Rings with polynomial identities and finite dimensional representations of algebras
著者
書誌事項
Rings with polynomial identities and finite dimensional representations of algebras
(Colloquium publications / American Mathematical Society, v. 66)
American Mathematical Society, c2020
- : hardcover
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注記
Other authors: Antonio Giambruno, Claudio Procesi, Amitai Regev
Includes bibliographical references (p. 605-621) and indexes
内容説明・目次
内容説明
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.
目次
Introduction
Foundations: Noncommutative algebra
Universal algebra
Symmetric functions and matrix invariants
Polynomial maps
Azumaya algebras and irreducible representations
Tensor symmetry
Combinatorial aspects of polynomial identities: Growth
Shirshov's theorem
$2\times2$ matrices
The structure theorems
Matrix identities
Structure theorems
Invariants and trace identities
Involutions and matrices
A geometric approach
Spectrum and dimension
The relatively free algebras: 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
The Golod-Shafarevich counterexamples
Bibliography
Index
Index of Symbols.
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