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.

几何物体的对称性，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，这个项目的焦点之一，是这些对称性的变形。另一个焦点是一类代数对象，李超代数。这些数学对象源于物理学中的超对称概念，并与之相关。PI将通过一个新的量子群研究李超代数表示形式的超对称性。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。这种方法很可能导致一种新的更高对称性，并应用于结理论。最近，PI和他的合作者发起了一个由量子对称对产生的正则基理论，推广了Lusztig和Kashiwara的经典结构。这些新的规范基与几何和o类密切相关。PI建议将各种几何结构和分类（局部为A类）增强为“A型对合”的统一主题。PI计划发展由量子对称对产生的规范基的成熟理论。这将导致共理想代数及其张量积模的分类，进而应用于超经典型O类的Koszul梯度提升。研究了在单位根处确定量子群和超群的O类不可约性质的新方法。最后，提出了仿射型量子共理想代数的新的几何和代数实现，以及它们的模和相关的正则基。