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

该奖项支持对Kauffman的结不变量和Hennings的3流形不变量的研究，以及Hennings不变量与Reshetikhin和Turaev描述的3流形不变量之间的联系。通过对与Faddeev-Reshetikhin-Takhtajan构造密切相关的双代数施加一定的条件，也将寻求量子Yang-Baxter方程的一般解类。首席研究员还将研究有限维准三角形Hopf代数和带状Hopf代数，这些代数强调从给定的结不变量推导出新的结不变量的构造。此外，还研究了与仿射群的量子化坐标环的Hopf代数对偶有关的积。量子群对于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“纯粹”数学的三个最古老的领域：代数、分析和几何，但它对研究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.