Link homology and categorification of quantum groups

链接量子群的同源性和分类

• 批准号: 1005750
• 负责人: Mikhail Khovanov
• 金额: $ 51.74万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005750&HistoricalAwards=false
• 关键词: Link; homology; categorification; quantum; groups

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.

量子链路不变量，包括Jones， Kauffman和其他链路多项式，可以从量子群中恢复，量子群是简单李代数的普合包络代数的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。链接同调将量子不变量提升了一个维度，可以看作是从链接协协范畴到多阶阿贝尔群范畴的函子。缠结的不变量变成了分配给缠结边界的三角化范畴之间的函子。分类导致缠结协数的不变量，它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，并在表示理论、辛拓扑和代数几何中出现。本论文旨在进一步阐明链路同调的结构，发现其中新的上同调运算，进一步将其与Hochschild同调联系起来，推广Rassmussen不变量，并将链路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分类，将它们实现为特定图解定义的环上的投影模的Grothendieck群，这也出现在整个表示理论中。PI认为，分类量子群将被证明在数学的几个领域中无处不在，并将在未来几年继续研究它们。在过去的30年里，数学家们发现了结理论和3流形（局部看起来像我们的空间，但具有不同的整体行为的物体）与代数和几何中大量复杂结构之间的深刻关系。许多这样的结构都与对称概念的阐释有关。一个给定的数学或物理对象的所有对称构成了所谓的群——一组可以组合和反转的对称。过去几十年的各种发展导致对群的概念进行了牵强的概括，包括德林菲尔德和金博发现的量子群。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，这是更复杂的物体，量子群可以通过忘记大部分信息来恢复。这些被分类的量子群被期望与四维拓扑密切相关。四维拓扑学研究的对象局部看起来像我们的空间加时间方向，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，包括表示理论、同调代数、拓扑和几何。