ABSTRACT
One of the earliest surveys on nonassociative algebras is the article by Shirshov [Shi58] that introduced the phrase “rings that are nearly associative.” The first book in the English language devoted to a systematic study of nonassociative algebras is Schafer [Sch66]. A comprehensive exposition of the work of the Russian school is Zhevlakov, Slinko, Shestakov, and Shirshov [ZSS82]. A collection of open research problems in algebra, including many problems on nonassociative algebra, is the Dniester Notebook [FKS93]; the survey article by Kuzmin and Shestakov [KS95] is from the same period. Three books on Jordan algebras that contain substantial material on general nonassociative algebras are Braun and Koecher [BK66], Jacobson [Jac68], and McCrimmon [McC04]. Recent research appears in the Proceedings of the International Conferences on Nonassociative Algebra and Its Applications [Gon94], [CGG00], [SSS06]. The present chapter provides very limited information on Lie algebras, since they are the subject of Chapter 70. The last section (Section 69.9) presents three applications of computational linear algebra to the study of polynomial identities for nonassociative algebras: Pseudorandom vectors in a nonassociative algebra, the expansion matrix for a nonassociative operation, and the representation theory of the symmetric group.
