Algebras that are nearly associative

代数结构理论起源于19世纪中期。 真实的数域已经扩展到包括-1的平方根，这就产生了复数域。 这两个系统都满足交换性ab = ba和结合性（ab）c = a（bc）。 复数（真实的数上的2维）然后被扩展到不可交换的四元数（4维）。 然后将四元数扩展到不结合的八元数（8维）。 像交换性和结合性这样的性质被称为代数的多项式恒等式。 当代数学中最重要的代数结构是结合代数（包括初等线性代数的矩阵）和李代数，以19世纪最有影响力的数学家之一Sophus Lie命名。 李代数是非结合的，并且与连续群密切相关，连续群在理论物理中的对称性研究中起着重要作用。另一类非结合结构是约旦代数，以世纪早期量子力学的创始人之一帕斯夸尔·约旦的名字命名。李代数和若当代数满足多项式恒等式，这些恒等式是结合性的推广，因此它们通常被称为“近结合代数”。 我的研究项目的主要重点是非结合结构的多项式恒等式的研究;方法涉及计算机代数，特别是具有数亿条目的大型矩阵的计算。第二个重点是构造非结合结构的泛结合包络代数。我特别感兴趣的是扩展经典理论的二进制代数结构乘法涉及两个以上的因素。这些新结构有望在纯数学和理论物理学中有许多应用。

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.