Semi-infinite Lakshmibai--Seshadri paths and level-zero extremal weight modules over twisted quantum affine algebras

TL;DR

This paper introduces semi-infinite Lakshmibai--Seshadri paths to model crystal bases of level-zero extremal weight modules over twisted quantum affine algebras, establishing an isomorphism between these paths and the modules' crystal bases.

Contribution

It constructs semi-infinite LS paths for level-zero weights and proves their crystal structure is isomorphic to the crystal basis of the corresponding extremal weight modules.

Findings

Semi-infinite LS paths are isomorphic to crystal bases of extremal modules.

Established a new combinatorial model for level-zero extremal weight modules.

Extended the theory to twisted quantum affine algebras.

Abstract

In this paper, we study level-zero extremal weight modules over twisted quantum affine algebras. To this end, we introduce semi-infinite Lakshmibai--Seshadri paths associated with a level-zero dominant integral weight $\lambda$. We then show that the set $\tfrac{\infty}{2}\mathrm{LS}(\lambda)$ of semi-infinite LS paths of shape $\lambda$ is isomorphic, as a crystal, to the crystal basis $\mathcal{B}(\lambda)$ of the corresponding level-zero extremal weight module $V(\lambda)$.