Braid group actions of quantum Borcherds-Bozec algebras

TL;DR

This paper constructs Lusztig symmetries for quantum Borcherds-Bozec algebras, establishing a braid group action that generalizes symmetries in quantum groups with applications to their representations.

Contribution

It introduces Lusztig symmetries for quantum Borcherds-Bozec algebras and proves they satisfy braid group relations, extending the theory of quantum group symmetries.

Findings

Lusztig symmetries constructed for quantum Borcherds-Bozec algebras

Symmetries satisfy braid group relations

Provides a new framework for algebraic symmetries in quantum algebras

Abstract

In this paper, we construct the Lusztig symmetries for quantum Borcherds-Bozec algebra $U_q(\mathscr g)$ and its weight module $M\in \mathcal O$, on which the generators with real indices of $U_q(\mathscr g)$ act nilpotently. We show that these symmetries satisfy the defining relations of the braid group, associated to the Weyl group $W$ of $U_q(\mathscr g)$, which gives a braid group action.