Quantum unipotent subgroup and Dual canonical basis
Quantum unipotent subgroup and Dual canonical basis
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DOI：
10.1215/21562261-1550976
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发表时间：
2010-10
影响因子：
0.6
通讯作者：
Yoshiyuki Kimura
中科院分区：
文献类型：
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作者：
Yoshiyuki Kimura
Geiss-Leclerc-Schroer定义了与Weyl群元相联系的酉子群的坐标环$C[N(W)]$上的簇代数结构，证明了簇单项式包含在Lusztig的对偶半标准基$S^*$中.我们给出了其结果的量子化设置，并提出了将量子簇代数与对偶正则基{B}^{UP}$联系起来的猜想。特别地，我们证明了${C}[N(W)]$的量子模拟$O_{q}[N(W)]$具有来自${B}^{up}$的导出基，它包含量子旗子式并且满足关于$O_{q}[N(W)]$的‘$Q$-中心’的因式分解性质。这将Caldero的结果从ADE情形推广到任意对称的Kac-Moody李代数。
Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical basis $S^{*}$. We give a set up for the quantization of their results and propose a conjecture which relates the quantum cluster algebras to the dual canonical basis ${B}^{up}$. In particular, we prove that the quantum analogue $O_{q}[N(w)]$ of ${C}[N(w)]$ has the induced basis from ${B}^{up}$, which contains quantum flag minors and satisfies a factorization property with respect to the `$q$-center' of $O_{q}[N(w)]$. This generalizes Caldero's results from ADE cases to an arbitary symmetrizable Kac-Moody Lie algebra.