Alexei Kanel-Belov, Yakov Karasik, Louis Halle Rowen

Chapman and Hall/CRC

Published
October 23, 2015

Reference
- 418 Pages

ISBN 9781498720083 - CAT# K25525

Series: Chapman & Hall/CRC Monographs and Research Notes in Mathematics

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- Presents a tighter formulation of Zubrilin’s theory
- Contains a more direct proof of the Wehrfritz–Beidar theorem
- Adds more details to the proof of Kemer’s difficult PI-representability theorem
- Develops several newer techniques, such as the "pumping procedure"

**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, Poincaré–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*.