## Polynomial Identities and Asymptotic MethodsOne of the main objectives of this book is to show how one can combine methods of ring theory, combinatorics, and representation theory of groups with an analytical approach in order to study the polynomial identities satisfied by a given algebra. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory. A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras that occur naturally in mathematics. The core of the book is the proof that the sequence of codimensions of any PI-algebra has integral exponential growth--the PI-exponent of the algebra. Later chapters apply these results to further subjects, such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent. Results are extended to graded algebras and algebras with involution. The book concludes with a study of the numerical invariants and their asymptotics in the class of Lie algebras. Even in algebras that are close to being associative, the behavior of the sequences of codimensions can be wild. |

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### Contents

Polynomial Identities and PIAlgebras | 1 |

5nRepresentations | 43 |

Group Gradings and Group Actions | 61 |

Codimension and Colength Growth | 87 |

Matrix Invariants and Central Polynomials | 119 |

The PiExponent of an Algebra | 143 |

Polynomial Growth and Low Piexponent | 165 |

Classifying Minimal Varieties | 193 |

Computing the Exponent of a Polynomial | 215 |

GIdentities and GI 5nAction | 255 |

Superalgebras Algebras and Codimension Growth | 283 |

Lie Algebras and Nonassociative Algebras | 307 |

Appendix A The GeneralizedSixSquare Theorem | 333 |

Bibliography | 341 |

Index | 349 |

### Common terms and phrases

5n-module algebra with involution algebraically closed field associative algebra asymptotically Capelli identity central polynomial characteristic zero cn(A codimensions compute Corollary corresponding decomposition define DEFINITION denote elements exists exponent exponential F-algebra f(xi field F finite dimensional algebra finite dimensional superalgebra follows free algebra G-graded G-identity graded identity Grassmann algebra Grassmann envelope group algebra Hence homogeneous homomorphism hook ideal idempotent integer involution irreducible isomorphic Jacobson radical k x k Lemma Let f Lie algebra linear combination Math matrices matrix algebra minimal superalgebra Mk(F module monomial multilinear polynomial nilpotent non-trivial non-zero nth cocharacter obtain partition permutation Pi-algebra polynomial growth polynomial identity polynomially bounded Proposition prove Recall Regev relatively free algebra ring sequence simple superalgebra space of multilinear standard identity subalgebra subspace Suppose symmetric group T-ideal variables verbally prime T-ideals write Young diagram Young tableau Z2-grading Zaicev