded tensor product
3.5 Anticommutative algebras
3.6 Covariant extension of an algebra
3.7 Derivations and skew derivations
3.8 Comments and exercises
3.9 Solutions to selected exercises
4 The tensor algebra of a module
General remarks
4.1 The tensor algebra
4.2 Functorial properties
4.3 The tensor algebra of a free module
4.4 Covariant extension of a tensor algebra
4.5 Derivations and skew derivations on a tensor algebra
4.6 Comments and exercises
4.7 Solutions to selected exercises
5 The exterior algebra of a module General remarks
5.1 The exterior algebra
5.2 Functorial properties
5.3 The exterior algebra of a free module
5.4 The exterior algebra of a direct sum
5.5 Covariant extension of an exterior algebra
5.6 Skew derivations on an exterior algebra
5.7 Pfaffians
5.8 Comments and exercises
5.9 Solutions to selected exercises
6 The symmetric algebra of a module General remarks
6.1 The symmetric algebra
6.2 Functorial properties
6.3 The symmetric algebra of a free module
6.4 The symmetric algebra of a direct sum
6.5 Covariant extension of a symmetric algebra
6.6 Derivations on a symmetric algebra
6.7 Differential operators
6.8 Comments and exercises
7 Coalgebras and Hopf algebras General remarks
7.1 A fresh look at algebras
7.2 Coalgebras
7.3 Graded coalgebras
7.4 Tensor products of coalgebras
7.5 Modified tensor products of coalgebras
7.6 Commutative and skew-commutative coalgebras
7.7 Linear forms on a coalgebra
7.8 Hopf algebras
7.9 Tensor products of Hopf algebras
7.10 E(M) as a (modified) Hopf algebra
7.11 The Grassmann algebra of a module
7.12 S(M) as a Hopf algebra
7.13 Comments and exercises
7.14 Solutions to selected exercises
8 Graded duality
General remarks
8.1 Modules of linear forms
8.2 The graded dual of a graded module
8.3 Graded duals of algebras and coalgebras
8.4 Graded duals of Hopf algebras
8.5 Comments and exercises
8.6 Solutions to selected exercises
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Tags: Douglas Geoffrey Northcott, Multilinear Algebra