Generalized cluster structures related to the Drinfeld double of GLn$GL_n$
Generalized cluster structures related to the Drinfeld double of GLn$GL_n$
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DOI：
10.1112/jlms.12542
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发表时间：
2020-04
期刊：
影响因子：
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通讯作者：
M. Gekhtman;M. Shapiro;A. Vainshtein
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文献类型：
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作者：
M. Gekhtman;M. Shapiro;A. Vainshtein
证明了在Gekhtman，Shapiro和Vainshtein(Int.数学课。Res.Note，2022年，将出现，arxiv：1912.00453)是完整的，并与双层上的标准泊松-李结构兼容。此外，我们还证明了当n=4$n=4$时，尽管它们具有相同的相容Poisson结构和相同的冻结变量集合，但这种结构不同于已知的Drinfeld对上的正则广义星团结构。进一步，我们证明了Gekhtman，Shapiro和Vainshtein(Int.数学课。Res.Note，2022年即将出现，Arxiv：1912.00453)具有类似的兼容性和完备性属性。
We prove that the regular generalized cluster structure on the Drinfeld double of GLn$GL_n$ constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for n=4$n=4$ this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties.