Rings with Polynomial Identities and Finite Dimensional Representations of Algebras

Cover
Title page
Preface
The plan of the book
Differences with other books
Introduction
0.1. Two classical problems
Part 1 . Foundations
Chapter 1. Noncommutative algebra
1.1. Noncommutative algebras
1.2. Semisimple modules
1.3. Finite-dimensional algebras
1.4. Noetherian rings
1.5. Localizations
1.6. Commutative algebra
Chapter 2. Universal algebra
2.1. Categories and functors
2.2. Varieties of algebras
2.3. Algebras with trace
2.4. The method of generic elements
2.5. Generalized identities
2.6. Matrices and the standard identity
Chapter 3. Symmetric functions and matrix invariants
3.1. Polarization
3.2. Symmetric functions
3.3. Matrix functions and invariants
3.4. The universal map into matrices
Chapter 4. Polynomial maps
4.1. Polynomial maps
4.2. The Schur algebra of the free algebra
Chapter 5. Azumaya algebras and irreducible representations
5.1. Irreducible representations
5.2. Faithfully flat descent
5.3. Projective modules
5.4. Separable and Azumaya algebras
Chapter 6. Tensor symmetry
6.1. Schur–Weyl duality
6.2. The symmetric group
6.3. The linear group
6.4. Characters
Part 2 . Combinatorial aspects of polynomial identities
Chapter 7. Growth
7.1. Exponential bounds
7.2. The ?⊗? theorem
7.3. Cocharacters of a PI algebra
7.4. Proper polynomials
7.5. Cocharacters are supported on a (?,ℓ) hook
7.6. Application: A theorem of Kemer
Chapter 8. Shirshov’s Height Theorem
8.1. Shirshov’s height theorem
8.2. Some applications of Shirshov’s height theorem
8.3. Gel’fand–Kirillov dimension
Chapter 9. 2×2 matrices
9.1. 2×2 matrices
9.2. Invariant ideals
9.3. The structure of generic 2×2 matrices
Part 3 . The structure theorems
Chapter 10. Matrix identities
10.1. Basic identities
10.2. Central polynomials
10.3. The theorem of M. Artin on Azumaya algebras
10.4. Universal splitting
Chapter 11. Structure theorems
11.1. Nil ideals
11.2. Semisimple and prime PI algebras
11.3. Generic matrices
11.4. Affine algebras
11.5. Representable algebras
Chapter 12. Invariants and trace identities
12.1. Invariants of matrices
12.2. Representations of algebras with trace
12.3. The alternating central polynomials
Chapter 13. Involutions and matrices
13.1. Matrices with involutions
13.2. Symplectic and orthogonal case
Chapter 14. A geometric approach
14.1. Geometric invariant theory
14.2. The universal embedding into matrices
14.3. Semisimple representations of CH algebras
14.4. Geometry of generic matrices
14.5. Using Cayley–Hamilton algebras
14.6. The unramified locus and restriction maps
Chapter 15. Spectrum and dimension
15.1. Krull dimension
15.2. A theorem of Schelter
Part 4 . The relatively free algebras
Chapter 16. The nilpotent radical
16.1. The Razmyslov–Braun–Kemer theorem
16.2. The theorem of Lewin
16.3. ?-ideals of identities of block-triangular matrices
16.4. The theorem of Bergman and Lewin
Chapter 17. Finite-dimensional and affine PI algebras
17.1. Strategy
17.2. Kemer’s theory
17.3. The trace algebra
17.4. The representability theorem, Theorem 17.1.1
17.5. The abstract Cayley–Hamilton theorem
Chapter 18. The relatively free algebras
18.1. Rationality and a canonical filtration
18.2. Complements of commutative algebra and invariant theory
18.3. Applications to PI algebras
18.4. Model algebras
Chapter 19. Identities and superalgebras
19.1. The Grassmann algebra
19.2. Superalgebras
19.3. Graded identities
19.4. The role of the Grassmann algebra
19.5. Finitely generated PI superalgebras
19.6. The trace algebra
19.7. The representability theorem, Theorem 19.7.4
19.8. Grassmann envelope and finite-dimensional superalgebras
Chapter 20. The Specht problem
20.1. Standard and Capelli
20.2. Solution of the Specht’s problem
20.3. Verbally prime ?-ideals
Chapter 21. The PI-exponent
21.1. The asymptotic formula
21.2. The exponent of an associative PI algebra
21.3. Growth of central polynomials
21.4. Beyond associative algebras
21.5. Beyond the PI exponent
Chapter 22. Codimension growth for matrices
22.1. Codimension growth for matrices
22.2. The codimension estimate for matrices
Chapter 23. Codimension growth for algebras satisfying a Capelli identity
23.1. PI algebras satisfying a Capelli identity
23.2. Special finite-dimensional algebras
Appendix A. The Golod–Shafarevich counterexamples
Bibliography
Index
Index of Symbols
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