丛代数与几何晶体及其在量子群和代数群表示论中的应用

Quantum affine algebras form a class of infinite-dimensional quantum groups. We will study the equations satisfied by the q-characters of minimal affinizations of quantum affine algebras and show that these equations correspond to mutation equations of some cluster algebras. We will show that the minimal affinizations correspond to cluster variables of these cluster algebras. By using the equations satisfied by q-characters, we will give a new algorithm to compute the q-characters of minimal affinizations. As an application, we will give a formula for decomposing minimal affinizations of type G2 as Uq(g)-modules into irreducible Uq(g)-modules. Crystal bases and canonical bases of quantum groups were introduced by Kashiwara and Lusztig. Berenstein and Kazhdan introduced the corresponding concept geometric crystals for algebraic groups. We will try to find a new geometric crystal structure on B^k, where B is the Borel subgroup of an algebraic group G. In order to connect Poisson structures and group actions on algebraic varieties, we will introduce Poisson crystals. We will show that the category of Poisson crystals is a monoidal category.

量子仿射代数是一类无限维量子群，我们将研究量子仿射代数的极小仿射化模的q-特征满足的方程，然后将这些方程对应到一些丛代数的突变方程，将极小仿射化模对应于这些丛代数的丛变量。利用q-特征满足的方程，我们将给出计算极小仿射化模的q-特征的新算法。应用这些结果，我们将研究G2型极小仿射化模作为Uq(g)-模分解成不可约Uq(g)-模的公式。我们将研究量子仿射代数的比极小仿射化模范围更广的模。量子群的晶体基和典范基由Kashiwara以及Lusztig提出，Berenstein和Kazhdan对代数群提出了相应的几何晶体的概念。我们将研究B^k上的新的几何晶体结构，这里B是代数群G的Borel子群。为了将Poisson结构与群在代数簇上的作用联系起来，我们将提出Poisson晶体的概念。我们将证明Poisson晶体范畴是一个monoidal范畴。

我们研究了 A, B 型量子仿射代数的极小仿射化模，蛇模的 q-特征满足的方程。证明了极小仿射化模，素蛇模是 Hernandez 和 Leclerc 定义的丛代数的丛变量。我们得到了 G2 型 Uq(^g) 的极小仿射化模 (作为 Uq(g) 模) 分解成不可约 Uq(g) 模的分解公式。我们研究了 G2 型量子仿射代数的极小仿射化模满足的 M-系统的方程。项目组成员纪影丹研究了 R-unipotent 半群代数。项目发表了5篇Sci论文，完成了预期的目标。