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

该奖项支持考夫曼纽结不变量和Hennings的3-流形不变量的研究，以及Hennings的不变量与Reshetikhin和Turaev所描述的3-流形不变量之间的联系。还将通过在与Faddeev-Reshetikhin-Takhtajan结构密切相关的双代数上施加某些条件来寻求量子Yang-Baxter方程的一般解。主要研究人员还将研究有限维拟三角Hopf代数和带状Hopf代数的类，重点是从给定的纽结不变量得到新的纽结不变量的构造。此外，还将研究与仿射群的量子化坐标环的Hopf代数对偶有关的乘积。量子群对数学家和物理学家来说都是一个新的研究领域。在数学方面，它结合了代数、分析和几何这三个最古老的纯数学领域，但它对致力于共形量子场论的物理学家非常感兴趣。

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.