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 R s [ R be a G-graded ring. We describe all types of gradings on R ifG is torsion free and R is Artinian semisimple. If R is a matrix algebra over an algebraically closed field F, then we give a description of all G-gradings on R provided that G is an abelian group. In the case of an abelian group G we also classify all finite-dimensional graded simple algebras and finite-dimensional graded division algebras over an algebraically closed field of characteristic zero. ᮊ 2001Academic Press

In this paper we describe all group gradings by a finite Abelian group G of several types of simple Jordan and Lie algebras over an algebraically closed field F of characteristic zero.  2004 Published by Elsevier Inc.

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