Quantum Supergroups and Canonical Bases

Drinfeld and Jimbo introduced quantum groups as deformations of the universal enveloping algebras associated to semi-simple and Kac-Moody Lie algebras. The study of the structure of these algebras lead to the discovery of canonical bases by Lusztig. These bases have many desirable properties, and recently have played an important role in the development of categorified quantum groups.

This dissertation studies the structures and canonical bases for quantum supergroups of anisotropic type; that is, quantum groups associated to Kac-Moody Lie superalgebras with no isotropic odd roots. We proceed by utilizing the framework of
covering quantum groups, which are algebras with two parameters, namely the quantum parameter q and half parameter pi squaring to 1, which interpolates between a Drinfeld-Jimbo quantum group (the pi=1 case) and a quantum supergroup (the pi=-1 case). A version of these algebras was first introduced by Hill-Wang in the context of categorifications of quantum groups.

We develop analogues of several classical quantum group constructions, including a (braided) Hopf algebra structure and a quasi-R-matrix which intertwines two coproducts. We also define an analogue of the BGG category O and show that the integrable modules in O are completely reducible.The covering quantum group admits a bar-involution and a bar-invariant integral form. We construct canonical bases for integrable modules of O and for the half-quantum covering group, generalizing Kashiwara's grand loop and globalization constructions. This canonical basis is then extended to a modified form of the quantum covering group, as defined by Lusztig. Specializing our constructions to pi to -1 yields the first examples of canonical bases known for quantum supergroups.