Duality between representations of Lie superalgebras and Lie algebras via Kazhdan-Lusztig theory

超表示理论传统上被认为与通常的李代数理论有很大的不同，主要是因为简单李超代数的Weyl群不足以控制不可约表示的结构。PI打算制定和建立一个新的直接联系，称为超对偶，李超代数和李代数的表示理论之间。超对偶断言模范畴在适当的无限极限中的某些等价性，以及李超代数和李代数的Kazhdan-Lusztig多项式的识别。一个主要的代数工具是Kazhdan-Lusztig理论的Fock空间公式。超对偶有望为各种类型的单李超代数上的模范畴和单位根上的量子超群上的模范畴的不可约特征标的Kazhdan-Lusztig型刻画提供一种新的途径。自然界中对称性有不同的表现形式，例如圆、球或五个正多面体之一等。用来描述对称性的数学语言通常涉及群或其无穷小对应物的概念，如李代数。表示论是研究群和李代数的一种方法，通过用矩阵来表示它们。另一方面，不同的对称性可以相互关联。为了寻找万物的统一理论，物理学家们提出了弦论作为候选理论。超对称性为这种考虑增加了另一个看不见的维度，而李超代数的研究对于理解超对称性至关重要。我们关于超对偶性的项目可以被认为是提供了一种精确的新方法，将超对称性与通常意义上的对称性联系起来。这有助于提供一个令人信服的证据来支持超对称性的想法，并可能应用于弦论。

The super representation theory is traditionally regarded as fairly different from the usual one for Lie algebras largely because the Weyl group for a simple Lie superalgebra does not suffice to control the structures of the irreducible representations. The PI intends to formulate and establish a new direct link, termed as super duality, between the representation theories of Lie superalgebras and of Lie algebras. The super duality asserts certain equivalences of module categories in a suitable infinite limit and the identification of the Kazhdan-Lusztig polynomials for Lie superalgebras and Liealgebras. A main algebraic tool is a Fock space formulation of the Kazhdan-Lusztig theory. The super duality is expected to provide a new approach toward the Kazhdan-Lusztig type conjectures on irreducible characters for module categories over simple Lie superalgebras of various types and for module categories over quantum supergroups at roots of unity.There are different manifestations of symmetries in nature, which one can find in, for example, a circle, a sphere, or one of the five regular polyhedra, and others. The mathematical language used to describe symmetries often involves the concept of groups or their infinitesimal counterparts such as Lie algebras. Representation Theory is a way of studying the groups and Lie algebras by expressing them in terms of matrices. On the other hand, different symmetries can be related to each other. In search for a unified theory of everything, physicists have proposedString Theory as a candidate theory. Supersymmetry adds another invisible dimension to such considerations and the study of Lie superalgebras is crucial to understanding the supersymmetry. Our project on Super Duality can be regarded as providing a precise and new way of relating supersymmetry to symmetry in the usual sense. This helps to provide a convincing evidence supporting the idea of supersymmetry and may have applications to String Theory.