Link homology and categorification of quantum groups

量子链接不变量，包括琼斯，考夫曼和其他链接多项式，可以从量子群中恢复，量子群是简单李代数的通用包络代数的Hopf代数变形。这些不变量可以从缠结范畴扩展到量子群表示范畴的函子。缠结的不变量是表示的同态。链同调将量子不变量提升了一个维度，可以看作是从链配边范畴到多阶阿贝尔群范畴的函子。缠结的不变量成为分配给缠结边界的三角化范畴之间的函子。范畴化导致了缠结配边的不变量，这些不变量在这些函子之间的自然变换中取值。以这种方式出现的范畴对量子群表示的张量积进行了分类，并出现在表示论、辛拓扑和代数几何中。该建议旨在进一步阐明链同调的结构，发现其中新的上同调运算，进一步将它们与Hochschild同调联系起来，推广Rassmussen不变量，并将链同调与量子群的分类联系起来。不到两年前发现的量子群的范畴化，将它们实现为某些图解定义的环上的投射模范畴的格罗滕迪克群，这也出现在整个表示论中。PI相信，分类量子群将被证明在数学的几个领域无处不在，并将在未来几年继续研究它们。在过去的30年里，数学家发现了纽结理论和三维流形（局部看起来像我们的空间，但具有不同的整体行为的对象）与代数和几何中大量复杂结构之间的深刻关系。这些结构中的许多都与对称性概念的阐明有关。一个给定的数学或物理对象的所有对称性构成了所谓的群--一个可以组合和反转的对称性的集合。在过去的几十年里，各种各样的发展导致了群的概念的牵强的推广，包括Drinfeld和Jimbo发现的量子群。在PI参与的方向上，最近的进一步进展导致了所谓的分类量子群的发现，这是更复杂的对象，量子群可以通过忘记大部分信息来恢复。这些分类的量子群预计将密切相关的拓扑结构在四维。四维拓扑学研究局部看起来像我们的空间加上时间方向的物体，但可能具有复杂的全局结构。这条研究路线有望进一步将数学的许多领域联系在一起，包括表示论、同调代数、拓扑和几何。

Quantum link invariants, including the Jones, Kauffman, and other link polynomials, can be recovered from quantum groups, which are Hopf algebra deformations of the universal enveloping algebras of simple Lie algebras. These invariants can be extended to functors from the category of tangles to categories of quantum group representations. The invariant of a tangle is a homomorphism of representations. Link homology lifts quantum invariants one dimension up and can be viewed as functors from the category of link cobordisms to the category of multi-graded abelian groups. The invariant of a tangle becomes a functor between triangulated categories assigned to the boundaries of the tangle. Categorification leads to invariants of tangle cobordisms which take values in natural transformations between these functors. The categories that appear in this way categorify tensor products of quantum group representations and appear thoughout representation theory, symplectic topology, and algebraic geometry. The proposal aims to further elucidate the structure of link homology, discover new cohomological operations in them, further tie them up with Hochschild homology, generalize the Rassmussen invariant, and relate link homology with the categorification of quantum groups. Categorification of quantum groups, discovered less that two years ago, realizes them as Grothendieck groups of categories of projective modules over certain diagrammatically defined rings, which also appear throughout representation theory. The PI believes that categorified quantum groups will prove ubiquitous in several areas of mathematics and will continue studying them for the next few years.Mathematicians discovered in the past 30 years deep relations between the theory of knots and 3-manifolds (objects that locally look like our space but have a different global behaviour) and a plethora of sophisticated structures in algebra and geometry. Many of these structures have to do with the elucidation of the notion of symmetry. All symmetries of a given mathematical or physical object constitute what is known as a group - a collection of symmetries that can be composed and reversed. Various developments in the past decades led to a far-fetched generalizations of the notion of a group, including the discovery of quantum groups by Drinfeld and Jimbo. More recent further progress in the direction, in which the PI was involved, resulted in the discovery of so-called categorified quantum groups, which are even more sophisticated objects, from which quantum groups can be recovered by forgetting most of the information. These categorified quantum groups are expected to be intimately related to the topology in four dimensions. Four-dimensional topology studies objects that locally look like our space plus the time direction, but may have complicated global structure. This line of research is expected to further tie together many areas of mathematics, including representation theory, homological algebra, topology and geometry.