Graded quiver varieties, quantum cluster algebras and dual canonical basis
Graded quiver varieties, quantum cluster algebras and dual canonical basis
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DOI：
10.1016/j.aim.2014.05.014
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发表时间：
2012-05
影响因子：
1.7
通讯作者：
Yoshiyuki Kimura;Fan Qin
中科院分区：
文献类型：
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作者：
Yoshiyuki Kimura;Fan Qin
受到 Nakajima 之前工作的启发，我们考虑了非循环分级箭袋品种上的反常滑轮，并从表示理论的角度研究了傅立叶-佐藤-德利涅变换。我们获得了具有特定系数的非循环量子簇代数的变形幺半群分类。特别是，只要（量子）簇代数中存在无环种子，（量子）正性猜想就会得到验证。在论文的第二部分中，我们引入了新的量子化，并表明我们设置中的所有量子簇单项式都属于相应量子单能子群的双正则基。这一结果概括了 Lampe 和 Hernandez-Leclerc 之前的工作，从 Kronecker 和 Dynkin 颤动情况到非循环情况。本文的傅里叶变换部分为第二作者的论文提供了关键的输入，他在该论文中构造了具有任意系数和量化的非循环量子簇代数的基础。
Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier–Sato–Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra.In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez–Leclerc from the Kronecker and Dynkin quiver case to the acyclic case.The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.