Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases
Twist Automorphisms on Quantum Unipotent Cells and Dual Canonical Bases
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DOI：
10.1093/imrn/rnz040
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发表时间：
2017-01
影响因子：
1
通讯作者：
Yoshiyuki Kimura;Hironori Oya
中科院分区：
文献类型：
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作者：
Yoshiyuki Kimura;Hironori Oya
本文构造了量子单幂元胞上的twist自同构，它是单幂元胞上的Berenstein-Fomin-Zelevinsky twist自同构的量子类似.我们证明了这些量子扭曲自同构保持量子单能胞的对偶正则基。此外，我们证明了在对称情形下，预投射代数的表示的量子扭自同构是由合合函子描述的。这是盖勒-勒克莱尔-施勒厄描述的量子类比，盖勒-勒克莱尔-施勒厄的结果在我们的证明中是必不可少的。因此，我们证明了量子扭曲自同构与量子簇单项式是相容的。还验证了特定量子扭曲自同构的6-周期性。
In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein–Fomin–Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells. Moreover, we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Geiß–Leclerc–Schröer’s description, and Geiß–Leclerc–Schröer’s results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.