Braid group action on extended crystals

In the paper, we prove that there exists a braid group action on the extended crystal $\widehat{B}(\infty)$ of finite type. The extended crystal $\widehat{B}(\infty)$ and its braid group action are investigated from the viewpoint of crystal similarity. We then interpret the braid group action on $\widehat{B}(\infty)$ in the Hernandez-Leclerc category $\mathscr{C}_\mathfrak{g}^0$.