We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin-Zelevinsky twist map is constructed and investigated for all double Bruhat cells. The results are valid over base fields of arbitrary characteristic and the deformation parameter is only assumed to be a non-root of unity.

中文翻译：

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Berenstein-Zelevinsky 量子簇代数猜想

我们证明了 Berenstein-Zelevinsky 猜想，即所有有限维简单代数群的双 Bruhat 单元的量化坐标环承认具有 [4] 指定的初始种子的量子簇代数结构。我们进一步证明了相应的上量子簇代数与构造的量子簇代数重合，并表现出大量的显式量子种子。在此过程中，从非对易 UFD 的角度对量子双 Bruhat 细胞的性质进行了详细研究，并为所有双 Bruhat 细胞构建和研究了 Fomin-Zelevinsky 扭曲图的量子模拟。结果在任意特征的基场上都是有效的，并且变形参数仅被假定为非单位根。