论文标题
高阶Kirillov-Reshetikhin模块，虚构模块和$ u_q（a_n^{（1）}）$的单体分类
Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$
论文作者
论文摘要
我们研究了量子仿射$ \ lie {sl} _ {n+1} $的量子模块的家族，其drinfeld多项式仅在Dynkin图的一个节点上支持。我们确定了这个家族中的所有主要模块，并证明了一个独特的分解定理。 Prime模块的Drinfeld多项式编码信息来自与$ M \ le n $相关的基本模块的张量张量产品的点。这些主要模块是Mukhin和Young研究的蛇模块的特殊类。我们将模块与Hernandez和Leclerc的工作联系起来，并定义了类别$ \ MATHSCR C^ - $的概括。这自然导致相应的粒度环的通胀概念。在上一节中，我们显示了（高阶）基里洛夫（Reshetikhin）模块的张量产物，其双重模块始终包含其Jordan-Holder系列中的假想模块，并为其Drinfeld多项式提供了明确的公式。加上\ cite {hl13a}的结果，这给出了群集变量产物的示例，而群集变量不在群集单元中。我们还讨论了我们的工作的联系与\ cite {lm18}的工作产生的示例。 最后，我们使用我们的方法为$ d_4 $类型的假想模块提供了一个家庭，这些模块不是由$ a_r $带有$ r \ le 3 $ in $ d_4 $的$ a_r $产生的。
We study the family of irreducible modules for quantum affine $\lie{sl}_{n+1}$ whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to $A_m$ with $m\le n$. These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category $\mathscr C^-$. This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov--Reshetikhin module with its dual always contains an imaginary module in its Jordan--Holder series and give an explicit formula for its Drinfeld polynomial. Together with the results of \cite{HL13a} this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of \cite{LM18}. Finally, we use our methods to give a family of imaginary modules in type $D_4$ which do not arise from an embedding of $A_r$ with $r\le 3$ in $D_4$.
