Representation theory and quantum symmetric pairs

表示论和量子对称对

• 批准号: 1405131
• 负责人: Weiqiang Wang
• 金额: $ 18万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-06-15 至 2017-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405131&HistoricalAwards=false
• 关键词: Representation; theory; quantum; symmetric; pairs

The symmetries of a geometric object, such as the rotations preserving a cube or a sphere, can be described by the mathematical notion of a group. Quantum groups, one of the focal points of this project, are deformations of these symmetries. Another focal point is a family of algebraic objects, Lie superalgebras. These mathematical objects arose from and connect to the idea of supersymmetry in physics. The PI will study supersymmetry in the form of Lie superalgebra representations through a novel quantum group. The PI is in the process of formulating and expanding a new approach to representations of classical Lie algebras and establishing connections to geometry. This approach is likely to lead to a new kind of higher symmetry and to have applications to knot theory.Recently the PI and his collaborators have initiated a theory of canonical bases arising from quantum symmetric pairs, generalizing the classic constructions of Lusztig and Kashiwara. These new canonical bases are intimately related to geometry and category O. The PI proposes to enhance various geometric constructions and categorification (which are of locally type A) to a unifying theme of 'type A with involution'. The PI plans to develop a full-fledged theory of canonical basis arising from quantum symmetric pairs. This should lead to a categorification of the coideal algebras and their tensor product modules, which in turn has applications to Koszul graded lift of category O of super classical type. The PI intends to develop new methods to determine the irreducible characters in category O for quantum groups and supergroups at roots of unity. Finally, the PI proposes to give new geometric and algebraic realizations of quantum coideal algebras of affine type, their modules, and the associated canonical bases.

几何对象的对称性，例如保持立方体或球体的旋转，可以用群的数学概念来描述。 量子群是这个项目的焦点之一，是这些对称性的变形。 另一个焦点是一族代数对象，李超代数。 这些数学对象起源于物理学中的超对称性，并与之相联系。 PI将通过一个新的量子群以李超代数表示的形式研究超对称性。 PI正在制定和扩展一种新的方法来表示经典李代数，并建立与几何的联系。 这种方法可能会导致一种新的更高的对称性，并有应用到纽结理论。最近，PI和他的合作者已经发起了一个理论的规范基础产生的量子对称对，推广经典的建设Lusztig和Kashiwara。 这些新的标准基与几何和范畴O密切相关。 PI建议将各种几何构造和分类（局部类型A）增强为“对合A型”的统一主题。 PI计划开发一个完整的理论规范基础所产生的量子对称对。 这将导致共理想代数和它们的张量积模的分类，这反过来又适用于超经典类型的O类Koszul分次提升。 PI打算开发新的方法来确定在单位根上的量子群和超群在O范畴中的不可约特征标。 最后，PI提出给出仿射型量子余理想代数、它们的模和相关的规范基的新的几何和代数实现。