标题： 关于简单代数群的部分紧化量子cluster结构以及Berenstein--Zelevinsky猜想的完全证明 显示英文标题

标题： Partially compactified quantum cluster structures on simple algebraic groups and the full Berenstein--Zelevinsky conjecture

摘要： 在坐标环上部分紧化簇代数的构造是通过利用簇覆盖上的余维数 2 论证来处理的。 量子情形下与此类似的结果是非常期望的，但尚未找到。 本文提出了一种通用方法，用于从局部化的量子簇代数结构构造量化坐标环上的部分紧化量子簇代数结构。 作为一个应用，我们为每个连通且单连通的复简单代数群的量化坐标环构造了一个部分紧化量子簇代数结构。 在此过程中，我们完全解决了伯恩斯坦-泽尔廷猜想，即所有量子双Bruhat胞腔都具有由任意符号字索引的种子对应的量子簇代数结构，并证明了所有这些种子之间都可以通过突变相互连接。 显示英文摘要

摘要： The construction of partially compactified cluster algebras on coordinate rings is handled by using codimension 2 arguments on cluster covers. An analog of this in the quantum situation is highly desirable but has not been found yet. In this paper, we present a general method for the construction of partially compactified quantum cluster algebra structures on quantized coordinate rings from that of quantum cluster algebra structures on localizations. As an application, we construct a partially compactified quantum cluster algebra structure on the quantized coordinate ring of every connected, simply connected complex simple algebraic group. Along the way, we settle in full the Berenstein--Zelevinsky conjecture that all quantum double Bruhat cells have quantum cluster algebra structures associated to seeds indexed by arbitrary signed words, and prove that all such seeds are linked to each by mutations.