辫子簇的幺范畴化与量子化
Monoidal Categorification and Quantization of Braid Varieties
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中文总结 AI 辅助
针对单链型单连通代数群的正辫子字，本文构造辫子簇量子簇代数的幺范畴化，将其与局域化量子环等同，验证特殊化同态恢复原簇代数，还建立了双弦簇变量的量子网格子式与广义量子T-系统。
中文摘要 AI 辅助
设"G"为单连通的单链型单代数群。对每个正辫子字"β"，以及附属于"Q"-数据的完全强对偶数据，我们构造了辫子簇"X(β)"的量子簇代数的显式基幺范畴化。我们将该代数同时等同于局域化量子格罗滕迪克环和玻色子扩张代数的局域化≥1阶子代数。在这些等同关系下，量子簇单项式同时对应实单模和归一化全局基元。我们构造了"q^{1/2}=1"处的特殊化同态并证明其恢复"CC[X(β)]"；特别地，所得积分形式是平坦量子形变。我们还给出了双弦附簇变量的内在Lusztig参数，识别了对应的量子网格子式，并建立了广义量子"T"-系统。
英文摘要
Let \(G\) be a simple and simply connected algebraic group of simply-laced type. For each positive braid word \(β\), and for the complete strong duality datum attached to a \(Q\)-datum, we construct an explicit based monoidal categorification of the quantum cluster algebra of the braid variety \(X(β)\). We identify this algebra with both a localized quantum Grothendieck ring and a localized level-\(\geq1\) subalgebra of the bosonic extension algebra. Under these identifications, quantum cluster monomials correspond simultaneously to real simple modules and normalized global basis elements. We construct the specialization homomorphism at \(q^{1/2}=1\) and prove that it recovers \(\CC[X(β)]\); in particular, the resulting integral form is a flat quantum deformation. We also give intrinsic Lusztig parameters for cluster variables attached to double strings, identify the corresponding quantum grid minors, and establish generalized quantum \(T\)-systems.
发表机构
- Harbin Engineering University（哈尔滨工程大学）
- Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）
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