{"title":"𝐺𝐿_{𝑛}上的大量团簇结构","authors":"M. Gekhtman, M. Shapiro, A. Vainshtein","doi":"10.1090/memo/1486","DOIUrl":null,"url":null,"abstract":"We continue the study of multiple cluster structures in the rings of regular functions on \n\n \n \n G\n \n L\n n\n \n \n GL_n\n \n\n, \n\n \n \n S\n \n L\n n\n \n \n SL_n\n \n\n and \n\n \n \n M\n a\n \n t\n n\n \n \n Mat_n\n \n\n that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group \n\n \n \n G\n \n \\mathcal {G}\n \n\n corresponds to a cluster structure in \n\n \n \n \n O\n \n (\n \n G\n \n )\n \n \\mathcal {O}(\\mathcal {G})\n \n\n. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of \n\n \n \n A\n n\n \n A_n\n \n\n type, which includes all the previously known examples. Namely, we subdivide all possible \n\n \n \n A\n n\n \n A_n\n \n\n type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on \n\n \n \n S\n \n L\n n\n \n \n SL_n\n \n\n compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of \n\n \n \n S\n \n L\n n\n \n \n SL_n\n \n\n equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":"13 8","pages":""},"PeriodicalIF":4.6000,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A Plethora of Cluster Structures on 𝐺𝐿_{𝑛}\",\"authors\":\"M. Gekhtman, M. Shapiro, A. 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Namely, we subdivide all possible \\n\\n \\n \\n A\\n n\\n \\n A_n\\n \\n\\n type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on \\n\\n \\n \\n S\\n \\n L\\n n\\n \\n \\n SL_n\\n \\n\\n compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of \\n\\n \\n \\n S\\n \\n L\\n n\\n \\n \\n SL_n\\n \\n\\n equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. 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引用次数: 1
摘要
我们继续研究 G L n GL_n 、S L n SL_n 和 M a t n Mat_n 上与泊松里结构和泊松均质结构相容的正则函数环中的多重簇结构。根据我们最初的猜想,半简单复群 G \mathcal {G} 上 Poisson-Lie 结构的 Belavin-Drinfeld 分类中的每一类都对应于 O ( G ) \mathcal {O}(\mathcal {G}) 中的一个簇结构。在这里,我们为 A n A_n 类型的贝拉文-德林费尔德(BD)数据的一个大子集证明了这一猜想,其中包括所有之前已知的例子。也就是说,我们将所有可能的 A n A_n 类型 BD 数据细分为定向和非定向两种。我们进一步挑选出满足特定组合条件(我们称之为非周期性)的 BD 数据,并证明对于任何此类定向 BD 数据,都存在与相应的泊松李括号兼容的规则簇结构。事实上,我们将非周期性条件扩展到了成对的定向 BD 数据,并证明了一个更一般的结果,即在 S L n SL_n 上存在一个与泊松括号相容的正则簇结构,该泊松括号与 S L n SL_n 的两个副本的左右作用同质,而这两个副本配备了两个不同的泊松-李括号。类似的结果也适用于非周期性的无取向 BD 数据,但相应的规则簇结构的分析更为复杂,在此不再赘述。如果不满足非周期性条件,则必须用广义簇结构来替代兼容簇结构。我们将在今后的出版物中讨论这些情况。
We continue the study of multiple cluster structures in the rings of regular functions on
G
L
n
GL_n
,
S
L
n
SL_n
and
M
a
t
n
Mat_n
that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group
G
\mathcal {G}
corresponds to a cluster structure in
O
(
G
)
\mathcal {O}(\mathcal {G})
. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of
A
n
A_n
type, which includes all the previously known examples. Namely, we subdivide all possible
A
n
A_n
type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on
S
L
n
SL_n
compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of
S
L
n
SL_n
equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications.
期刊介绍:
ACS Applied Bio Materials is an interdisciplinary journal publishing original research covering all aspects of biomaterials and biointerfaces including and beyond the traditional biosensing, biomedical and therapeutic applications.
The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrates knowledge in the areas of materials, engineering, physics, bioscience, and chemistry into important bio applications. The journal is specifically interested in work that addresses the relationship between structure and function and assesses the stability and degradation of materials under relevant environmental and biological conditions.