Description

Nonassociative algebra is an important area of modern mathematics with numerous applications in natural sciences. For instance, Lie algebras first appeared as infinitesimal symmetries of differential equations, similar to finite symmetry groups in Galois Theory of algebraic equations. Jordan algebras were first introduced by Pascual Jordan (1933) to formalize the notion of an algebra of observables in quantum mechanics. Alternative algebras naturally appear in the context of composition algebras and projective planes. Superalgebras are needed to describe bosons and fermions in the theory of elementary particles. Various other classes of nonassociative algebras have been studied and are typically defined by polynomial identities.

This course is intended for graduate students in mathematics, both pure and applied.

Objectives

To introduce students to general nonassociative algebras, as algebras defined by polynomial identities, and to prominent particular classes, such as Lie algebras and superalgebras, Jordan algebras, and alternative algebras.

Prerequisites

Undergraduate courses in linear algebra (such as MATH 2051) and abstract algebra (MATH 3320).

Texts

• Schafer, Richard D. An Introduction to Nonassociative Algebras, Dover, 2017
•  Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. Rings that are nearly associative, Academic Press, 1982