Perfect basis theory for quantum Borcherds-Bozec algebras

TL;DR

This paper develops a perfect basis theory for quantum Borcherds-Bozec algebras and their modules, establishing isomorphisms between perfect graphs and crystal bases, advancing the understanding of their algebraic and combinatorial structures.

Contribution

It introduces the perfect basis theory for quantum Borcherds-Bozec algebras and proves the isomorphism of their perfect graphs with crystal bases.

Findings

Perfect basis theory is established for quantum Borcherds-Bozec algebras.

The perfect graphs of these bases are shown to be isomorphic to known crystal bases.

The results connect algebraic bases with combinatorial crystal structures.

Abstract

In this paper, we develop the perfect basis theory for quantum Borcherds-Bozec algebras $U_{q}(\mathfrak g)$ and their irreducible highest weight modules $V(\lambda)$. We show that the lower perfect graph (resp. upper perfect graph) of every lower perfect basis (resp. upper perfect basis) of $U_{q}^{-}(\mathfrak g)$ (resp. $V(\lambda)$) is isomorphic to the crystal $B(\infty)$ (resp. $B(\lambda)$).