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Computational Aspects of Polynomial Identities: Volume l, Kemer's
Theorems, 2nd Edition presents the underlying ideas in recent
polynomial identity (PI)-theory and demonstrates the validity of
the proofs of PI-theorems. This edition gives all the details
involved in Kemer's proof of Specht's conjecture for affine
PI-algebras in characteristic 0. The book first discusses the
theory needed for Kemer's proof, including the featured role of
Grassmann algebra and the translation to superalgebras. The authors
develop Kemer polynomials for arbitrary varieties as tools for
proving diverse theorems. They also lay the groundwork for
analogous theorems that have recently been proved for Lie algebras
and alternative algebras. They then describe counterexamples to
Specht's conjecture in characteristic p as well as the underlying
theory. The book also covers Noetherian PI-algebras,
Poincare-Hilbert series, Gelfand-Kirillov dimension, the
combinatoric theory of affine PI-algebras, and homogeneous
identities in terms of the representation theory of the general
linear group GL. Through the theory of Kemer polynomials, this
edition shows that the techniques of finite dimensional algebras
are available for all affine PI-algebras. It also emphasizes the
Grassmann algebra as a recurring theme, including in Rosset's proof
of the Amitsur-Levitzki theorem, a simple example of a finitely
based T-ideal, the link between algebras and superalgebras, and a
test algebra for counterexamples in characteristic p.
A comprehensive study of the main research done in polynomial
identities over the last 25 years, including Kemer's solution to
the Specht problem in characteristic O and examples in the
characteristic p situation. The authors also cover codimension
theory, starting with Regev's theorem and continuing through the
Giambruno-Zaicev exponential rank. The "best" proofs of classical
results, such as the existence of central polynomials, the tensor
product theorem, the nilpotence of the radical of an affine
PI-algebra, Shirshov's theorem, and characterization of group
algebras with PI, are presented.
Some of the important advances in Polynomial Identity (PI) theory
in the last twenty years have remained accessible only to experts,
limiting the exposure of advanced aspects of PI-theory to the
general mathematical community. This book's main objective is to
describe these breakthroughs in full, starting with Shirshov's
theorem, discussing Kemer's solution of Specht's conjecture in
characteristic zero, and completing with a proof of the theorem.
The authors detail the theory needed for this proof in the early
chapters of the book. Later chapters discuss related topics such as
counterexamples to Specht's conjecture in characteristic p,
Noetherian PI-algebras, Poincare-Hilbert series, Gelfand-Kirillov
dimension, the combinatoric theory of affine PI-algebras, the
ideals of identities, multilinear identities in terms of
representation theory, and trace identities.
Computational Aspects of Polynomial Identities: Volume l, Kemer's
Theorems, 2nd Edition presents the underlying ideas in recent
polynomial identity (PI)-theory and demonstrates the validity of
the proofs of PI-theorems. This edition gives all the details
involved in Kemer's proof of Specht's conjecture for affine
PI-algebras in characteristic 0. The book first discusses the
theory needed for Kemer's proof, including the featured role of
Grassmann algebra and the translation to superalgebras. The authors
develop Kemer polynomials for arbitrary varieties as tools for
proving diverse theorems. They also lay the groundwork for
analogous theorems that have recently been proved for Lie algebras
and alternative algebras. They then describe counterexamples to
Specht's conjecture in characteristic p as well as the underlying
theory. The book also covers Noetherian PI-algebras,
Poincare-Hilbert series, Gelfand-Kirillov dimension, the
combinatoric theory of affine PI-algebras, and homogeneous
identities in terms of the representation theory of the general
linear group GL. Through the theory of Kemer polynomials, this
edition shows that the techniques of finite dimensional algebras
are available for all affine PI-algebras. It also emphasizes the
Grassmann algebra as a recurring theme, including in Rosset's proof
of the Amitsur-Levitzki theorem, a simple example of a finitely
based T-ideal, the link between algebras and superalgebras, and a
test algebra for counterexamples in characteristic p.
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