Algebras that are nearly associative

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The theory of algebraic structures originated in the middle of the 1800's. The field of real numbers had already been extended to include the square root of -1, and this produced the field of complex numbers. Both of these systems satisfy commutativity, ab = ba, and associativity, (ab)c = a(bc). The complex numbers (2-dimensional over the real numbers) were then extended to the quaternions (4-dimensional) which are not commutative. The quaternions were then extended to the octonions (8-dimensional) which are not associative. Properties such as commutativity and associativity are known as polynomial identities for algebras. The most important algebraic structures in contemporary mathematics are associative algebras (including the matrices of elementary linear algebra), and Lie algebras, named after Sophus Lie, one of the most influential mathematicians of the 19th century. Lie algebras are nonassociative and are closely related to continuous groups, which play an essential role in the study of symmetry in theoretical physics. Another class of nonassociative structures are Jordan algebras, named after Pascual Jordan, one of the founders of quantum mechanics in the early 20th century. Lie and Jordan algebras satisfy polynomial identities which are generalizations of associativity, and hence they are often called "algebras that are nearly associative". The primary focus of my research program is the study of polynomial identities for nonassociative structures; the methodology involves computer algebra, and especially calculations with large matrices having hundreds of millions of entries. The secondary focus is the construction of universal associative enveloping algebras for nonassociative structures. I am especially interested in extending the classical theory of binary algebraic structures to multiplications which involve more than two factors. These new structures promise to have many applications in pure mathematics and theoretical physics.