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.
