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On the Drinfeld coproduct 关于 Drinfeld 共积
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-03-26 DOI: 10.4310/pamq.2024.v20.n1.a6
Ilaria Damiani
{"title":"On the Drinfeld coproduct","authors":"Ilaria Damiani","doi":"10.4310/pamq.2024.v20.n1.a6","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a6","url":null,"abstract":"This paper provides a construction of the Drinfeld coproduct $Delta_v$ on an affine quantum Kac–Moody algebra or on a quantum affinization $mathcal{U}$ through the exponentials of some locally nilpotent derivations, thus proving that this “coproduct” with values in a suitable completion of $mathcal{U} oplus mathcal{U}$ is well defined. For the affine quantum algebras, $Delta_v$ is also obtained as “$t$-equivariant limit” of the Drinfeld–Jimbo coproduct $Delta$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313834","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 $1$-dimensional formal group over the prismatization of $operatorname{Spf}:mathbb{Z}_p$ $operatorname{Spf}:mathbb{Z}_p$棱镜化上的1$维形式群
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-03-26 DOI: 10.4310/pamq.2024.v20.n1.a7
Vladimir Drinfeld
{"title":"A $1$-dimensional formal group over the prismatization of $operatorname{Spf}:mathbb{Z}_p$","authors":"Vladimir Drinfeld","doi":"10.4310/pamq.2024.v20.n1.a7","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a7","url":null,"abstract":"Let $Sigma$ denote the prismatization of $operatorname{Spf}:mathbb{Z}_p$. The multiplicative group over $Sigma$ maps to the prismatization of $mathbb{G}_m times operatorname{Spf}:mathbb{Z}_p$. We prove that the kernel of this map is the Cartier dual of some $1$-dimensional formal group over $Sigma$. We obtain some results about this formal group (e.g., we describe its Lie algebra). We give a very explicit description of the pullback of the formal group to the quotient stack $Q/mathbb{Z}^times_p$, where $Q$ is the $q$-de Rham prism.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313602","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
The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits 与刚性零势轨道相关的无多重性原始理想数
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-03-26 DOI: 10.4310/pamq.2024.v20.n1.a12
Alexander Premet, David I. Stewart
{"title":"The number of multiplicity-free primitive ideals associated with the rigid nilpotent orbits","authors":"Alexander Premet, David I. Stewart","doi":"10.4310/pamq.2024.v20.n1.a12","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a12","url":null,"abstract":"Let $G$ be a simple algebraic group defined over $mathbb{C}$ and let $e$ be a rigid nilpotent element in $g = operatorname{Lie} (G)$. In this paper we prove that the finite $W$-algebra $U(mathfrak{g}, e)$ admits either one or two $1$-dimensional representations. Thanks to the results obtained earlier this boils down to showing that the finite $W$-algebras associated with the rigid nilpotent orbits of dimension 202 in the Lie algebras of type $E_8$ admit exactly two 1‑dimensional representations. As a corollary, we complete the description of the multiplicity-free primitive ideals of $U(mathfrak{g})$ associated with the rigid nilpotent $G$-orbits of $mathfrak{g}$. At the end of the paper, we apply our results to enumerate the small irreducible representations of the related reduced enveloping algebras.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313847","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
Congruences for Hasse-Witt matrices and solutions of $p$-adic KZ equations 哈塞-维特矩阵的同余式和 p$-adic KZ方程的解
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-03-26 DOI: 10.4310/pamq.2024.v20.n1.a13
Alexander Varchenko, Wadim Zudilin
{"title":"Congruences for Hasse-Witt matrices and solutions of $p$-adic KZ equations","authors":"Alexander Varchenko, Wadim Zudilin","doi":"10.4310/pamq.2024.v20.n1.a13","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a13","url":null,"abstract":"We prove general Dwork-type congruences for Hasse–Witt matrices attached to tuples of Laurent polynomials.We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of Knizhnik–Zamolodchikov (KZ) equations, the solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application we show that the $p$-adic KZ connection associated with the family of hyperelliptic curves $y^2 = (t - z_1) dotsc (t - z_{2g+1})$ has an invariant subbundle of rank $g$. Notice that the corresponding complex KZ connection has no nontrivial subbundles due to the irreducibility of its monodromy representation.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140313854","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
Seshadri stratifications and Schubert varieties: a geometric construction of a standard monomial theory 塞沙德里分层与舒伯特变体：标准单项式理论的几何构造
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-03-26 DOI: 10.4310/pamq.2024.v20.n1.a5
Rocco Chirivì, Xin Fang, Peter Littelmann
{"title":"Seshadri stratifications and Schubert varieties: a geometric construction of a standard monomial theory","authors":"Rocco Chirivì, Xin Fang, Peter Littelmann","doi":"10.4310/pamq.2024.v20.n1.a5","DOIUrl":"https://doi.org/10.4310/pamq.2024.v20.n1.a5","url":null,"abstract":"A standard monomial theory for Schubert varieties is constructed exploiting (1) the geometry of the Seshadri stratifications of Schubert varieties by their Schubert subvarieties and (2) the combinatorial LS‑path character formula for Demazure modules. The general theory of Seshadri stratifications is improved by using arbitrary linearization of the partial order and by weakening the definition of balanced stratification.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140314087","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
On the paving size of a subfactor 关于子因子的铺设尺寸
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-01-30 DOI: 10.4310/pamq.2023.v19.n5.a6
Sora Popin
{"title":"On the paving size of a subfactor","authors":"Sora Popin","doi":"10.4310/pamq.2023.v19.n5.a6","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a6","url":null,"abstract":"Given an inclusion of $mathrm{II}_1$ factors $N subset M$ with finite Jones index, $[M:N] lt infty$, we prove that for any $F subset M$ finite and $varepsilon gt 0$, there exists a partition of $1$ with $r leq lceil 16 varepsilon^{-2} rceil cdot {lceil 4 [M:N] varepsilon}^{-2} rceil$ projections $p_1, dotsc , p_r in N$ such that ${lVert sum^r_{i=1} p_i xp_i - E_{N^prime cap M} (x) rVert} leq varepsilon {lVert x - E_{N^prime cap M} (x) rVert}, forall x in F$ (where $lceil beta rceil$ denotes the least integer $geq beta$). We consider a series of related invariants for $N subset M$, generically called <i>paving size.</i>","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647274","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
Covering complexity, scalar curvature, and quantitative $K$-theory 覆盖复杂性、标量曲率和定量 $K$ 理论
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-01-30 DOI: 10.4310/pamq.2023.v19.n6.a13
Hao Guo, Guoliang Yu
{"title":"Covering complexity, scalar curvature, and quantitative $K$-theory","authors":"Hao Guo, Guoliang Yu","doi":"10.4310/pamq.2023.v19.n6.a13","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a13","url":null,"abstract":"We establish a relationship between a certain notion of covering complexity of a Riemannian spin manifold and positive lower bounds on its scalar curvature. This makes use of a pairing between quantitative operator $K$-theory and Lipschitz topological $K$-theory, combined with an earlier vanishing theorem for the quantitative higher index.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656191","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
The mathematical work of H. Blaine Lawson, Jr. 小布莱恩-劳森（H. Blaine Lawson, Jr.
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-01-30 DOI: 10.4310/pamq.2023.v19.n6.a1
Robert Bryant, Jeff Cheeger, Paulo Lima-Filho, Jonathan Rosenberg, Brian White
{"title":"The mathematical work of H. Blaine Lawson, Jr.","authors":"Robert Bryant, Jeff Cheeger, Paulo Lima-Filho, Jonathan Rosenberg, Brian White","doi":"10.4310/pamq.2023.v19.n6.a1","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a1","url":null,"abstract":"In this article, we celebrate the 80th birthday and remarkable career of H. Blaine Lawson, Jr. For more than half a century, Lawson has been a leading figure in mathematics. His work, a masterful combination of differential geometry, topology, algebraic geometry and analysis, has been enormously influential. He has made numerous fundamental contributions to diverse areas involving these subjects. He can be seen as a true “Renaissance man,” combining profound mathematical insight with a remarkable talent for expressing his discoveries with elegance and clarity. Roughly speaking, Lawson has changed the focus of his research every 10 to 15 years, in each instance, illuminating new fields of study with his unique insight and perspective. In the narrative that follows, we will endeavor, albeit with notable omissions, to showcase his most significant achievements. The order of presentation is essentially chronological. We will conclude with a concise overview of his highly influential expository work.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656187","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
The descendant colored Jones polynomials 后代彩色琼斯多项式
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-01-30 DOI: 10.4310/pamq.2023.v19.n5.a2
Stavros Garoufalidis, Rinat Kashaev
{"title":"The descendant colored Jones polynomials","authors":"Stavros Garoufalidis, Rinat Kashaev","doi":"10.4310/pamq.2023.v19.n5.a2","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a2","url":null,"abstract":"We discuss two realizations of the colored Jones polynomials of a knot, one appearing in an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another formulated in terms of a sequence of elements of the Habiro ring appearing in recent work of D. Zagier and the first author on the Refined Quantum Modularity Conjecture.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647281","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
The generality of closed $mathrm{G}_2$ solitons 封闭 $mathrm{G}_2$ 孤子的普遍性
IF 0.7 4区 数学
Pure and Applied Mathematics Quarterly Pub Date : 2024-01-30 DOI: 10.4310/pamq.2023.v19.n6.a8
Robert L. Bryant
{"title":"The generality of closed $mathrm{G}_2$ solitons","authors":"Robert L. Bryant","doi":"10.4310/pamq.2023.v19.n6.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a8","url":null,"abstract":"The local generality of the space of solitons for the Laplacian flow of closed $mathrm{G}_2$-structures is analyzed, and it is shown that the germs of such structures depend, up to diffeomorphism, on $16$ functions of $6$ variables (in the sense of É. Cartan). The method is to construct a natural exterior differential system whose integral manifolds describe such solitons and to show that it is involutive in Cartan’s sense, so that Cartan–Kähler theory can be applied. Meanwhile, it turns out that, for the more special case of gradient solitons, the natural exterior differential system is not involutive, and the generality of these structures remains a mystery.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656368","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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