This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. 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 PIalgebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finitedimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of codimensions of any PIalgebra has integral exponential growth  the PIexponent of the algebra. Later chapters further apply these results to 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. The material is suitable for graduate students and research mathematicians interested in polynomial identity algebras. Readership Graduate students and research mathematicians interested in polynomial identity algebras. Reviews "Written by two of the leading experts in the theory of PIalgebras, the book is interesting and useful."  Vesselin Drensky for Zentralblatt MATH Table of Contents  Polynomial identities and PIalgebras
 \(S_n\)representations
 Group gradings and group actions
 Codimension and colength growth
 Matrix invariants and central polynomials
 The PIexponent of an algebra
 Polynomial growth and low PIexponent
 Classifying minimal varieties
 Computing the exponent of a polynomial
 \(G\)identities and \(G\wr S_n\)action
 Superalgebras, *algebras and codimension growth
 Lie algebras and nonassociative algebras
 The generalizedsixsquare theorem
 Bibliography
 Index
