We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.
{"title":"Actions of Compact and Discrete Quantum Groups on Operator Systems","authors":"Joeri De Ro, Lucas Hataishi","doi":"10.1093/imrn/rnae118","DOIUrl":"https://doi.org/10.1093/imrn/rnae118","url":null,"abstract":"We introduce the notion of an action of a discrete or compact quantum group on an operator system, and study equivariant operator system injectivity. We then prove a duality result that relates equivariant injectivity with dual injectivity of associated crossed products. As an application, we give a description of the equivariant injective envelope of the reduced crossed product built from an action of a discrete quantum group on an operator system.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"105 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez, Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters, following Kaletha, Minguez, Shin, and White.
在阿瑟早先的一本著作中，建立了准分裂正交群和交点群的表征的内视分类。后来，莫克给出了准分裂单元群的内视分类。之后，Kaletha、Minguez、Shin 和 White 又给出了通用参数的非准分裂单元群的内视分类。在本文中，我们继 Kaletha、Minguez、Shin 和 White 之后，证明了泛参数非准分裂奇特正交群的表征的内视分类。
{"title":"The Endoscopic Classification of Representations of Non-Quasi-Split Odd Special Orthogonal Groups","authors":"Hiroshi Ishimoto","doi":"10.1093/imrn/rnae113","DOIUrl":"https://doi.org/10.1093/imrn/rnae113","url":null,"abstract":"In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez, Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters, following Kaletha, Minguez, Shin, and White.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.
{"title":"On a Theorem of Lafforgue","authors":"Matthew Baker, Oliver Lorscheid","doi":"10.1093/imrn/rnae114","DOIUrl":"https://doi.org/10.1093/imrn/rnae114","url":null,"abstract":"We give a new proof, along with some generalizations, of a folklore theorem (attributed to Laurent Lafforgue) that a rigid matroid (i.e., a matroid with indecomposable basis polytope) has only finitely many projective equivalence classes of representations over any given field.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"41 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that projective hyperkähler manifolds of K3$^{[n]}$-type admitting a non-trivial symplectic birational self-map of finite order are isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3 surfaces. Motivated by this result, we analyze the reflections on the movable cone of moduli spaces of sheaves and determine when they come from a birational involution.
{"title":"On Symplectic Birational Self-Maps of Projective Hyperkähler Manifolds of K3$^{[n]}$-Type","authors":"Yajnaseni Dutta, Dominique Mattei, Yulieth Prieto-Montañez","doi":"10.1093/imrn/rnae112","DOIUrl":"https://doi.org/10.1093/imrn/rnae112","url":null,"abstract":"We prove that projective hyperkähler manifolds of K3$^{[n]}$-type admitting a non-trivial symplectic birational self-map of finite order are isomorphic to moduli spaces of stable (twisted) coherent sheaves on K3 surfaces. Motivated by this result, we analyze the reflections on the movable cone of moduli spaces of sheaves and determine when they come from a birational involution.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141191124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a stability condition $sigma $ on a triangulated category, Dimitrov–Katzarkov introduced the notion of a $sigma $-exceptional collection. In this paper, we study full $sigma $-exceptional collections in the derived category of an acyclic quiver. In particular, we prove that any stability condition $sigma $ on the derived category of a Dynkin quiver admits a full $sigma $-exceptional collection.
{"title":"Full Exceptional Collections and Stability Conditions for Dynkin Quivers","authors":"Takumi Otani","doi":"10.1093/imrn/rnae110","DOIUrl":"https://doi.org/10.1093/imrn/rnae110","url":null,"abstract":"For a stability condition $sigma $ on a triangulated category, Dimitrov–Katzarkov introduced the notion of a $sigma $-exceptional collection. In this paper, we study full $sigma $-exceptional collections in the derived category of an acyclic quiver. In particular, we prove that any stability condition $sigma $ on the derived category of a Dynkin quiver admits a full $sigma $-exceptional collection.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"130 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Theodore D Drivas, Daniil Glukhovskiy, Boris Khesin
We consider pairs of point vortices having circulations $Gamma _{1}$ and $Gamma _{2}$ and confined to a two-dimensional surface $S$. In the limit of zero initial separation $varepsilon $, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the “singular vortex pair” moves as a single-charged particle on the surface with a charge of order $1/varepsilon ^{2}$ in an magnetic field $B$ that is everywhere normal to the surface and of strength $|B|=Gamma _{1} +Gamma _{2}$. In the case $Gamma _{1}=-Gamma _{2}$, this gives another proof of Kimura’s conjecture [11] that singular dipoles follow geodesics.
{"title":"Singular Vortex Pairs Follow Magnetic Geodesics","authors":"Theodore D Drivas, Daniil Glukhovskiy, Boris Khesin","doi":"10.1093/imrn/rnae106","DOIUrl":"https://doi.org/10.1093/imrn/rnae106","url":null,"abstract":"We consider pairs of point vortices having circulations $Gamma _{1}$ and $Gamma _{2}$ and confined to a two-dimensional surface $S$. In the limit of zero initial separation $varepsilon $, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the “singular vortex pair” moves as a single-charged particle on the surface with a charge of order $1/varepsilon ^{2}$ in an magnetic field $B$ that is everywhere normal to the surface and of strength $|B|=Gamma _{1} +Gamma _{2}$. In the case $Gamma _{1}=-Gamma _{2}$, this gives another proof of Kimura’s conjecture [11] that singular dipoles follow geodesics.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $operatorname{overline{{mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.
{"title":"A Generalization of Operads Based on Subgraph Contractions","authors":"Denis Lyskov","doi":"10.1093/imrn/rnae096","DOIUrl":"https://doi.org/10.1093/imrn/rnae096","url":null,"abstract":"We introduce a generalization of the notion of operad that we call a contractad, whose set of operations is indexed by connected graphs and whose composition rules are numbered by contractions of connected subgraphs. We show that many classical operads, such as the operad of commutative algebras, Lie algebras, associative algebras, pre-Lie algebras, the little disks operad, and the operad of moduli spaces of stable curves $operatorname{overline{{mathcal{M}}}}_{0,n+1}$, admit generalizations to contractads. We explain that standard tools like Koszul duality and the machinery of Gröbner bases can be easily generalized to contractads. We verify the Koszul property of the commutative, Lie, associative, and Gerstenhaber contractads.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"64 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Henna Koivusalo, Jason Levesley, Benjamin Ward, Xintian Zhang
Let $psi :{mathbb{N}} to [0,infty )$, $psi (q)=q^{-(1+tau )}$ and let $psi $-badly approximable points be those vectors in ${mathbb{R}}^{d}$ that are $psi $-well approximable, but not $cpsi $-well approximable for arbitrarily small constants $c>0$. We establish that the $psi $-badly approximable points have the Hausdorff dimension of the $psi $-well approximable points, the dimension taking the value $(d+1)/(tau +1)$ familiar from theorems of Besicovitch and Jarník. The method of proof is an entirely new take on the Mass Transference Principle (MTP) by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named “delayed pruning” to construct a sufficiently large $liminf $ set and combine this with ideas inspired by the proof of the MTP to find a large $limsup $ subset of the $liminf $ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.
{"title":"The Dimension of the Set of $psi $-Badly Approximable Points in All Ambient Dimensions: On a Question of Beresnevich and Velani","authors":"Henna Koivusalo, Jason Levesley, Benjamin Ward, Xintian Zhang","doi":"10.1093/imrn/rnae101","DOIUrl":"https://doi.org/10.1093/imrn/rnae101","url":null,"abstract":"Let $psi :{mathbb{N}} to [0,infty )$, $psi (q)=q^{-(1+tau )}$ and let $psi $-badly approximable points be those vectors in ${mathbb{R}}^{d}$ that are $psi $-well approximable, but not $cpsi $-well approximable for arbitrarily small constants $c>0$. We establish that the $psi $-badly approximable points have the Hausdorff dimension of the $psi $-well approximable points, the dimension taking the value $(d+1)/(tau +1)$ familiar from theorems of Besicovitch and Jarník. The method of proof is an entirely new take on the Mass Transference Principle (MTP) by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named “delayed pruning” to construct a sufficiently large $liminf $ set and combine this with ideas inspired by the proof of the MTP to find a large $limsup $ subset of the $liminf $ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"48 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alessio D’Alì, Martina Juhnke-Kubitzke, Melissa Koch
Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we prove that two symmetric edge polytopes are unimodularly equivalent precisely when they share the same graphical matroid. The second goal is to show that one can construct a generalized symmetric edge polytope starting from every regular matroid. Just like in the usual case, we are able to find combinatorial ways to describe the facets and an explicit regular unimodular triangulation of any such polytope. Finally, we show that the Ehrhart theory of the polar of a given generalized symmetric edge polytope is tightly linked to the structure of the lattice of flows of the dual regular matroid.
{"title":"On a Generalization of Symmetric Edge Polytopes to Regular Matroids","authors":"Alessio D’Alì, Martina Juhnke-Kubitzke, Melissa Koch","doi":"10.1093/imrn/rnae107","DOIUrl":"https://doi.org/10.1093/imrn/rnae107","url":null,"abstract":"Starting from any finite simple graph, one can build a reflexive polytope known as a symmetric edge polytope. The first goal of this paper is to show that symmetric edge polytopes are intrinsically matroidal objects: more precisely, we prove that two symmetric edge polytopes are unimodularly equivalent precisely when they share the same graphical matroid. The second goal is to show that one can construct a generalized symmetric edge polytope starting from every regular matroid. Just like in the usual case, we are able to find combinatorial ways to describe the facets and an explicit regular unimodular triangulation of any such polytope. Finally, we show that the Ehrhart theory of the polar of a given generalized symmetric edge polytope is tightly linked to the structure of the lattice of flows of the dual regular matroid.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bourgain in his seminal paper of 1986 about the analysis of maximal functions associated to convex bodies has estimated in a sharp way the $L^{2}$-operator norm of the maximal function associated to a kernel $Kin L^{1},$ with differentiable Fourier transform $widehat{K}.$ We formulate the extension to Bourgain’s $L^{2}$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^{p}$-boundedness of maximal functions on graded Lie groups.
{"title":"L 2-Maximal Functions on Graded Lie Groups","authors":"Duván Cardona","doi":"10.1093/imrn/rnae105","DOIUrl":"https://doi.org/10.1093/imrn/rnae105","url":null,"abstract":"Bourgain in his seminal paper of 1986 about the analysis of maximal functions associated to convex bodies has estimated in a sharp way the $L^{2}$-operator norm of the maximal function associated to a kernel $Kin L^{1},$ with differentiable Fourier transform $widehat{K}.$ We formulate the extension to Bourgain’s $L^{2}$-estimate in the setting of maximal functions on graded Lie groups. Our criterion is formulated in terms of the group Fourier transform of the kernel. We discuss the application of our main result to the $L^{p}$-boundedness of maximal functions on graded Lie groups.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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