Representation theory and quantum symmetric pairs

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，它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概念，并与之联系在一起。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正在形成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。这种方法可能导致一种新的更高的对称性，并在纽结理论中得到应用。最近，Pi和他的合作者提出了一种源于量子对称对的正则基理论，推广了Lusztig和Kashiwara的经典构造。这些新的正则基与几何和范畴O密切相关。PI建议将各种几何结构和范畴(局部类型为A)增强为一个统一的主题，即“带对合的类型A”。PI计划发展一个由量子对称对产生的完整的正则基理论。这将导致对余理想代数及其张量积模的分类，这反过来又可应用于超古典型范畴O的Koszul分次提升。PI打算发展新的方法来确定量子群和单位根处的超群在范畴O中的不可约特征标。最后，PI建议给出仿射型量子余理想代数、它们的模以及相关的标准基的新的几何和代数实现。

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.