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

传统上认为李超代数的超表示理论与一般的李超代数的超表示理论有很大的不同，这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。在李超代数的表示理论和李代数的表示理论之间，PI打算表述和建立一种新的直接联系，称为超对偶。给出了李超代数和李超代数的模范畴在适当无限极限下的某些等价性和Kazhdan-Lusztig多项式的辨识。一个主要的代数工具是Kazhdan-Lusztig理论的Fock空间公式。该超对偶性有望为各种类型的简单李超代数上的模范畴和量子超群上的模范畴在单位根上的不可约性质的Kazhdan-Lusztig型猜想提供新的途径。自然界中对称性有不同的表现形式，例如，可以在圆、球或五个正多面体中的一个等中找到。用于描述对称性的数学语言通常涉及群的概念或其无限小的对应概念，如李代数。表示论是用矩阵来表示群和李代数的一种研究方法。另一方面，不同的对称性可以相互关联。在寻找万物统一理论的过程中，物理学家提出了弦理论作为候选理论。超对称为这些考虑增加了另一个看不见的维度，研究李超代数对于理解超对称至关重要。我们的超对偶性项目可以被视为提供了一种将超对称与通常意义上的对称联系起来的精确的新方法。这有助于提供一个令人信服的证据来支持超对称的想法，并可能应用于弦理论。

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.