Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a6
Ilaria Damiani
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 $Delta$.
{"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":"31 1","pages":""},"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}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a7
Vladimir Drinfeld
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.
{"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":"29 1","pages":""},"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}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a12
Alexander Premet, David I. Stewart
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.
{"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":"47 1","pages":""},"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}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a13
Alexander Varchenko, Wadim Zudilin
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.
{"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":"13 1","pages":""},"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}
Pub Date : 2024-03-26DOI: 10.4310/pamq.2024.v20.n1.a5
Rocco Chirivì, Xin Fang, Peter Littelmann
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.
利用（1）舒伯特子变量对舒伯特变量的塞沙德里分层的几何性质和（2）德马祖模的组合 LS 路径特征公式，构建了舒伯特变量的标准单项式理论。通过使用部分阶的任意线性化和弱化平衡分层的定义，改进了塞沙德里分层的一般理论。
{"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":"13 1","pages":""},"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}
Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a6
Sora Popin
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 paving size.
{"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":"20 1","pages":""},"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}
Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a13
Hao Guo, Guoliang Yu
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.
{"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":"38 1","pages":""},"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}
Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a1
Robert Bryant, Jeff Cheeger, Paulo Lima-Filho, Jonathan Rosenberg, Brian White
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.
{"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":"157 1","pages":""},"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}
Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a11
Dennis Sullivan
$defspinc{operatorname{spin}^mathrm{c}}$We give a 1965 era proof of the title assuming $M$ is $spinc$. The fact that any oriented four manifold is $spinc$ is a challenging result from 1995 whose interesting argument by Teichner–Vogt is analyzed and used in the appendix to show an analogous integral lift result about the top Wu class in $dim 4k$. This will be used in future work to study related complex structures on higher dimensional open manifolds.
{"title":"Any oriented non-closed connected $4$-manifold can be spread holomorphically over the complex projective plane minus a point","authors":"Dennis Sullivan","doi":"10.4310/pamq.2023.v19.n6.a11","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a11","url":null,"abstract":"$defspinc{operatorname{spin}^mathrm{c}}$We give a 1965 era proof of the title assuming $M$ is $spinc$. The fact that any oriented four manifold is $spinc$ is a challenging result from 1995 whose interesting argument by Teichner–Vogt is analyzed and used in the appendix to show an analogous integral lift result about the top Wu class in $dim 4k$. This will be used in future work to study related complex structures on higher dimensional open manifolds.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139659454","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}
Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a2
Stavros Garoufalidis, Rinat Kashaev
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.
我们讨论了结的彩色琼斯多项式的两种实现方式，一种出现在第二作者 1994 年关于从五边形特性解中获得的统一根量子 R 矩阵的一项未被注意的工作中，另一种则出现在 D. Zagier 和第一作者关于精炼量子模块性猜想的最新工作中，以哈比罗环元素序列的形式制定。
{"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":"24 1","pages":""},"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}
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