Rings with Polynomial Identities and Finite Dimensional Representations of Algebras

Eli Aljadeff, Antonio Giambruno, Claudio Procesi, Amitai Regev (Autoren)

Buch | Softcover

630 Seiten

2020
American Mathematical Society (Verlag)
978-1-4704-5174-5 (ISBN)

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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.

Eli Aljadeff, Technion-Israel Institute of Technology, Haifa, Israel Antonio Giambruno, Universita di Palermo, Italy Claudio Procesi, Universita di Roma ""La Sapienza"", Italy Amitai Regev, The Weitzmann Institute of Science, Rehovot, Israel

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.

Erscheinungsdatum	15.01.2021
Reihe/Serie	Colloquium Publications
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	1340 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-5174-3 / 1470451743
ISBN-13	978-1-4704-5174-5 / 9781470451745
Zustand	Neuware