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Internal symmetry of the $L_{leqslant 3}$ algebra arising from a Lie pair 由李对产生的$L_{leqslant 3}$代数的内部对称性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a16
Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He
{"title":"Internal symmetry of the $L_{leqslant 3}$ algebra arising from a Lie pair","authors":"Dadi Ni, Jiahao Cheng, Zhuo Chen, Chen He","doi":"10.4310/pamq.2023.v19.n4.a16","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a16","url":null,"abstract":"$defDerL{operatorname{Der}(L)}$A Lie pair is an inclusion $A$ to $L$ of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stiénon, and Xu introduced a canonical $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$ whose unary bracket is the Chevalley–Eilenberg differential arising from every Lie pair $(L,A)$. In this note, we prove that to such a Lie pair there is an associated Lie algebra action by $operatorname{Der}(L)$ on the $L_{leqslant 3}$ algebra $Gamma (wedge^bullet A^vee otimes L/A)$. Here $DerL$ is the space of derivations on the Lie algebroid $L$, or infinitesimal automorphisms of $L$. The said action gives rise to a larger scope of gauge equivalences of Maurer–Cartan elements in $Gamma (wedge^bullet A^vee otimes L/A)$, and for this reason we elect to call the $DerL$-action internal symmetry of $Gamma (wedge^bullet A^vee otimes L/A)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"263 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Which Hessenberg varieties are GKM? 哪些海森伯格变种是GKM?
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a8
Rebecca Goldin, Julianna Tymoczko
{"title":"Which Hessenberg varieties are GKM?","authors":"Rebecca Goldin, Julianna Tymoczko","doi":"10.4310/pamq.2023.v19.n4.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a8","url":null,"abstract":"Hessenberg varieties $mathcal{H}(X,H)$ form a class of subvarieties of the flag variety $G/B$, parameterized by an operator $X$ and certain subspaces $H$ of the Lie algebra of $G$. We identify several families of Hessenberg varieties in type $A_{n-1}$ that are $T$-stable subvarieties of $G/B$, as well as families that are invariant under a subtorus $K$ of $T$. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the $T$-stable Hessenberg varieties, we identify several that are <i>GKM spaces</i>, meaning $T$ acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type $A_{n-1}$ and in general Lie type.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"195 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Dedekind sums via Atiyah–Bott–Lefschetz
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-11-20 DOI: 10.4310/pamq.2023.v19.n4.a3
Ana Cannas da Silva
{"title":"Dedekind sums via Atiyah–Bott–Lefschetz","authors":"Ana Cannas da Silva","doi":"10.4310/pamq.2023.v19.n4.a3","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a3","url":null,"abstract":"This paper, written for differential geometers, shows how to deduce the reciprocity laws of Dedekind and Rademacher, as well as $n$-dimensional generalizations of these, from the Atiyah–Bott–Lefschetz formula, by applying this formula to appropriate elliptic complexes on weighted projective spaces.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"194 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Hilbert reciprocity using $K$-theory localization 利用K理论定位的希尔伯特互易
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-04-07 DOI: 10.4310/pamq.2023.v19.n2.a1
Oliver Braunling
{"title":"Hilbert reciprocity using $K$-theory localization","authors":"Oliver Braunling","doi":"10.4310/pamq.2023.v19.n2.a1","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a1","url":null,"abstract":"Usually the boundary map in $K$-theory localization only gives the tame symbol at $K_2$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at $p=2$. Our idea is to pinch singularities near the ramification locus. This fattens up $K$-theory and makes the wild symbol visible as a boundary map.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"198 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unitary matrix models, free fermions, and the giant graviton expansion 统一矩阵模型，自由费米子，和巨大的引力子膨胀
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-01-01 DOI: 10.4310/pamq.2023.v19.n1.a12
S. Murthy
{"title":"Unitary matrix models, free fermions, and the giant graviton expansion","authors":"S. Murthy","doi":"10.4310/pamq.2023.v19.n1.a12","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a12","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70527062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Indefinite theta series: the case of an $N$-gon 不定级数:$N$- on的情况
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-01-01 DOI: 10.4310/pamq.2023.v19.n1.a8
Jens Funke, S. Kudla
{"title":"Indefinite theta series: the case of an $N$-gon","authors":"Jens Funke, S. Kudla","doi":"10.4310/pamq.2023.v19.n1.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a8","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Sum expressions for $p$-adic Hecke $L$-functions of totally real fields 全实域的$p$-进Hecke $L$-函数的和表达式
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-01-01 DOI: 10.4310/pamq.2023.v19.n2.a7
Lu Zhao
{"title":"Sum expressions for $p$-adic Hecke $L$-functions of totally real fields","authors":"Lu Zhao","doi":"10.4310/pamq.2023.v19.n2.a7","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a7","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Künneth formulas for path homology 路径同调的第n个公式
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-01-01 DOI: 10.4310/pamq.2023.v19.n2.a10
Fang Li, Ting Yu
{"title":"Künneth formulas for path homology","authors":"Fang Li, Ting Yu","doi":"10.4310/pamq.2023.v19.n2.a10","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a10","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings 扭曲多正则束上的度量与扭曲正则环的有限生成
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-01-01 DOI: 10.4310/pamq.2023.v19.n2.a2
Bojie He, Xiangyu Zhou
{"title":"Metrics on twisted pluricanonical bundles and finite generation of twisted canonical rings","authors":"Bojie He, Xiangyu Zhou","doi":"10.4310/pamq.2023.v19.n2.a2","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n2.a2","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A proof of van der Waerden’s Conjecture on random Galois groups of polynomials 关于多项式随机伽罗瓦群的van der Waerden猜想的证明
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2023-01-01 DOI: 10.4310/pamq.2023.v19.n1.a3
M. Bhargava
{"title":"A proof of van der Waerden’s Conjecture on random Galois groups of polynomials","authors":"M. Bhargava","doi":"10.4310/pamq.2023.v19.n1.a3","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n1.a3","url":null,"abstract":"","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70526632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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