Combinatorics and Arithmetic for Physics

Crystal Structure of Localized Quantum Unipotent Coordinate Category

21 nov. 2024, 14:00

50m

Centre de conférences Marilyn et James Simons (IHES)

Orateur

Toshiki Nakashima (Sophia University, Tokyo)

Description

For a monoidal category $\mathcal{T}$, if there exists a "real commuting family $(C_i,R_{C_i},{\varphi }_i{)}_{i\in I}$", we can define a localization $\tilde{\mathcal{T}}$ of $\mathcal{T}$ by $(C_i,R_{C_i},{\varphi }_i{)}_{i\in I}$.
Let $R=R(\mathfrak{g})$ be the quiver Hecke algebra(=KLR algebra) associated with a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and ${\mathcal{C}}_w$ 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 $(C_i,R_{C_i},{\varphi }_i{)}_{i\in I}$. Thus, we get its localization ${\tilde{\mathcal{C}}}_w$, 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 ${\tilde{\mathcal{C}}}_{w_0}=\widetilde{R\text{-gmod}}$ holds a crystal structure and is isomorphic to the cellular crystal ${\mathbb{B}}_{i_1\dots i_N}$ where $i_1\dots i_N$ 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 ${\tilde{\mathcal{C}}}_w$ also holds a crystal structure, and it is isomorphic to the cellular crystal ${\mathbb{B}}_{i_1\dots i_m}$ associated with a reduced word $i_1\cdots i_m$ of $w$.

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