Rings With Polynomial Identities and Finite Dimensional Representations of Algebras (Colloquium Publications)

A polynomial identity for an algebra (or a ring) $A$ is a polynomial in noncommutative variables that vanishes under any evaluation in $A$. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem. Cover 1 Title page 4 Preface 10 The plan of the book 11 Differences with other books 12 Introduction 14 0.1. Two classical problems 14 Part 1 . Foundations 18 Chapter 1. Noncommutative algebra 20 1.1. Noncommutative algebras 20 1.2. Semisimple modules 31 1.3. Finite-dimensional algebras 32 1.4. Noetherian rings 41 1.5. Localizations 43 1.6. Commutative algebra 45 Chapter 2. Universal algebra 50 2.1. Categories and functors 50 2.2. Varieties of algebras 59 2.3. Algebras with trace 69 2.4. The method of generic elements 74 2.5. Generalized identities 79 2.6. Matrices and the standard identity 82 Chapter 3. Symmetric functions and matrix invariants 88 3.1. Polarization 88 3.2. Symmetric functions 93 3.3. Matrix functions and invariants 99 3.4. The universal map into matrices 111 Chapter 4. Polynomial maps 122 4.1. Polynomial maps 122 4.2. The Schur algebra of the free algebra 128 Chapter 5. Azumaya algebras and irreducible representations 138 5.1. Irreducible representations 138 5.2. Faithfully flat descent 147 5.3. Projective modules 152 5.4. Separable and Azumaya algebras 159 Chapter 6. Tensor symmetry 178 6.1. Schur–Weyl duality 178 6.2. The symmetric group 182 6.3. The linear group 185 6.4. Characters 192 Part 2 . Combinatorial aspects of polynomial identities 202 Chapter 7. Growth 204 7.1. Exponential bounds 204 7.2. The A⊗B theorem 208 7.3. Cocharacters of a PI algebra 211 7.4. Proper polynomials 215 7.5. Cocharacters are supported on a (k,l) hook 218 7.6. Application: A theorem of Kemer 223 Chapter 8. Shirshov’s Height Theorem 226 8.1. Shirshov’s height theorem 226 8.2. Some applications of Shirshov’s height theorem 234 8.3. Gel’fand–Kirillov dimension 236 Chapter 9. 2×2 matrices 244 9.1. 2×2 matrices 244 9.2. Invariant ideals 259 9.3. The structure of generic 2×2 matrices 267 Part 3 . The structure theorems 276 Chapter 10. Matrix identities 278 10.1. Basic identities 278 10.2. Central polynomials 283 10.3. The theorem of M. Artin on Azumaya algebras 291 10.4. Universal splitting 295 Chapter 11. Structure theorems 300 11.1. Nil ideals 300 11.2. Semisimple and prime PI algebras 304 11.3. Generic matrices 310 11.4. Affine algebras 313 11.5. Representable algebras 315 Chapter 12. Invariants and trace identities 326 12.1. Invariants of matrices 326 12.2. Representations of algebras with trace 336 12.3. The alternating central polynomials 344 Chapter 13. Involutions and matrices 358 13.1. Matrices with involutions 358 13.2. Symplectic and orthogonal case 361 Chapter 14. A geometric approach 372 14.1. Geometric invariant theory 372 14.2. The universal embedding into matrices 381 14.3. Semisimple representations of CH algebras 382 14.4. Geometry of generic matrices 391 14.5. Using Cayley–Hamilton algebras 397 14.6. The unramified locus and restriction maps 400 Chapter 15. Spectrum and dimension 406 15.1. Krull dimension 406 15.2. A theorem of Schelter 410 Part 4 . The relatively free algebras 416 Chapter 16. The nilpotent radical 418 16.1. The Razmyslov–Braun–Kemer theorem 418 16.2. The theorem of Lewin 426 16.3. T-ideals of identities of block-triangular matrices 430 16.4. The theorem of Bergman and Lewin 433 Chapter 17. Finite-dimensional and affine PI algebras 442 17.1. Strategy 442 17.2. Kemer’s theory 444 17.3. The trace algebra 460 17.4. The representability theorem, Theorem 17.1.1 467 17.5. The abstract Cayley–Hamilton theorem 470 Chapter 18. The relatively free algebras 478 18.1. Rationality and a canonical filtration 478 18.2. Complements of commutative algebra and invariant theory 487 18.3. Applications to PI algebras 494 18.4. Model algebras 496 Chapter 19. Identities and superalgebras 500 19.1. The Grassmann algebra 500 19.2. Superalgebras 506 19.3. Graded identities 514 19.4. The role of the Grassmann algebra 517 19.5. Finitely generated PI superalgebras 525 19.6. The trace algebra 533 19.7. The representability theorem, Theorem 19.7.4 537 19.8. Grassmann envelope and finite-dimensional superalgebras 540 Chapter 20. The Specht problem 542 20.1. Standard and Capelli 542 20.2. Solution of the Specht’s problem 544 20.3. Verbally prime T-ideals 546 Chapter 21. The PI-exponent 554 21.1. The asymptotic formula 554 21.2. The exponent of an associative PI algebra 558 21.3. Growth of central polynomials 562 21.4. Beyond associative algebras 563 21.5. Beyond the PI exponent 566 Chapter 22. Codimension growth for matrices 568 22.1. Codimension growth for matrices 568 22.2. The codimension estimate for matrices 573 Chapter 23. Codimension growth for algebras satisfying a Capelli identity 586 23.1. PI algebras satisfying a Capelli identity 586 23.2. Special finite-dimensional algebras 606 Appendix A. The Golod–Shafarevich counterexamples 610 Bibliography 618 Index 636 Index of Symbols 642 Back Cover 645 Noncommutative algebra -- Universal algebra -- Symmetric functions and matrix invariants -- Polynomial maps -- Azumaya algebras and irreducible representations -- Tensor symmetry -- Growth -- Shirshov's theorem -- 2 x 2 matrices -- Matrix identities -- Structure theorems -- Invariants and trace identities -- Involutions and matrices -- A geometric approach -- Spectrum and dimension -- The nilpotent radical -- Finite dimensional and affine PI algebras -- The relatively free algebras -- Identities and superalgebras -- The Specht problem -- The PI-exponent -- Codimension growth for matrices -- Codimension growth for algebras satisfying a Capelli identity