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

该奖项支持考夫曼结不变量的研究 和Hennings的3-maniforld不变量，以及 Hennings '不变量和3-流形不变量， 列舍季欣和图拉耶夫 问题的一般解类 量子杨-巴克斯特方程也将寻求通过施加 一个双代数上的某些条件密切相关的 Faddeev-Reshetikhin-Takhtajan建筑。 校长 研究人员还将研究有限维的类 拟三角Hopf代数与带状Hopf代数 强调新的纽结不变量是 从给定的。 此外，与 仿射量子化坐标环的Hopf代数对偶 团体将进行调查。 量子群是一个新的研究领域， 数学家和物理学家 在数学方面，它 结合了"纯"数学的三个最古老的领域， 代数，分析和几何，但它是非常感兴趣的， 研究共形量子场论的物理学家

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.