For additional information and updates on this book, visit www.ams.org/bookpages/surv-122 Library of Congress Cataloging-in-Publication Data Giambruno, A. Polynomial identities and asymptotic methods / Antonio Giambruno, Mikhail Zaicev. p. cm.-(Mathematical surveys and monographs ; v. 122) Includes biblographical references and index. ISBN 0-8218-3829-6 (alk. paper) 1. Pi-algebras. 2. Rings (Algebra). I. Zaicev, Mikhail. II. Title. III. Mathematical surveys and monographs ; no. 122.
Let A be a PI-algebra over a field F. We study the asymptotic behavior of the sequence of codimensions c n (A) of A. We show that if A is finitely generated over F then Inv(A)=lim n Ä n -c n (A) always exists and is an integer. We also obtain the following characterization of simple algebras: A is finite dimensional central simple over F if and only if Inv(A)=dim A.
Academic Press
Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. If c G n (A), n = 1, 2, . . ., is the sequence of graded codimensions of A, we prove that exp G (A) = limn→∞ n c G n (A), the G-exponent of A, exists and is an integer. This result was proved in [1] and [9], in case G is abelian.
Let $A$ be a finite dimensional algebra over a field\ud
of characteristic zero graded by a finite abelian group $G$. Here we\ud
study a growth function related to the graded polynomial identities\ud
satisfied by $A$ by computing the exponential rate of growth of the\ud
sequence of graded codimensions of $A$. We prove that the\ud
$G$-exponent of $A$ exists and is an integer related in an explicit\ud
way to the dimension of a suitable semisimple subalgebra of $A$
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