From Rings and Modules to Hopf Algebras : One Flew Over the Algebraist's Nest

This textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups. Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory — whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld–Lusztig double of a group algebra, appearing for the first time in a textbook at this level. Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course — with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level. Preface Contents Chapter 1 Prerequisites and Preliminaries 1.1 Prerequisites 1.1.1 Groups: π-components of elements 1.1.2 Nilpotent groups 1.1.3 Complements on Sylow subgroups 1.1.4 Solvability and the Schur–Zassenhaus Theorem 1.2 Preliminary: the Language of Categories 1.2.1 What is a category? 1.2.2 First examples 1.2.3 Monomorphisms, epimorphisms 1.2.4 Functors 1.2.5 Examples 1.2.6 Yet another example: presheaves 1.2.7 Faithful and full functors 1.2.8 Morphisms of functors 1.2.9 Isomorphisms of functors 1.2.10 Morphisms of functors: Yoneda’s Lemma 1.2.11 Universals 1.2.12 Adjoint functors and adjunctions 1.2.13 Equivalences of categories 1.2.14 Complements: Going on and on 1.2.15 Horizontal composition Part I Rings and Modules Chapter 2 Rings, Polynomials, Divisibility 2.1 Rings, Morphisms, Modules 2.1.1 Morphisms 2.1.2 Subrings 2.1.3 Endomorphisms of abelian groups and modules 2.2 Polynomials and Power Series 2.2.1 Generalities 2.2.2 Infinite sums and products 2.2.3 Derivatives 2.2.4 Logarithm and exponential Definition 2.2.14 Proposition 2.2.15 Proposition 2.2.16 Remark 2.2.17 2.2.5 Euclidean division Proposition 2.2.18 (Euclidean division by a monic polynomial) Remarks 2.2.19 Corollary 2.2.20 Corollary 2.2.21 Exercise 2.2.22 Remark 2.2.23 Theorem 2.2.24 (Cayley–Hamilton) Lemma 2.2.25 Exercise 2.2.26 2.3 Canonical Morphisms 2.3.1 Prime ring and characteristic 2.3.2 Universal property of the polynomial ring 2.4 Ideals 2.4.1 Left, right, two-sided ideals 2.4.2 Ideals and morphisms 2.4.3 Chinese Remainder Theorem 2.5 Factorial Domains, Principal Ideal Domains, Euclidean Domains 2.5.1 Divisors and irreducible elements 2.5.2 Factorial domains Definition 2.5.4 Remark 2.5.5 Lemma 2.5.6 (Gauß’ Lemma) Example 2.5.7 Proposition-Definition 2.5.8 Lemma 2.5.9 2.5.3 Principal ideal domains Definition 2.5.10 Proposition 2.5.11 Exercise 2.5.12 Lemma 2.5.13 Remark 2.5.14 2.5.4 Euclidean rings Definition and first properties Definition 2.5.15 Examples 2.5.16 Remark 2.5.17 Exercise 2.5.18 Proposition 2.5.19 Lemma 2.5.20 Remark 2.5.21 Complement without proofs: quadratic extensions of Theorem 2.5.22 Remark 2.5.23 Theorem 2.5.24 About Euclidean rings: Euclid’s algorithm 2.5.5 Case of and application Remark 2.5.25 Theorem 2.5.26 Example 2.5.27 Exercise 2.5.28 2.6 Roots of Unity, Cyclotomic Polynomials 2.7 More Exercises Chapter 3 Polynomial Rings in Several Indeterminates 3.1 Universal Property, Substitutions 3.1.1 First particular case 3.1.2 Second particular case: evaluation function 3.1.3 Third particular case: substitution 3.1.4 Fourth particular case: specialization 3.2 Symmetric Polynomials 3.2.1 Definition and fundamental theorem 3.2.2 Newton formulae 3.2.3 Symmetric fractions 3.2.4 Antisymmetric polynomials 3.3 Resultant and Discriminant 3.3.1 Resultant of two polynomials 3.3.2 First properties 3.3.3 Resultant and roots 3.3.4 A geometric application 3.3.5 Discriminant 3.4 More Exercises Chapter 4 More on Modules 4.1 Several Equivalent Definitions 4.1.1 Two definitions of “module” 4.1.2 Morphisms 4.2 Submodules 4.2.1 Generalities 4.2.2 Direct sums 4.2.3 Quotients 4.2.4 Kernels, images, cokernels, coimages 4.2.5 Exact sequences 4.2.6 Ideals and modules 4.3 Torsion Elements, Torsion Submodule 4.3.1 Cyclic modules 4.3.2 Torsion and torsion free elements 4.4 Free and Generating Systems, Free Modules 4.4.1 Free systems, generating systems, bases 4.4.2 A property of free modules 4.4.3 Projective modules 4.5 Sums and Products 4.5.1 Direct sums (coproducts) and products 4.5.2 Split exact sequences 4.6 R-Linear and Abelian R-Linear Categories 4.6.1 Initial, terminal, null objects 4.6.2 R-linear categories 4.6.3 R-linear functors 4.6.4 An example: stable category 4.6.5 Kernels and cokernels 4.6.6 Canonical decomposition of a morphism 4.6.7 A bunch of definitions for abelian categories 4.6.8 Grothendieck group 4.6.9 Functors between abelian R-linear categories 4.7 More Exercises 4.8 Tensor Products 4.8.1 Definition of the tensor product 4.8.2 Functoriality and other properties of the tensor product 4.8.3 Exact sequences, Hom R and ⊗R 4.8.4 Tensor product and duality 4.8.5 Extension of scalars 4.8.6 Extending scalars for an algebra 4.8.7 Trace and restriction of scalars 4.8.8 Complement: Kronecker product of matrices 4.9 Tensor, Symmetric and Exterior Algebras 4.9.1 Symmetric and alternating squares 4.9.2 Tensor algebra 4.9.3 Symmetric algebra 4.9.4 Exterior algebra 4.10 More on Algebras 4.10.1 Generalities about algebras 4.10.2 Left and right modules 4.10.3 Tensor product of left with right modules 4.10.4 Tensor product and bimodules Exercise 4.10.14 Proposition 4.10.15 4.10.5 A famous adjunction Proposition 4.10.16 mod Exercises 4.10.17 mod 4.11 Modules Over a Matrix Algebra 4.11.1 An equivalence of categories 4.11.2 An application: the Skolem–Noether theorem 4.12 More Exercises Chapter 5 On Representations of Finite Groups 5.1 Generalities on Representations 5.1.1 Introduction 5.1.2 Representations on a category 5.2 Set-Representations 5.2.1 Union and product 5.2.2 Transitive representations 5.2.3 Classification of transitive representations 5.2.4 Burnside’s marks 5.2.5 Induction and restriction 5.2.6 Generalized transfer 5.3 Linear Representations 5.3.1 Generalities 5.3.2 The group algebra 5.3.3 Induction and restriction for finite group algebras 5.3.3.1 Restriction 5.3.3.2 Induction Definition 5.3.3.3 Universal property of induction 5.3.3.4 Induction and tensor product 5.3.3.5 Another definition of induction 5.3.4 Mackey’s formula 5.3.5 Trace on induced modules 5.3.6 Generalized tensor induction 5.3.7 Complement: fixed and cofixed points 5.4 Projective Representations, Twisted Group Algebras 5.4.1 Preliminary: fragments on cohomology 5.4.2 Projective representations, k×-groups, twisted group algebras 5.4.3 Above a stable module for a normal subgroup 5.5 More Exercises Part II Integral Domains, Polynomials, Fields Chapter 6 Prime and Maximal Ideals, Integral Domains 6.1 Definition and First Examples 6.2 Examples in Polynomial Rings 6.2.1 Generalities 6.2.2 Example of maximal ideals of Z[X] 6.3 Nilradical and Radical 6.3.1 Characterizations 6.3.2 Local rings 6.3.3 Finite-dimensional algebras over a field 6.4 Integral Domains, Fields of Fractions 6.4.1 Construction of field of fractions 6.4.2 Universal property of the field of fractions 6.5 Localizations 6.5.1 Localizations on rings 6.5.2 Localizations on modules 6.5.3 Local properties of modules 6.5.4 On localization and projectivity 6.6 Irreducibility Criteria in R[X] 6.6.1 Primitive and irreducible polynomials 6.6.2 Reduction modulo a prime ideal 6.6.3 Case of R[X] for R factorial 6.6.4 Content and primitive part 6.6.5 Example–Exercise: the decimal numbers 6.6.6 An application: automorphisms of k(X) 6.6.7 Eisenstein criterion 6.6.8 More on irreducible elements in k(X 6.7 Transfer Properties 6.7.1 Transfer of some properties to polynomial rings 6.7.2 Yet another proof of the Cayley–Hamilton theorem 6.8 More Exercises Chapter 7 Fields, Division Rings 7.1 Finite Subgroups of the Multiplicative Group of a Field 7.2 Algebraic Extensions 7.2.1 First properties 7.2.2 Algebraic closure 7.3 Splitting Polynomials, Normal Extensions 7.4 Separable Polynomials, Separable Extensions 7.5 Norm and Traces for Normal Separable Extensions 7.6 Short Introduction to Galois Theory 7.6.1 Quick overview 7.6.2 The Galois group as a permutation group 7.6.3 The generic equation 7.7 Finite Fields 7.8 Quaternions 7.8.1 Rings of quaternions 7.8.2 The quaternion group of order 8 7.9 More Exercises Part III Finitely Generated Modules Chapter 8 Integrality, Noetherianity 8.1 Integrality Over a Ring 8.1.1 Definition and characterization 8.1.2 Integral extensions 8.1.3 Integrality and localization 8.1.4 Integral closure and field extensions 8.2 Complement: Jacobson Rings, Hilbert’s Nullstellensatz 8.2.1 On maximal ideals of polynomial algebras 8.2.2 Application to algebraic varieties 8.3 Noetherian Rings and Modules 8.3.1 Noetherian modules 8.3.2 Noetherian rings 8.3.3 Hilbert’s Basis Theorem 8.3.4 Localization over Noetherian rings 8.3.5 More exercises Chapter 9 Finitely Generated Projective Modules 9.1 Rank and Basis of a Finitely Generated Free Module 9.1.1 Rank: another proof 9.1.2 The dual of a free module of finite rank 9.1.3 About finitely generated torsion-free modules 9.2 Finitely Generated Projective Modules 9.2.1 Characterization, dual 9.2.2 Projective morphisms 9.2.3 A series of characterizations 9.2.4 The case of local rings 9.3 More Exercises Chapter 10 Finitely Generated Modules Over Dedekind Domains 10.1 Fractional Modules, Dedekind Domains 10.1.1 Projective modules and fractional modules 10.1.2 Dedekind domains: definition 10.1.3 Arithmetic with ideals in Dedekind domains 10.2 Finitely Generated Projective Modules Over Dedekind Domains 10.3 Remarkable Decompositions of Finitely Generated Torsion Modules 10.3.1 The p-primary decomposition and applications 10.3.2 On quotients of fractional ideals 10.3.3 The main theorems about torsion modules 10.3.4 Proofs of the main theorems 10.3.5 Completely reducible modules 10.4 Adapted Pseudo-Bases 10.4.1 The main theorem 10.4.2 On characteristic ideals of finitely generated torsion modules 10.5 Applications to Abelian Groups and Endomorphisms of Vector Spaces 10.5.1 Invariants 10.5.2 Completely reducible endomorphisms 10.6 More Exercises Chapter 11 Complement on Dedekind Domains 11.1 Characterizations of Dedekind Domains 11.2 Rings of Integers of Number Fields are Dedekind 11.2.1 On integral closures of Noetherian rings 11.2.2 Integral closures of Dedekind domains 11.3 More Exercises Part IV Characteristic Zero Linear Representations of Finite Groups Chapter 12 Monoidal Categories: An Introduction 12.1 First Definitions and Examples 12.1.1 The case of k vect: duality 12.1.2 Right and left duals 12.2 Monoidal Functors 12.3 Braided Categories 12.3.1 Definition, braid relation 12.3.2 Actions of braids 12.3.2.1 Action of braid groups Chapter 13 Characteristic 0 Representations 13.1 The Main Tool: 1|G|∑g∈G g 13.1.1 Fixed points 13.1.2 Maschke’s Theorem 13.1.3 Spectrum 13.2 Characters 13.2.1 First properties 13.2.2 Orthogonality relations and first applications 13.2.3 Splitting fields 13.2.4 From permutations to characters 13.3 Structure of the Group Algebra 13.3.1 A product of endomorphism algebras 13.3.2 Canonical decomposition of a representation 13.4 Group Determinant 13.5 Center and Action of Cyclotomic Galois Groups 13.5.1 About centers 13.5.2 Extending and restricting scalars 13.5.3 Action of (Z/eGZ)× and of Galois groups of cyclotomic extensions 13.6 More on a Splitting Field and First Applications 13.6.1 Character tables 13.6.2 Products of groups 13.6.3 Abelian groups 13.7 Some Arithmetical Properties of Characters 13.7.1 Characters and integrality 13.7.1.1 About representations and normal subgroups 13.7.2 Applications: two theorems of Burnside 13.8 More Exercises Chapter 14 PlayingWith the Base Field 14.1 Analysis of an Irreducible Module 14.1.1 An example: the quaternion group Q8 14.1.2 Scalars of an irreducible module 14.1.3 Schur indices 14.2 Complements on Rationality 14.2.1 Group of characters and rationality 14.2.2 Reflections and rationality 14.2.3 Questions of rationality over R 14.3 More Exercises Chapter 15 Induction and Restriction: Some Applications to Finite Groups 15.1 Induction and Normal Subgroups 15.1.1 Clifford theory 15.1.1.1 Nothing happens above the inertial group 15.1.1.2 From H to the inertial group 15.1.2 Some applications to nilpotent groups 15.2 Induction and Restriction for Class Functions 15.2.1 Formulas 15.2.2 A result of Camille Jordan 15.2.3 Application of induction to an example 15.2.4 Tensor inducing class functions 15.3 More Exercises Chapter 16 Brauer’s Theorem and Some Applications 16.1 Artin’s Theorem 16.2 Brauer’s Characterization of Characters 16.2.1 Statement and first consequences 16.2.2 Proof of Brauer’s Theorem 16.3 Fusion and Isometries 16.3.1 π-elements and class functions 16.3.2 π-Control subgroups 16.4 Some Fundamental Theorems about Finite Groups 16.4.1 Existence of a normal π-complement 16.4.2 π-trivial intersection subgroups 16.4.3 Frobenius groups 16.5 More Exercises Chapter 17 Graded Representations and Characters 17.1 Graded Vector Spaces, Algebras, Modules 17.1.1 Graded vector spaces 17.1.2 Graded algebras and modules 17.1.3 Nakayama’s Lemma 17.1.4 Free graded modules 17.1.5 Polynomial algebras and Noether parameters 17.2 Graded Characters of Graded kG-Modules 17.2.1 Notation and definitions 17.2.2 Isotypic components of the symmetric algebra 17.2.3 Computations with power series 17.2.4 A (very) simple example 17.2.5 Complement: reflection groups 17.3 More Exercises Chapter 18 The Drinfeld–Lusztig Double of a Group Algebra 18.1.1 The semidirect product of kG and its dual 18.1.2 Center of DkG 18.1.3 DkG as a symmetric algebra 18.1.4 A description of DkG mod 18.1.5 A category equivalent to DkG mod as an abelian category 18.1.6 Description of X(s, S) and its character 18.1 The Drinfeld–Lusztig Double as an Algebra: Generalities 18.2 The Drinfeld–Lusztig Double Algebra in Characteristic Zero 18.2.1 Structure of the Drinfeld–Lusztig double algebra 18.2.2 Idempotents and characters 18.3 Hopf Algebras: An Introduction from Scratch 18.3.1 Notation for multiple tensor products 18.3.2 Algebras, coalgebras, bialgebras, Hopf algebras 18.3.3 On algebra representations of a Hopf algebra 18.4 More Structure on the Drinfeld–Lusztig Double 18.4.1 Universal R-matrix for DkG 18.4.2 Generalized cocommutativity and twist on DkG mod 18.4.3 “Concrete” description of DkG mod 18.5 Action of GL2 (Z/eGZ) 18.5.1 The automorphisms S, Ω, Δn 18.5.2 Action of GL2 (Z/eGZ) 18.5.3 The Verlinde formula 18.6 More Exercises References Index