Nonassociative algebras in physics.

*(English)*Zbl 0840.17001
Hadronic Press Monographs in Mathematics. Palm Harbor, FL: Hadronic Press, Inc. xiv, 271 p. (1994).

The monograph under review is mainly devoted to trying to convince the reader of the importance of some nonassociative algebras in physics. As written by the authors in the introductory remarks to Chapter 4: “…, attempts have been made to use nonassociative (non-Lie) algebras, especially the octonion algebra, in the theory of relativistic-invariant equations and some other topics … . The uniqueness and exceptional properties of octonions also seem to have very important consequences for the physical world. We think that this highly exceptional mathematical system, where many different structures of mathematics are overlapping, serves as the foundation of highly determined, perhaps the only possible physical world …”.

Therefore, the monograph is devoted to survey and present results, without proofs but with many references provided, about the application to physics of some algebraic systems closely related to the algebra of octonions (Cayley-Dickson algebras, sedenions, Moufang loops and Malcev algebras, …), stressing the latest contributions of the authors.

The monograph is divided in six chapters. The first one is a review of some nonassociative algebras and their applications to physics. Chapter 2 is devoted to deformations of algebras. Chapter 3 presents the Cayley-Dickson algebras (over the reals), together with the concept of representations and the algebras of sedenions. Chapter 4 deals with an approach to the Dirac equation and self-dual Yang-Mills equations using octonions and sedenions and representations of Clifford algebras associated to them. The aim of Chapter 5 is to present Moufang analytical loops, which are an extension of Lie groups, and their tangent algebras, which are Malcev algebras, and to the actions of these by means of a concept of birepresentation which does not coincide with the usual notion of representation for algebras in a variety. Finally, Chapter 6 provides some conjectures on the potential future applications of nonassociative algebras in physics.

Therefore, the monograph is devoted to survey and present results, without proofs but with many references provided, about the application to physics of some algebraic systems closely related to the algebra of octonions (Cayley-Dickson algebras, sedenions, Moufang loops and Malcev algebras, …), stressing the latest contributions of the authors.

The monograph is divided in six chapters. The first one is a review of some nonassociative algebras and their applications to physics. Chapter 2 is devoted to deformations of algebras. Chapter 3 presents the Cayley-Dickson algebras (over the reals), together with the concept of representations and the algebras of sedenions. Chapter 4 deals with an approach to the Dirac equation and self-dual Yang-Mills equations using octonions and sedenions and representations of Clifford algebras associated to them. The aim of Chapter 5 is to present Moufang analytical loops, which are an extension of Lie groups, and their tangent algebras, which are Malcev algebras, and to the actions of these by means of a concept of birepresentation which does not coincide with the usual notion of representation for algebras in a variety. Finally, Chapter 6 provides some conjectures on the potential future applications of nonassociative algebras in physics.

Reviewer: Alberto Elduque (Zaragoza)

##### MSC:

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17A01 | General theory of nonassociative rings and algebras |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

17D10 | Mal’tsev rings and algebras |

17A35 | Nonassociative division algebras |

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |