Crystal bases and canonical bases for quantum Borcherds-Bozec algebras

TL;DR

This paper develops the theory of crystal and canonical bases for quantum Borcherds-Bozec algebras, establishing their properties, connections, and introducing primitive canonical bases that coincide with lower global bases.

Contribution

It reconstructs crystal basis theory with modified Kashiwara operators and introduces primitive canonical bases, linking them to lower global bases in quantum Borcherds-Bozec algebras.

Findings

Crystal basis theory reconstructed with modified Kashiwara operators.

Important lemmas proved during grand-loop argument.

Primitive canonical bases shown to coincide with lower global bases.

Abstract

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.