Abstract crystals for quantum Borcherds-Bozec algebras

TL;DR

This paper develops a new theory of abstract crystals for quantum Borcherds-Bozec algebras, providing foundational tools and characterizations for their crystal structures, which are crucial in understanding their representation theory.

Contribution

It introduces a novel construction of abstract crystals for quantum Borcherds-Bozec algebras, differing from Bozec's approach, and proves key theorems including crystal embedding and characterizations.

Findings

Established a new construction of abstract crystals for quantum Borcherds-Bozec algebras.

Proved the crystal embedding theorem for these algebras.

Provided characterizations of ${B}(ty)$ and ${B}( l)$ crystals.

Abstract

In this paper, we develop the theory of abstract crystals for quantum Borcherds-Bozec algebras. Our construction is different from the one given by Bozec. We further prove the crystal embedding theorem and provide a characterization of ${B}(\infty)$ and ${B}(\lambda)$ as its application, where ${B}(\infty)$ and ${B}(\lambda)$ are the crystals of the negative half part of the quantum Borcherds-Bozec algebra $U_q(\mathfrak g)$ and its irreducible highest weight module $V(\lambda)$, respectively.