Algebras that are nearly associative

近结合代数

• 批准号: 153128-2011
• 负责人: Bremner, Murray
• 金额: $ 0.8万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569397
• 关键词: Algebras; nearly; associative

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.

代数结构的理论起源于1800年代中期。 实际数字的字段已经扩展到包括-1的平方根，这产生了复数的字段。 这两个系统都满足了换向性，AB = BA和关联性，（AB）C = A（BC）。 然后将复数（对实数上的二维）扩展到不合理的四季度（4维）。 然后将四元组扩展到非关联的八元（8维）。 诸如通勤和关联之类的特性称为代数的多项式身份。 当代数学中最重要的代数结构是联想代数（包括基本线性代数的矩阵）和以19世纪最有影响力的数学家之一命名的Lie代数。 Lie代数是非缔合性的，并且与连续组密切相关，该组在理论物理学的对称性研究中起着至关重要的作用。另一类非社交结构是约旦代数，以帕斯基·约旦（Pascual Jordan）的名字命名，他是20世纪初期量子力学的创始人之一。 Lie和Jordan代数满足多项式身份，这些身份是关联性的概括，因此通常被称为“几乎是关联的代数”。 我的研究计划的主要重点是研究非缔合结构的多项式身份。该方法涉及计算机代数，尤其是具有数亿个条目的大型矩阵的计算。次要重点是建造用于非缔合结构的通用联想包围代数。我特别有兴趣将二进制代数结构的经典理论扩展到涉及两个以上因素的乘法。这些新结构有望在纯数学和理论物理学中有许多应用。