Mutations of Alternative Algebras

Around 1978, a mutation of associative algebras was introduced to generalize the formalism of classical mechanics as well as quantum mechanics.
This volume presents the first book devoted to a self-contained and detailed treatment of the mathematical theory of mutation algebras, which is based on research in this subject over the past fifteen years. The book also deals with a broader class of algebras, mutations of alternative algebras, which are a natural generalization of mutations of associative algebras. A complete structure theory, including automorphisms, derivations and certain representations, is given for mutations of artinian alternative algebras, and, in particular, of Cayley--Dickson algebras. Since the mutation algebras do not form a variety, the structure theory explored in this volume takes quite a different approach from the standard theory of nonassociative algebras and provides an important interplay with the theory of noncommutative (associative) algebras through mutation parameters. New simple algebras and open problems presented in this book will stimulate additional research and applications in the area.
This book will be valuable to graduate students, mathematicians and physicists interested in applications of algebras.


Inhalt
I Nonassociative Algebras.- 1. Definitions and basic results.- 2. Simple and central simple algebras.- 3. Alternative algebras.- 4. Malcev-admissible algebras.- 5. Jordan-admissible algebras.- II Mutations of Associative Algebras.- 1. Introduction.- 2. Identities in mutation algebras.- 3. Simple and prime mutation algebras.- 4. Mutations of simple associative algebras.- 5. Radicals of mutation algebras.- III Mutations of Alternative Algebras.- 1. Introduction.- 2. Basic results.- 3. Identities.- 4. Mutations of Cayley-Dickson algebras.- 5. Simple and prime mutations of alternative algebras.- 6. Mutations of artinian alternative algebras.- IV Automorphisms and Derivations.- 1. Introduction.- 2. Jordan homomorphisms and derivations.- 3. Automorphisms of mutations of associative algebras.- 4. Derivations of mutations of associative algebras.- 5. Composition algebras: The associative case.- 6. Composition algebras: The octonion case.- V The Mutation Variety and Strict Representations.- 1. Basic concepts.- 2. Identities for the mutation variety.- 3. Additional properties of the mutation variety.- 4. Strict representations of mutation algebras.- References.- Symbol Index.

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