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9781470451745 Academic Inspection Copy

Rings with Polynomial Identities and Finite Dimensional Representationsof Algebras

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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.
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