Block decomposition for quantum affine algebras by the associated simply-laced root system
Block decomposition for quantum affine algebras by the associated simply-laced root system
复制标题
通过相关的简单根系进行量子仿射代数的分块分解
DOI：
--
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
E. Park
中科院分区：
文献类型：
--
作者：
M. Kashiwara;Myungho Kim;Se;E. Park
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $\mathscr{C}_\mathfrak{g}$ be the category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules. We write $\mathscr{C}_\mathfrak{g}^0$ for the monoidal subcategory of $\mathscr{C}_\mathfrak{g}$ introduced by Hernandez-Leclerc. In this paper, we give the block decompositions of $\mathscr{C}_\mathfrak{g}$ and $\mathscr{C}_\mathfrak{g}^0$ for all untwisted and twisted quantum affine algebras by using the associated simply-laced finite type root system. We first define a certain abelian group $\mathcal{W}$ (resp. $\mathcal{W}_0$) arising from simple modules of $ \mathscr{C}_\mathfrak{g}$ (resp. $\mathscr{C}_\mathfrak{g}^0$) by using the invariant $\Lambda^\infty$ introduced in the previous work by the authors. The groups $\mathcal{W}$ and $\mathcal{W}_0$ have the subsets $\Delta$ and $\Delta_0$ determined by the fundamental representations in $ \mathscr{C}_\mathfrak{g}$ and $\mathscr{C}_\mathfrak{g}^0$ respectively. We prove that the pair $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, \Delta_0)$ is an irreducible simply-laced root system of finite type and the pair $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}, \Delta) $ is isomorphic to the direct sum of infinite copies of $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, \Delta_0)$ as a root system. We next show that there exist direct decompositions of $\mathscr{C}_\mathfrak{g}$ and $\mathscr{C}_\mathfrak{g}^0$ parameterized by elements of $\mathcal{W}$ and $\mathcal{W}_0$ respectively, and prove that these decompositions are their block decompositions.
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $\mathscr{C}_\mathfrak{g}$ be the category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules. We write $\mathscr{C}_\mathfrak{g}^0$ for the monoidal subcategory of $\mathscr{C}_\mathfrak{g}$ introduced by Hernandez-Leclerc. In this paper, we give the block decompositions of $\mathscr{C}_\mathfrak{g}$ and $\mathscr{C}_\mathfrak{g}^0$ for all untwisted and twisted quantum affine algebras by using the associated simply-laced finite type root system. We first define a certain abelian group $\mathcal{W}$ (resp. $\mathcal{W}_0$) arising from simple modules of $ \mathscr{C}_\mathfrak{g}$ (resp. $\mathscr{C}_\mathfrak{g}^0$) by using the invariant $\Lambda^\infty$ introduced in the previous work by the authors. The groups $\mathcal{W}$ and $\mathcal{W}_0$ have the subsets $\Delta$ and $\Delta_0$ determined by the fundamental representations in $ \mathscr{C}_\mathfrak{g}$ and $\mathscr{C}_\mathfrak{g}^0$ respectively. We prove that the pair $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, \Delta_0)$ is an irreducible simply-laced root system of finite type and the pair $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}, \Delta) $ is isomorphic to the direct sum of infinite copies of $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, \Delta_0)$ as a root system. We next show that there exist direct decompositions of $\mathscr{C}_\mathfrak{g}$ and $\mathscr{C}_\mathfrak{g}^0$ parameterized by elements of $\mathcal{W}$ and $\mathcal{W}_0$ respectively, and prove that these decompositions are their block decompositions.
影响因子：
3.9
作者：
H. Nakajima
通讯作者：
H. Nakajima