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Let $U_{q}^{-}(\mathfrak g)$ be the negative half of a quantumBorcherds-Bozec algebra $U_{q}(\mathfrak g)$ and $V(\lambda)$ be theirreducible highest weight module with $\lambda \in P^{+}$. In this paper, weinvestigate the structures, properties and their close connections betweencrystal 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 importantlemmas, which play crucial roles in the later developments of the paper. Next,based on the theory of canonical bases on quantum Bocherds-Bozec algebras, weintroduce the notion of primitive canonical bases and prove that primitivecanonical bases coincide with lower global bases.

Crystal bases and canonical bases for quantum Borcherds-Bozec algebras