Algebra VI: Combinatorial and Asymptotic Methods of Algebra. Non-Associative Structures (Encyclopaedia of Mathematical Sciences (57))

This monograph contains two self-contained surveys of key aspects of algebra, complete with definitions and simple properties and 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. "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 Grobner 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."--BOOK JACKET "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 employ the notions of Grobner 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."--Jacket From the fields, commutative rings and groups studied in university mathematics courses, through Lie groups and algebras to category theory, this text shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. A.i. Kostrikin, I.r. Shafarevich, (eds.). Translation Of: Algebra 7, Issued In Serial: Itogi Nauki I Tekhniki. Serii͡a Sovremennye Problemy Matematiki. Fundamentalʹnye Napravlenii͡a. Includes Bibliographical References And Indexes.