论文标题
量子簇代数的泊松结构和第二个量化
Poisson structure and second quantization of quantum cluster algebras
论文作者
论文摘要
以群集代数兼容的泊松结构在其量化中起关键作用的现象（即量子群集代数），我们引入了量子群集代数的第二个量化，这意味着量子群集代数的第二个量化，这意味着量子群集群集Algebra的兼容Poisson结构之间的对应关系。基于此观察结果，我们发现量子群集代数具有双量子群集代数，因此它们的第二个量化基本上是相同的。 例如，我们在\ s5.2.1中给出了$ fun _ {\ mathbb c}（sl_ {q}（2））$的第二个量化的群集代数$ a_ {p，q}（s sl（2））$ in \ s5.2.1中，并表明它是一个非琐碎的量化量化，这可能是一个非琐碎的量化，这可能是一个平行量化的两个参数量化的量子。此外，我们获得了具有具有非平凡次数量化的系数的一类量子群集代数。它的一种特殊类型是量子群集代数，其几乎主要系数具有额外的条件。 最后，我们证明没有系数的量子簇代数的兼容泊松结构始终是局部标准的泊松结构。之后，结果表明，没有系数的量子群集代数的第二个量化实际上是微不足道的。
Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the second quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its secondly quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses dual quantum cluster algebras such that their second quantization is essentially the same. As an example, we give the secondly quantized cluster algebra $A_{p,q}(SL(2))$ of $Fun_{\mathbb C}(SL_{q}(2))$ in \S5.2.1 and show that it is a non-trivial second quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. Furthermore, we obtain a class of quantum cluster algebras with coefficients which possess a non-trivial second quantization. Its one special kind is quantum cluster algebras with almost principal coefficients with an additional condition. Finally, we prove that the compatible Poisson structures of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the second quantization of a quantum cluster algebra without coefficients is in fact trivial.
