标题： 表示箭图杨-巴克斯特代数的算法 显示英文标题

标题： Algorithms for representations of quiver Yangian algebras

摘要： 在本文中，我们旨在详细回顾构建箭图杨代数晶体表示的算法。 箭图杨代数被认为描述了BPS代数在环绕分次卡勒-丘三折的D膜系统中的BPS态上的作用。 这些代数的晶体模来源于相应三折的唐纳森-托马斯不变量的熔化晶体模型。 尽管这一主题最初是代数几何与有效超对称场论的交叉点，但等变分次作用大大简化了应用计算。 因此，该算法实现的唯一先决条件是线性代数。 借助任何符号计算系统，可以很容易地将其教授给机器。 此外，这些算法可以推广到环面和椭圆代数，并用于这些代数的各种数值实验。 我们通过$\mathsf{Y}(\mathfrak{sl}_2)$、$\mathsf{Y}(\widehat{\mathfrak{gl}}_{1})$和$\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$的简单情况来说明算法的应用。 显示英文摘要

摘要： In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm's implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of $\mathsf{Y}(\mathfrak{sl}_2)$, $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1})$ and $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$.