Canonical bases of Borcherds-Cartan type

Abstract

We study the canonical basis for the negative part $\mathbf{U}^{-}$ of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras $\mathbf{U}^{-}$ associated to two different matrices satisfying certain conditions may coincide (6.3). We show that the canonical bases coincide provided that the algebras $\mathbf{U}^{-}$ coincide (Theorem 6.3.5). We also answer partially a question by Lusztig in [L3] (Theorem 7.1.1).