Mathematical Sciences: Knot and 3-Manifold Invariants Derived from Finite-Dimensional Hopf Algebras and Bialgebra Constructions Arising in Quantum Groups

数学科学：由有限维 Hopf 代数和量子群中出现的双代数构造导出的结和 3 流形不变量

• 批准号: 9308106
• 负责人: David Radford
• 金额: $ 6万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9308106&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Knot; Manifold; Invariants

This award supports research on Kauffman's knot invariants and Hennings' 3-maniforld invariants, and the connection between Hennings' invariants and the 3-manifold invariants described by Reshetikhin and Turaev. General classes of solutions to the quantum Yang-Baxter equation will also be sought by imposing certain conditions on a bialgebra closely associated to the Faddeev-Reshetikhin-Takhtajan construction. The principal investigator will also study the classes of finite-dimensional quasitriangular Hopf algebras and ribbon Hopf algebras emphasizing constructions from which new knot invariants are derived from given ones. In addition, products related to the Hopf algebra dual of the quantized coordinate ring of affine groups will be investigated. Quantum groups are a new area of research for both mathematicians and physicists. On the mathematical side, it combines three of the oldest areas of "pure" mathematics, algebra, analysis and geometry, yet it is of great interest to physicists working on conformal quantum field theory.

该奖项支持有关Kauffman的结和Hennings的3个Maniforld不变性的研究，以及Hennings的不变性与Reshetikhin和Turaev所描述的3个Manifold不变性之间的联系。 还将通过与Faddeev-Reshetikhin-Takhtajan构建的双子施加某些条件，从而寻求量子阳边方程解决方案的一般类别。 首席研究者还将研究有限维的准二元霍普夫代数和色带Hopf代数强调结构，从而从给定的结构中得出了新的结。 此外，将研究与仿射组的量化坐标环相关的产品。 量子组是数学家和物理学家的新研究领域。 在数学方面，它结合了“纯”数学，代数，分析和几何学的三个最古老的领域，但对于从事保形量子量理论的物理学家来说，它引起了极大的兴趣。