Kirillov–Reshetikhin crystal-base conjecture for quantum affine algebras

Let $\mathfrak{g}$ be one of the affine Lie algebras considered in the paper, let $I_0$ be its classical index set, and fix $i\in I_0$ and $s\in\mathbb{Z}_{\ge 1}$. For the Drinfeld polynomials specified by the type of $\mathfrak{g}$, let $W^{(i)}_s(\zeta)$ denote the corresponding Kirillov–Reshetikhin module over $U_q^{\prime}(\mathfrak{g})$. A crystal base is a crystal basis of such a module, and a perfect crystal is one satisfying the usual perfectness conditions.

Kirillov–Reshetikhin crystal-base conjecture. For some $\zeta^{(i)}_s\in\mathbb{C}(q)^\times$, the module $W^{(i)}_s(\zeta^{(i)}_s)$ has a crystal base $\mathcal{B}^{i,s}$ that is a perfect $U_q^{\prime}(\mathfrak{g})$-crystal of level $s$.

This conjecture concerns the existence of crystal bases for the important finite-dimensional Kirillov–Reshetikhin modules. The paper assumes it for simply-laced quantum affine algebras and constructs perfect crystals for corresponding twisted algebras; its general resolution is not supplied here.