Appears in collection : Combinatorics and Arithmetic for Physics - 2024

For a monoidal category ${\mathcal T}$, if there exists a "real commuting family $(C_i,R_{C_i},\phi_i)_{i\in I}$", we can define a localization $\widetilde {{\mathcal T}}$ of ${\mathcal T}$ by $(C_i,R_{C_i},\phi_i)_{i\in I}$. Let $R=R({\mathfrak g})$ be the quiver Hecke algebra(=KLR algebra) associated with a symmetrizable Kac-Moody Lie algebra ${\mathfrak g}$ and ${\mathscr C}_w$ the subcategory of $R$-gmod(=the category of graded finite-dimensional $R$-modules) associated with a Weyl group element $w$, which is a monoidal category with a real commuting family $(C_i,R_{C_i},\phi_i)_{i\in I}$. Thus, we get its localization $\widetilde{\mathscr C}_w$, which is called a ``localized quantum unipotent coordinate category" associated with $w$. In the former half of the talk, we shall present that for a (semi-)simple ${\mathfrak g}$ and the longest element $w_0$, the family of self-dual simple modules in $\widetilde{\mathscr C}_{w_0}=\widetilde{R\hbox{-gmod}}$ holds a crystal structure and is isomorphic to the cellular crystal ${\mathbb B}_{i_1\ldots i_N}$ where $i_1\ldots i_N$ is an arbitrary reduced word of $w_0$. Furthermore, in the latter half of the talk, the latest result, which is a joint work with M. Kashiwara, will be presented that for a general symmetrizable Kac-Moody Lie algebra ${\mathfrak g}$ and a general Weyl group element $w$, the family of self-dual simple modules in the localized category $\widetilde{\mathscr C}_{w}$ also holds a crystal structure, and it is isomorphic to the cellular crystal ${\mathbb B}_{i_1\ldots i_m}$ associated with a reduced word $i_1\cdots i_m$ of $w$.