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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