Categorification of Quantum Groups

该奖项由2009年《美国复苏和再投资法案》(公法111-5)资助。该奖项建立在Beilinson、Lusztig和MacPherson的几何结构的基础上，PI与Mikhail Khovanov合作将量子sl(N)归类，以及任意Kac-Moody李代数的“下三角”子代数的普适包络代数的量子形变。虽然这些分类完全是组合的，但它们利用了各种图解演算，强调了拓扑学和代数之间的新的相互作用。该方案旨在进一步发展范畴量子群的理论，研究它们的Hopf结构(余积和对极)，进一步发展它们的表示理论，对其他Kac-Moody李代数的整个量子包络代数进行分类，并了解它们与几何表示理论的关系。量子群范畴化的潜在应用包括研究量子包络代数的正性性质，Khovanov同调的表示理论解释以及Witten-Reshetikhin-Turaev量子3-流形不变量的范畴。量子群普遍存在于数学和理论物理中。这些Hopf代数为理解量子链不变量，如Jones多项式、Kauffman和HOMFLY-PT多项式，以及Witten-Reshetikhin-Turaev量子三维流形不变量提供了表示理论框架。它们还与统计力学、量子场论和仿射李代数有关。最近的工作表明，量子群的表示理论，以及量子群本身，都是一种更丰富的代数结构的影子，这种代数结构被称为范畴化量子群。这些结构被克雷恩和弗伦克尔推测存在，并构成了这一提议的焦点。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).Building off geometric constructions of Beilinson, Lusztig, and MacPherson, the PI in collaboration with Mikhail Khovanov categorified quantum sl(n), as well as the quantum deformation of the universal enveloping algebra of the ``lower-triangular'' subalgebra for an arbitrary Kac-Moody Lie algebra. While these categorifications are completely combinatorial, they utilize various diagrammatic calculi emphasizing a new interplay between topology and algebra. The proposal aims to further develop the theory of categorified quantum groups, study their Hopf structure (comultiplication and antipode), further develop their representation theory, categorify the entire quantum enveloping algebra for other Kac-Moody Lie algebras, and understand the relationship to geometric representation theory. Potential applications of categorified of quantum groups include the study of positivity properties for quantum enveloping algebras, a representation theoretic explanation of Khovanov homology and a categorification of the Witten-Reshetikhin-Turaev quantum 3-manifold invariants.Quantum groups are prevalent throughout mathematics and theoretical physics. These Hopf algebras provide the representation theoretic framework for understanding quantum link invariants such as the Jones polynomial, Kauffman and HOMFLY-PT polynomial, as well as the Witten-Reshetikhin-Turaev quantum 3-manifold invariants. They also relate to statistical mechanics, quantum field theory, and affine Lie algebras. Recent work suggests that the representation theory of quantum groups, and the quantum groups themselves, are shadows of a much richer algebraic structure known as categorified quantum groups. These structures were conjectured to exist by Crane and Frenkel and form the focal point of this proposal.