Orateur
Description
For a monoidal category $(\tau, \circ)$, if there exists a“real commuting family $(C_i, R_{Ci}, \phi_i)_{i\epsilon I}$”, we can define a localization $\tilde{\tau}$ of $\tau$ by $(C_i, R_{Ci}, \phi_i)_{i\epsilon I}$. Let $R = R(\mathfrak{g})$ be the quiver Hecke algebra (=KLR algebra) associated with a simple Lie algebra $\mathfrak{g}$ and $R$-gmod the category of finite-dimensional graded $R$-modules, which is a monoidal category with a real commuting family $(C_i, R_{Ci}, \phi_i)_{i\epsilon I}$. Thus, we get its localization $\tilde{R}$-gmod. It has been shown that $R$-gmod categorifies the unipotent quantum coordinate ring $A_q(\mathfrak{g})$, that is, the Grothendieck ring $K$($R$-gmod) is isomorphic to $A_q(\mathfrak{g})$. For the localized category $\tilde{R}$-gmod, its Grothendieck ring $K(\tilde{R}-gmod)$ defines the localized (unipotent) quantum coordinate ring $\widetilde{A_q(\mathfrak{g})}$.
We shall give a certain crystal structure on the set of self-dual simple objects $\mathbb{B}(\tilde{R}-gmod)$ in $\tilde{R}$-gmod-gmod. We also give the isomorphism of crystals from $\mathbb{B}(\tilde{R}-gmod)$ to the cellular crystal $\mathbb{B}_{\textrm{i}}=B_{i_{1}}\otimes \cdot \cdot \cdot B_{i_{N}}$ for an arbitrary reduced word $\mathbf{i}=i_{1}\cdot\cdot\cdot i_{N}$ of the longest Weyl group element. This result can be seen as a localized version for the categorification of the crystal base $B(∞)$ for the subalgebra $U_{q}^{-}(\mathfrak{g})(\cong A_{q}(\mathfrak{g}))$ of the quantum algebra $U_{q}(\mathfrak{g})$, given by Lauda-Vazirani.
