Realization of the crystal bases of level-zero representations of quantum affine algebras as algebraic cycles

在（数学）物理学的各个领域，如粒子物理学、弦理论和统计力学，仿射李代数作为自然对称出现;量子仿射代数被引入作为仿射李代数的通用包络代数的q-变形（或量子变形）。向量空间上的线性作用）在研究粒子或弦的状态方面非常有用。我们的主要研究结果是用凸多面体的形式对Verma模的晶体基（即，最普遍的最高权模）的A型量子仿射代数。

In various areas of (mathematical) physics, such as particle physics, string theory, and statistical mechanics, affine Lie algebras appear as a natural symmetry; a quantum affine algebra is introduced as a q-deformations (or a quantum deformation) of the universal enveloping algebra of an affine Lie algebra.The study of representations (i.e., linear actions on vector spaces) of quantum affine algebras are very useful in examining the states of particles or strings.The main result of our research is an explicit combinatorial description, in terms of convex polytopes, of the crystal bases of Verma modules (i.e., the most universal highest weight modules) for type A quantum affine algebras.