Description

Book Synopsis
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.

Table of Contents
  • 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.

Rings with Polynomial Identities and Finite

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    A Paperback by Eli Aljadeff, Antonio Giambruno, Claudio Procesi

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      Publisher: MP-AMM American Mathematical
      Publication Date: 12/30/2020 12:00:00 AM
      ISBN13: 9781470451745, 978-1470451745
      ISBN10: 1470451743
      Also in:
      Algebraic geometry

      Description

      Book Synopsis
      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.

      Table of Contents
      • 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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