Categorification of Quantum Groups

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。建立在Beilinson，Lusztig和MacPherson的几何结构基础上，PI与Mikhail Khovanov合作分类量子sl（n），以及任意Kac-Moody李代数的“下三角”子代数的通用包络代数的量子变形。 虽然这些代数运算是完全组合的，但它们利用各种图解演算，强调拓扑和代数之间的新相互作用。该提案旨在进一步发展分类量子群的理论，研究它们的Hopf结构（余乘和对极），进一步发展它们的表示理论，为其他Kac-Moody李代数分类整个量子包络代数，并理解与几何表示理论的关系。量子群的分类的潜在应用包括研究量子包络代数的正性性质、Khovanov同调的表示论解释和Witten-Reshetikhin-Turaev量子3-流形不变量的分类。这些Hopf代数提供了理解量子链路不变量的表示理论框架，如Jones多项式，Kauffman和HOMFLY-PT多项式，以及Witten-Reshetikhin-Turaev量子3-流形不变量。 它们也涉及到统计力学、量子场论和仿射李代数。 最近的工作表明，量子群的表示论和量子群本身，是一个更丰富的代数结构的影子，称为分类量子群。 这些结构是由起重机和Frenkel证明存在的，并形成了本提案的重点。

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.