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 Lie algebras. 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 proposed String 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.
