Handbook of Algebra (eBook)

M. Hazewinkel (Herausgeber)

eBook Download: EPUB

2006 | 1. Auflage
542 Seiten
Elsevier Science (Verlag)
978-0-08-046249-3 (ISBN)

Infos

Algebra, as we know it today, consists of many different ideas, concepts and results. A reasonable estimate of the number of these different items would be somewhere between 50,000 and 200,000. Many of these have been named and many more could (and perhaps should) have a name or a convenient designation. Even the nonspecialist is likely to encounter most of these, either somewhere in the literature, disguised as a definition or a theorem or to hear about them and feel the need for more information. If this happens, one should be able to find enough information in this Handbook to judge if it is worthwhile to pursue the quest.

In addition to the primary information given in the Handbook, there are references to relevant articles, books or lecture notes to help the reader. An excellent index has been included which is extensive and not limited to definitions, theorems etc.

The Handbook of Algebra will publish articles as they are received and thus the reader will find in this third volume articles from twelve different sections. The advantages of this scheme are two-fold: accepted articles will be published quickly and the outline of the Handbook can be allowed to evolve as the various volumes are published.

A particularly important function of the Handbook is to provide professional mathematicians working in an area other than their own with sufficient information on the topic in question if and when it is needed.

- Thorough and practical source for information
- Provides in-depth coverage of new topics in algebra
- Includes references to relevant articles, books and lecture notes

Inhalt

Cover 
1 
Contents 22
Preface 6
Basic philosophy 6
Planning 7
The individual chapters 7
The present 8
The future 8
Outline of the Series 10
Philosophy and principles of the Handbook of Algebra 10
Section 1. Linear algebra. Fields. Algebraic number theory 11
Section 2. Category theory. Homological and homotopical algebra. Methods from logic 12
Section 3. Commutative and associative rings and algebras 13
Section 4. Other algebraic structures. Nonassociative rings and algebras. Commutative and associative algebras with extra structure 15
Section 5. Groups and semigroups 16
Section 6. Representation and invariant theory 18
Section 7. Machine computation. Algorithms. Tables 19
Section 8. Applied algebra 20
Section 9. History of algebra 20
List of Contributors 24
Section 2C. Algebraic K-theory 26
Higher Algebraic K-Theory 28
Introduction 30
Simplicial objects, classifying spaces, and spectra 32
Definitions of and relations between several higher algebraic K-theories (for rings) 36
Higher K-theory of exact, symmetric monoidal and Waldhausen categories 40
Some fundamental results and exact sequences in higher K-theory 49
Higher K-theory and connections to Galois, étale and motivic cohomology theories 59
Higher K-theory of rings of integers in local and global fields 69
Higher K-theory of orders, group-rings and modules over `EI'-categories 80
Equivariant higher algebraic K-theory together with relative generalizations 90
Acknowledgement 95
References 95
Section 3B. Associative Rings and Algebras 100
Filter Dimension 102
Introduction 104
Filter dimension of algebras and modules 105
The first filter inequality 108
Krull, Gelfand-Kirillov and filter dimensions of simple finitely generated algebras 110
Filter dimension of the ring of differential operators on a smooth irreducible affine algebraic variety (proof of Theorem 4.3) 114
Multiplicity for the filter dimension, holonomic modules over simple finitely generated algebras 116
Filter dimension and commutative subalgebras of simple finitely generated algebras and their division algebras 122
Filter dimension and isotropic subalgebras of Poisson algebras 127
References 130
Section 4E. Lie Algebras 132
Gelfand-Tsetlin Bases for Classical Lie Algebras 134
Introduction 136
Gelfand-Tsetlin bases for representations of gln 141
Weight bases for representations of oN and sp2n 162
Gelfand-Tsetlin bases for representations of oN 183
Acknowledgements 189
References 189
Section 4H. Rings and Algebras with Additional Structure 196
Hopf Algebras 198
Introduction 200
Part 1. Basic concepts 202
Coalgebras and comodules 202
Hopf algebras 207
Modules and comodules for Hopf algebras 210
Normal Hopf subalgebras, quotients and extensions 212
Special Hopf algebras 213
Twisting in Hopf algebras 216
Constructing new Hopf algebras from known ones 217
Part 2. Fundamental theorems 218
The fundamental theorem for Hopf modules 218
Integrals 219
Maschke's theorem 220
The antipode 221
The Nichols-Zoeller theorem 221
Kac-Zhu theorem 222
Part 3. Actions and coactions 222
Smash products, crossed products and invariants 222
Galois extensions 226
Duality theorems 229
Analogues of two theorems of E. Noether inner and outer actions
Part 4. Categories of representations of Hopf algebras 233
Rigid tensor categories and Hopf algebras 233
The FRT construction 237
Yetter-Drinfeld categories and the Drinfeld double 238
Hopf algebras in braided categories, biproducts and bosonizations 243
Part 5. Structure theory for special classes of Hopf algebras 246
Semisimple Hopf algebras 246
Pointed Hopf algebras 252
References 256
Difference Algebra 266
Introduction 268
Basic concepts of difference algebra 269
Difference modules 293
Difference field extensions 304
Difference Galois theory 339
References 355
Section 5A. Groups and Semigroups 360
Reflection Groups 362
Finite groups generated by reflections 365
Real reflection groups 374
Braid groups 387
Representation theory 393
Hints for further reading 398
References 402
Hurwitz Groups and Hurwitz Generation 410
Introduction 412
Triangle groups 413
Finite simple and quasi-simple groups which are Hurwitz 415
Low-dimensional representations of Hurwitz groups 422
Related results 435
Acknowledgements 440
Appendix 440
References 449
Braids, their Properties and Generalizations 452
Introduction 454
Historical remarks 454
Definitions and general properties 456
Garside normal form, center and conjugacy problem 463
Ordering of braids 466
Representations 467
Generalizations of braids 468
Homological properties 477
Connections with the other domains 480
Acknowledgements 483
References 483
Groups with Finiteness Conditions 492
Finiteness conditions, examples 494
Layer-finite groups 496
Groups with layer-finite periodic part 499
Generalizations of layer-finite groups 501
Layer-Chernikov groups 502
Generalized Chernikov groups 504
T0-groups 506
Phi-groups 508
Almost layer-finite groups 510
Periodic groups with minimality condition 511
Groups with finitely embedded involution 513
Frobenius groups and finiteness conditions 515
References 516
Subject Index 520

Textauszug

(iii) Derived categories. Let be an exact category and b(C) the (bounded) homotopy category of , i.e. the stable category of b(C) (see [63]). So, (Hb(C))=Chb(C) and morphisms are homotopy classes of bounded complexes. Let (C) be the full subcategory of b(C) consisting of acyclic complexes (see [63]). The derived category b(C) of is defined by b(C)=Hb(C)/A(C). A morphism of complexes in b(C) is called a quasi-isomorphism if its image in b(C) is an isomorphism. We could also define the unbounded derived category (C) from unbounded complexes (C).
Note that there exists a faithful embedding of in an Abelian category such that ⊂A is closed under extensions and the exact functor →A reflects exact sequences. So, a complex in (C) is acyclic iff its image in (A) is acyclic. In particular, a morphism in (C) is a quasi-isomorphism iff its image in (A) is a quasi-isomorphism. Hence, the derived category (C) is the category obtained from (C) formally inverting quasi-isomorphisms.

(iv) Stable derived categories and Waldhausen categories. Now let =M′(R). A complex M. in ′(R) is said to be compact if the functor Hom(M., –) commutes with arbitrary set-valued coproducts. Let (R)¯ denote the full subcategory of (M′(R)) consisting of compact objects. Then we have (R)¯⊂Db(M′(R))⊂D(M′(R)).
Define the stable derived category of bounded complexes ¯b(M′(R)) as the quotient category of b(M′(R)) with respect to (R)¯. A morphism of complexes in b(M′(R)) is called a stable quasi-isomorphism if its image in ¯b(M′(R)) is an isomorphism. The family of stable quasi-isomorphisms in =Chb(M′(R)) is denoted A.

(v) Theorem.

(1) (Chb(M′(R))) forms a set of weak equivalences and satisfies the saturation and extension axioms.

(2) b(M′(R)) together with the family of stable quasi-isomorphisms is a Waldhausen category.

4 Some fundamental results and exact sequences in higher K-theory

4.1 Resolution theorem

4.1.1. Resolution theorem for exact categories [114]. Let ⊂H be full exact subcategories of an Abelian category , both closed under extensions and inheriting their exact structure from . Suppose that (1) every object M of has a finite -resolution and (2) is closed under kernels in , i.e. if L → M N is an exact sequence in with ,N∈P, then L is also in. Then nP≅KnH for all n ≥ 0.

4.1.2. Remarks and examples.

(i) Let R be a regular Noetherian ring. Then by taking =M(R), =P(R) in 4.1.1, we have Kn(R) Gn(R) for all n ≥ 0.

(ii) Let R be any ring with identity and (R) the category of all R-modules having finite homological dimension (i.e. having a finite resolution by finitely generated projective R-modules), ¯s(R) the subcategory of modules in (R) having resolutions of length ≤ s. Then by 4.1.1 applied to (R)⊆H¯s(R)⊆H¯(R) we have n(R)≅Kn(H¯(R))≅Kn(H¯s(R)) for all s ≥ 1.

(iii) Let T = {Ti} be an exact connected sequence of functors from an exact category to an Abelian category, i.e. given an exact sequence 0 → M′ → M ′→ M″ → 0 in there exists a long exact sequence … → T2M″ → T1M′ → T1M →. Let be the full subcategory of T-acyclic objects (i.e. objects M such that Tn(M) = 0 for all n ≥ 1), and assume that for each ∈C, there is a map P M such that ∈P and that TnM = 0 for n sufficiently large. Then nP≅KnC∀n≥0 (see [114]).

(iv) As an example of (iii) let A, B be a Noetherian rings, f: A → B a homomorphism, B a flat A-module, then we have a homomorphism of K-groups: (B ⊗A ?)*: Gn(A) → Gn(B) (since B ⊗A ? is exact). Let B be of finite tordimension as a right A-module. Then by applying (iii) above, to =M(A), Ti(M) = ToriA(B, M) and taking as the full subcategory of (A) consisting of M such the TiM = 0 for i > 0, we have n(P)≈Gn(A).

(v) Let be an exact category and (C) the category whose objects are pairs (M, ν) with ∈C and ν is a nilpotent endomorphism of M. Let 0⊂C be an exact subcategory of such that every object of has a finite 0-resolution. Then every object of (C) has a finite (C0) resolution and so, by 4.1.1,

n(Nil(C0))≈Kn(Nil(C)).

4.2 Additivity theorem (for exact and Waldhausen categories)

4.2.1. Let , be exact categories. A sequence of functors F′ → F F″ from to is called an exact sequence of exact functors if 0 → F′(A) → F(A) → F″(A) → 0 is an exact sequence in for every ∈A.

Let , be Waldhausen categories. If F′(A) F(A) F″(A) is a cofibration sequence in and for every cofibration A A′ in , F(A) F′(A) F′(A′) → F(A′) is a cofibration in say that F′ F F″ a short exact sequence or a cofibration sequence of exact functors.

4.2.2. Additivity theorem. Let F′ F F″ be a short exact sequence of exact functors from to where both and are either exact categories or Waldhausen categories. Then *≃F′*+F″*:Kn(A)→Kn(B).

4.2.3. Remarks and...

Produkt-Details

• Erscheint lt. Verlag: 30.5.2006
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-08-046249-9 / 0080462499
• ISBN-13: 978-0-08-046249-3 / 9780080462493