Crystal bases and canonical bases for quantum Borcherds-Bozec algebras

Let $U_q^{-}(\mathfrak{g})$ be the negative half of a quantum Borcherds-Bozec algebra $U_q(\mathfrak{g})$ and $V(\lambda )$ be the irreducible highest weight module with $\lambda \in P^{+}$. In this paper, we investigate the structures, properties and their close connections between crystal bases and canonical bases of $U_q^{-}(\mathfrak{g})$ and $V(\lambda )$. We first re-construct crystal basis theory with modified Kashiwara operators. While going through Kashiwara’s grand-loop argument, we prove several important lemmas, which play crucial roles in the later developments of the paper. Next, based on the theory of canonical bases on quantum Bocherds-Bozec algebras, we introduce the notion of primitive canonical bases and prove that primitive canonical bases coincide with lower global bases.