Representations of infinite-dimensional Lie algebras and related topics

无限维李代数的表示及相关主题

• 批准号: 0303529
• 负责人: Edward Frenkel
• 金额: $ 39.78万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0303529&HistoricalAwards=false
• 关键词: Representations; infinite; dimensional; Lie; algebras

Principal Investigator: Edward Frenkel Proposal Number: 0303529Institution: University of California-BerkeleyAbstract: Representations of infinite-dimensional Lie algebras and related topicsThe principal investigator proposes to conduct research in the following areas: local Langlands correspondence for affine Kac--Moody algebras; vertex algebras and quantum groups; cohomology of the sheaves of differential operators on the moduli stacks of bundles on curves. In the traditional local Langlands correspondence one wishes to describe smooth representations of a reductive group over a local non-archimedian field, such as the field of formal Laurent power series over a finite field, in terms of the Galois group of the local field and the Langlands dual group. When we replace the finite field by the complex field, we are naturally led to loop groups and loop Lie algebras and the central extensions of the latter, i.e., the affine Kac-Moody algebras. The principal investigator wishes to describe the categories of Harish-Chandra modules over an affine Kac-Moody algebra in terms of geometric data associated to the dual group. More specifically, the principal investigator intends to prove that the derived category of a certain category of modules over an affine algebra is equivalent to the derived category of the category of quasicoherent sheaves on the Springer fiber of a nilpotent element of the Langlands dual Lie algebra. In addition, the principal investigator proposes the construction of an extension of the W-algebra associated to an arbitrary simple Lie algebra to a vertex algebra, which carries an action of the dual group by vertex algebra automorphisms. He intends to prove that the category of representations of this vertex algebra is equivalent to the category of representations of a quantum group associated to the Langland dual Lie algebra. Finally, he intends to compute the cohomology of the vacuum representation over an affine Kac-Moody algebra and to relate it to the cohomology of the sheaf of differential operators on the moduli stack of bundles on an algebraic curve.A lot of effort has been made over the last thirty years in the development of the Langlands Program which ties together seemingly unrelated structures in number theory, automorphic representations and algebraic geometry. The principal investigator expects that uncovering the Langlands duality patterns in the new setting of affine Kac-Moody algebras and more general vertex algebra will significantly enhance our understanding of the Langlands correspondence, which to this day remains a mystery. In particular, the Langlands correspondence is elevated in this case to the level of categories and therefore one can see a much finer structure than was previously possible. It is hoped that the interdisciplinary nature of this proposal will serve to advance discovery and understanding of representation theory of affine Kac-Moody algebras and vertex algebras by relating them to the Langlands Program, and at the same time will stimulate the development of the Langlands Program by bringing in new insights from geometry.

主要研究者：Edward Frenkel提案编号：0303529机构：加州大学伯克利分校摘要：表示的无限维李代数和相关topicsThe主要研究者提出进行研究在以下领域：局部Langlands对应仿射Kac-穆迪代数;顶点代数和量子群;上同调的微分算子层上的模栈的丛曲线。在传统的局部朗兰兹对应中，人们希望用局部域的伽罗瓦群和朗兰兹对偶群来描述局部非阿基米德域上的约化群的光滑表示，例如有限域上的形式洛朗幂级数的域。当我们用复数域代替有限域时，我们自然会得到循环群和循环李代数以及后者的中心扩张，即，仿射Kac-Moody代数主要研究者希望描述的范畴Harish-Chandra模在仿射Kac-Moody代数的几何数据相关联的对偶群。更具体地说，主要研究者打算证明仿射代数上的某类模的导出范畴等价于朗兰兹对偶李代数的幂零元的斯普林格纤维上的拟相干层范畴的导出范畴。此外，主要研究者提出了一个扩展的W-代数相关的任意单李代数的顶点代数，它进行了行动的对偶群顶点代数自同构的建设。他打算证明，这一类代表顶点代数是等价的一类代表的量子组相关的朗兰对偶李代数。最后，他打算计算仿射Kac-Moody代数上的真空表示的上同调，并将其与代数曲线上的丛的模栈上的微分算子层的上同调联系起来。在过去的三十年里，在朗兰兹纲领的发展中已经做出了很多努力，该纲领将数论中看似无关的结构联系在一起，自守表示和代数几何。首席研究员预计，在仿射Kac-Moody代数和更一般的顶点代数的新环境中揭示朗兰兹对偶模式将显着提高我们对朗兰兹对应的理解，直到今天仍然是一个谜。特别是，朗兰兹对应在这种情况下被提升到范畴的水平，因此人们可以看到比以前可能的更精细的结构。希望这个建议的跨学科性质将有助于通过将仿射Kac-Moody代数和顶点代数与朗兰兹纲领联系起来来推进对它们的表示理论的发现和理解，同时通过引入几何学的新见解来刺激朗兰兹纲领的发展。