Algebra VI

Dieser Band der Encyclopaedia enthält zwei Beiträge, einen zu kombinatorischen und asymptotischen Methoden in der Algebra und einen über nichtassoziative Strukturen. Beide sind Übersichtsartikel, die sich an einen breiteren Leserkreis als den der Spezialisten wenden. Obwohl die Themen der Algebra zuzurechnen sind, haben sie auch Beziehungen zur Topologie, Differentialgeometrie, Computeralgebra und Quantenphysik. Für Forscher und Studenten im Hauptstudium in der Mathematik, der Informatik und der theoretischen Physik wird das Buch eine wertvolle Unterstützung bei Ihrer Arbeit sein.

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This book contains two contributions: "Combinatorial and Asymptotic Methods in Algebra" by V.A. Ufnarovskij is a survey of various combinatorial methods in infinite-dimensional algebras, widely interpreted to contain homological algebra and vigorously developing computer algebra, and narrowly interpreted as the study of algebraic objects defined by generators and their relations. The author shows how objects like words, graphs and automata provide valuable information in asymptotic studies. The main methods emply the notions of Gröbner bases, generating functions, growth and those of homological algebra. Treated are also problems of relationships between different series, such as Hilbert, Poincare and Poincare-Betti series. Hyperbolic and quantum groups are also discussed. The reader does not need much of background material for he can find definitions and simple properties of the defined notions introduced along the way. "Non-Associative Structures" by E.N.Kuz'min and I.P.Shestakov surveys the modern state of the theory of non-associative structures that are nearly associative. Jordan, alternative, Malcev, and quasigroup algebras are discussed as well as applications of these structures in various areas of mathematics and primarily their relationship with the associative algebras. Quasigroups and loops are treated too. The survey is self-contained and complete with references to proofs in the literature. The book will be of great interest to graduate students and researchers in mathematics, computer science and theoretical physics.