论文标题
量子仿射代数和单体分类的类别
Categories over quantum affine algebras and monoidal categorification
论文作者
论文摘要
令$ u_q'（\ Mathfrak {g}）$为未介绍的仿射$ ade $ type的量子仿射代数，而$ \ mathcal {c} _ {\ Mathfrak {\ mathfrak {g}}}^0 $对于合适的无限序列$ \ wideHat {w} _0 = \ cdots s_ {i _ { - 1}} s_ {i_0} s_ {i_1} s_ {i_1} \ cdots $简单反思的$ $ \ MATHCAL {C} _ {\ MATHFRAK {g}}^0 $ for ALL $ A \ LE B \ in \ MATHBB {Z} \ SQCUP \ sqcup \ {\ PM \ pm \ infty \} $。与$ [a，b] $中的一个间隔的特定链条$ \ mathfrak {c} $相关联，我们构建了一个真正的简单通勤家庭$ m（\ m athfrak {c}）$ in $ \ \ \ m athcal {c} _ {c} _ {\ mathfrak {\ mathfrak {g}}^{a，b]^{[a，b]} $ kirillov，类别$ \ MATHCAL {C} _ {\ MATHFRAK {g}}}^{[a，b]} $提供了群集algebra $ k（\ Mathcal {C} _ {\ Mathfrak {\ Mathfrak {\ Mathfrak {g}}}}^^a，B] [a，b] [a，b] [a，B] $，其设置的clluce $ [M（\ Mathfrak {C}）] $。特别是，该结果给出了$ \ Mathcal {C} _ {\ Mathfrak {\ Mathfrak {g}}^ - $ $ hernandez-leclerc的肯定答案$ \ mathcal {c} _ {\ Mathfrak {g}}^0 $，因为它是$ \ Mathcal {C} _ {\ Mathfrak {\ Mathfrak {g}}}^{ - [ - \ \ infty，\ infty，\ infty]} $。
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $\mathcal{C}_{\mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q'(\mathfrak{g})$-modules. For a suitable infinite sequence $\widehat{w}_0= \cdots s_{i_{-1}}s_{i_0}s_{i_1} \cdots$ of simple reflections, we introduce subcategories $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ of $\mathcal{C}_{\mathfrak{g}}^0$ for all $a \le b \in \mathbb{Z}\sqcup\{ \pm \infty \}$. Associated with a certain chain $\mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(\mathfrak{C})$ in $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(\mathcal{C}_{\mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(\mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $\mathcal{C}_{\mathfrak{g}}^-$ by Hernandez-Leclerc since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,0]}$, and is also applicable to $\mathcal{C}_{\mathfrak{g}}^0$ since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,\infty]}$.
