Quantum supergroups II. Canonical basis

Abstract

Abstract. Following Kashiwara's algebraic approach, we construct crystal bases and canonical bases for quantum supergroups of anisotropic type and for their integrable modules.

Quantum supergroups IV: the modified form

“…In this section, we will recall the essential definitions and results from earlier papers. We caution the reader that while the results are largely restatements of results in [CHW1,CHW2], several of our conventions differ from those in loc. cit.…”

“…Together with Hill and Wang [CW,CHW1,CHW2], we defined and systematically developed the theory of covering quantum groups associated to an anisotropic Kac-Moody superalgebra. These algebras, whose definition is inspired by the results in [CW, HW], combine the structure and representation theories of a classical quantum group and a quantum supergroup by using a "half parameter" π, satisfying π 2 = 1, in the place of super signs; one recovers the classical quantum group under the specialization π = 1, and the quantum supergroup under the specialization π = −1.…”

“…In Section 4, we construct canonical bases for the family of modules F. The main result of this section is the construction of a canonical basis ofU which is simultaneously compatible with the canonical bases of the F. This basis is a generalization of the canonical basis of U − which was constructed in [CHW2].…”

Super Kac–Moody 2‐categories

“…Let be the symmetric bilinear form on the root lattice defined from . In this section, we recall the definition of the covering quantum group of Clark, Hill and Wang . Our exposition is based mostly on and .…”

“…Our exposition is based mostly on and . Note that our is the parameter denoted in , which is in . We write and in place of and ; we would also write for the generator although we won't actually need this here.…”