Appears in collection : Combinatorics and Arithmetic for Physics: special days 2023

For a monoidal category $(\mathcal T ,\circ)$, if there exists a “real commuting family $(C_i, R_{C_i}, \phi_i)_{i\in I}$ “, we can define a localization $\tilde\mathcal T$ 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 simple Lie algebra $\mathfrak g$ and $R-\mathrm{gmod}$ the category of finite-dimensional graded $R$-modules, 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 $\tilde R$-gmod. It has been shown that $R$-gmod categorifies the unipotent quantum coordinate ring $\mathcal A_q(\mathfrak g)$, that is, the Grothendieck ring $\mathcal K(R-\mathrm{gmod})$ is isomorphic to $\mathcal A_q(\mathfrak g)$. For the localized category $\tilde R-\mathrm{gmod}$, its Grothendieck ring $\mathcal K(\tilde R-\mathrm{gmod})$ defines the localized (unipotent) quantum coordinate ring $\widetilde{\mathcal A_q(\mathfrak g)}$.

We shall give a certain crystal structure on the set of self-dual simple objects $\mathbb B(\tilde R-\mathrm{gmod})$ in $\tilde R-\mathrm{gmod}$. We also give the isomorphism of crystals from $\mathbb B(\tilde R-\mathrm{gmod})$ to the cellular crystal $\mathbb B_i = B_i_1 \otimes \dots \otimes B_i_N$ for an arbitrary reduced word $\mathrm{i} = i_1 \dots i_N$ of the longest Weyl group element. This result can be seen as a localized version for the categorification of the crystal base $B(\infty)$ for the subalgebra $U^−_q (\mathfrak g)(\cong \mathcal A_q(\mathfrak g))$ of the quantum algebra $U_q(\mathfrak g)$, given by Lauda-Vazirani.