论文标题
编织小组在量子仿射代数的模块类别上行动
Braid group action on the module category of quantum affine algebras
论文作者
论文摘要
令$ \ mathfrak {g} _0 $为ADE类型的简单谎言代数，让$ u'_Q（\ Mathfrak {g}）$为相应的未介绍的量子仿射代数。我们表明，在Hernandez-leclerc的类别$ c _ {\ Mathfrak {g}}^0 $中，在Quantum grothendieck环$ k_t（\ Mathfrak {g}）$上存在辫子组$ b（\ mathfrak {g} _0）$的动作。专注于$ a_ {n-1} $的情况，我们构建了一个单型自动函数的家族$ \ {\ Mathscr {\ Mathscr {s} _i \} _ {i \ in \ in \ in \ Mathbb {z}} $ a intobalization $ t_n $ tyge a flite a flitiate a flunite-dimensional modules yecke yecive yecive yecive yecive yecive a $ a _ {\ infty} $。在Grothendieck ring $ k（t_n）$ $ t_n $和量子grothendieck环$ k_t（{a^{（1）} _ {n-1}}）$之间的同构中$ b（a_ {n-1}）$。我们研究了这些函子的进一步属性。
Let $\mathfrak{g}_0$ be a simple Lie algebra of type ADE and let $U'_q(\mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(\mathfrak{g}_0)$ on the quantum Grothendieck ring $K_t(\mathfrak{g})$ of Hernandez-Leclerc's category $C_{\mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors $\{\mathscr{S}_i\}_{i\in \mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t({A^{(1)}_{N-1}})$, the functors $\{\mathscr{S}_i\}_{1\le i\le N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.
