Pub Date : 2025-04-03DOI: 10.1016/j.exmath.2025.125687
Xiaofei Qi , Huimin Chen , Jinchuan Hou
We generalize the concept of homomorphisms between Hilbert C-modules to the concept of Jordan homomorphisms. Let be a Hilbert C-module over a C-algebra and be a map. Under the condition that is commutative and is -linear bounded, we show that is a Jordan homomorphism if and only if is a homomorphism. In addition, we also discuss the relationship between -unitary maps and automorphisms, and give some conditions under which is an automorphism if and only if is a -unitary map.
{"title":"Jordan homomorphisms on Hilbert C∗-modules","authors":"Xiaofei Qi , Huimin Chen , Jinchuan Hou","doi":"10.1016/j.exmath.2025.125687","DOIUrl":"10.1016/j.exmath.2025.125687","url":null,"abstract":"<div><div>We generalize the concept of homomorphisms between Hilbert C<span><math><msup><mrow></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-modules to the concept of Jordan homomorphisms. Let <span><math><mi>M</mi></math></span> be a Hilbert C<span><math><msup><mrow></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-module over a C<span><math><msup><mrow></mrow><mrow><mo>∗</mo></mrow></msup></math></span>-algebra <span><math><mi>A</mi></math></span> and <span><math><mrow><mi>φ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow></math></span> be a map. Under the condition that <span><math><mi>A</mi></math></span> is commutative and <span><math><mi>φ</mi></math></span> is <span><math><mi>ℂ</mi></math></span>-linear bounded, we show that <span><math><mi>φ</mi></math></span> is a Jordan homomorphism if and only if <span><math><mi>φ</mi></math></span> is a homomorphism. In addition, we also discuss the relationship between <span><math><mi>Φ</mi></math></span>-unitary maps and automorphisms, and give some conditions under which <span><math><mi>φ</mi></math></span> is an automorphism if and only if <span><math><mi>φ</mi></math></span> is a <span><math><mi>Φ</mi></math></span>-unitary map.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125687"},"PeriodicalIF":0.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808261","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 : 2025-04-03DOI: 10.1016/j.exmath.2025.125686
Jingsheng Wang, Pengtong Li
In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let be an LCA group, let be a uniform lattice in , let be an automorphism of , and let be an automorphism of . We show that the redundancy of a Gabor type unitary system indexed by equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.
{"title":"The redundancy of Gabor type unitary systems on locally compact abelian groups","authors":"Jingsheng Wang, Pengtong Li","doi":"10.1016/j.exmath.2025.125686","DOIUrl":"10.1016/j.exmath.2025.125686","url":null,"abstract":"<div><div>In this paper, we extend the redundancy theorem of J. Gabardo and D. Han for Gabor type unitary systems indexed by full-rank lattices in Euclidean spaces to the setting of locally compact abelian (LCA) groups. Let <span><math><mi>G</mi></math></span> be an LCA group, let <span><math><mi>Λ</mi></math></span> be a uniform lattice in <span><math><mi>G</mi></math></span>, let <span><math><mi>α</mi></math></span> be an automorphism of <span><math><mi>G</mi></math></span>, and let <span><math><mi>β</mi></math></span> be an automorphism of <span><math><mover><mrow><mi>G</mi></mrow><mrow><mo>̂</mo></mrow></mover></math></span>. We show that the redundancy of a Gabor type unitary system indexed by <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>Λ</mi><mo>)</mo></mrow><mo>×</mo><mi>β</mi><mrow><mo>(</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mo>⊥</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> equals the reciprocal of the density of this index set. As an application, we give a new proof of the famous time-frequency density theorem in Gabor analysis.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125686"},"PeriodicalIF":0.8,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143808262","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 : 2025-04-02DOI: 10.1016/j.exmath.2025.125685
Emilie Mai Elkiær
We show via an application of techniques from complex interpolation theory how the -pseudofunction algebras of a locally compact group can be understood as sitting between and . Motivated by this, we collect and review various characterizations of group amenability connected to the -pseudofunction algebra of Herz and generalize these to the symmetrized setting. Along the way, we describe the Banach space dual of the symmetrized pseudofunction algebras on associated with representations on reflexive Banach spaces.
我们通过应用复插值理论的技术，说明局部紧凑群 G 的 Lp 伪函数代数如何被理解为介于 L1(G) 和 C∗(G) 之间。受此启发，我们收集并回顾了与赫兹的 p 伪函数代数相关的各种群可亲性特征，并将这些特征推广到对称设置中。同时，我们还描述了与反身巴拿赫空间上的表征相关的 G 上对称伪函数代数的巴拿赫空间对偶。
{"title":"Symmetrized pseudofunction algebras from Lp-representations and amenability of locally compact groups","authors":"Emilie Mai Elkiær","doi":"10.1016/j.exmath.2025.125685","DOIUrl":"10.1016/j.exmath.2025.125685","url":null,"abstract":"<div><div>We show via an application of techniques from complex interpolation theory how the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-pseudofunction algebras of a locally compact group <span><math><mi>G</mi></math></span> can be understood as sitting between <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msup><mrow><mi>C</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Motivated by this, we collect and review various characterizations of group amenability connected to the <span><math><mi>p</mi></math></span>-pseudofunction algebra of Herz and generalize these to the symmetrized setting. Along the way, we describe the Banach space dual of the symmetrized pseudofunction algebras on <span><math><mi>G</mi></math></span> associated with representations on reflexive Banach spaces.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125685"},"PeriodicalIF":0.8,"publicationDate":"2025-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820722","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 : 2025-04-01DOI: 10.1016/j.exmath.2025.125680
Qiyao Bao , Deguang Han , Rui Liu , Jie Shen
Nonlinear framings naturally appear in many applications where nonlinear procedures are necessary. This paper examines two basic issues involving the linearization of Lipschitz framings. We first prove that every Lipschitz framing induces a linear framing which shares the same synthesis operator, and consequently every Banach space admitting a Lipschitz framing has the bounded approximation property. Secondly, we examine the projection-valued dilations of Lipschitz operator-valued measures on Banach spaces. We prove that every Lipschitz operator-valued measure can induce an operator-valued measure by linearization, and every -valued measure has a projection-valued measure dilation by establishing a nonlinear version of minimal dilation theory. As examples, we discuss a concrete construction of the minimal dilation for the special case when the measure space is , and how nonlinear sampling naturally induces a Lipschitz framing.
{"title":"Linearization of Lipschitz framings for Banach spaces","authors":"Qiyao Bao , Deguang Han , Rui Liu , Jie Shen","doi":"10.1016/j.exmath.2025.125680","DOIUrl":"10.1016/j.exmath.2025.125680","url":null,"abstract":"<div><div>Nonlinear framings naturally appear in many applications where nonlinear procedures are necessary. This paper examines two basic issues involving the linearization of Lipschitz framings. We first prove that every Lipschitz framing induces a linear framing which shares the same synthesis operator, and consequently every Banach space admitting a Lipschitz framing has the bounded approximation property. Secondly, we examine the projection-valued dilations of Lipschitz operator-valued measures on Banach spaces. We prove that every Lipschitz operator-valued measure can induce an operator-valued measure by linearization, and every <span><math><mrow><mi>Lip</mi><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></mrow></math></span>-valued measure has a projection-valued measure dilation by establishing a nonlinear version of minimal dilation theory. As examples, we discuss a concrete construction of the minimal dilation for the special case when the measure space is <span><math><mrow><mo>(</mo><mi>N</mi><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></math></span>, and how nonlinear sampling naturally induces a Lipschitz framing.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125680"},"PeriodicalIF":0.8,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143768160","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 : 2025-03-27DOI: 10.1016/j.exmath.2025.125679
Antigona Pajaziti , Mohammad Sadek
Let be an elliptic curve defined over and denote the reduction of modulo a prime of good reduction for . The divisibility of by an integer for a set of primes of density 1 is determined by the torsion subgroups of elliptic curves that are -isogenous to . In this work, we give explicit families of elliptic curves over together with integers such that the congruence class of modulo can be computed explicitly. In addition, we can estimate the density of primes for which each congruence class occurs. These include elliptic curves over whose torsion grows over a quadratic field where is determined by the -torsion subgroups in the -isogeny class of . We also exhibit elliptic curves over for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than 1 are divisible by given small integers.
{"title":"Divisibility of orders of reductions of elliptic curves","authors":"Antigona Pajaziti , Mohammad Sadek","doi":"10.1016/j.exmath.2025.125679","DOIUrl":"10.1016/j.exmath.2025.125679","url":null,"abstract":"<div><div>Let <span><math><mi>E</mi></math></span> be an elliptic curve defined over <span><math><mi>Q</mi></math></span> and <span><math><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub></math></span> denote the reduction of <span><math><mi>E</mi></math></span> modulo a prime <span><math><mi>p</mi></math></span> of good reduction for <span><math><mi>E</mi></math></span>. The divisibility of <span><math><mrow><mo>|</mo><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mo>|</mo></mrow></math></span> by an integer <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> for a set of primes <span><math><mi>p</mi></math></span> of density 1 is determined by the torsion subgroups of elliptic curves that are <span><math><mi>Q</mi></math></span>-isogenous to <span><math><mi>E</mi></math></span>. In this work, we give explicit families of elliptic curves <span><math><mi>E</mi></math></span> over <span><math><mi>Q</mi></math></span> together with integers <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> such that the congruence class of <span><math><mrow><mo>|</mo><msub><mrow><mover><mrow><mi>E</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mo>|</mo></mrow></math></span> modulo <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> can be computed explicitly. In addition, we can estimate the density of primes <span><math><mi>p</mi></math></span> for which each congruence class occurs. These include elliptic curves over <span><math><mi>Q</mi></math></span> whose torsion grows over a quadratic field <span><math><mi>K</mi></math></span> where <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>E</mi></mrow></msub></math></span> is determined by the <span><math><mi>K</mi></math></span>-torsion subgroups in the <span><math><mi>Q</mi></math></span>-isogeny class of <span><math><mi>E</mi></math></span>. We also exhibit elliptic curves over <span><math><mrow><mi>Q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than 1 are divisible by given small integers.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125679"},"PeriodicalIF":0.8,"publicationDate":"2025-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143747610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-25DOI: 10.1016/j.exmath.2025.125681
Paweł Goldstein , Zofia Grochulska , Piotr Hajłasz
Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais’ argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais’ argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.
{"title":"Gluing diffeomorphisms, bi-Lipschitz mappings and homeomorphisms","authors":"Paweł Goldstein , Zofia Grochulska , Piotr Hajłasz","doi":"10.1016/j.exmath.2025.125681","DOIUrl":"10.1016/j.exmath.2025.125681","url":null,"abstract":"<div><div>Cerf and Palais independently proved a remarkable result about extending diffeomorphisms defined on smooth balls in a manifold to global diffeomorphisms of the manifold onto itself. We explain Palais’ argument and show how to extend it to the class of homeomorphisms and bi-Lipschitz homeomorphisms. While Palais’ argument is surprising, it is elementary and short. However, its extension to bi-Lipschitz homeomorphisms and homeomorphisms requires deep results: the stable homeomorphism and the annulus theorems.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 4","pages":"Article 125681"},"PeriodicalIF":0.8,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143777148","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 : 2025-03-01DOI: 10.1016/j.exmath.2024.125600
Charles H. Conley , William Goode
We present a general method for describing the annihilators of modules of Lie algebras under certain conditions, which hold for some tensor modules of vector field Lie algebras. As an example, we apply the method to obtain an efficient proof of previously known results on the annihilators of the bounded irreducible modules of .
{"title":"An approach to annihilators in the context of vector field Lie algebras","authors":"Charles H. Conley , William Goode","doi":"10.1016/j.exmath.2024.125600","DOIUrl":"10.1016/j.exmath.2024.125600","url":null,"abstract":"<div><div>We present a general method for describing the annihilators of modules of Lie algebras under certain conditions, which hold for some tensor modules of vector field Lie algebras. As an example, we apply the method to obtain an efficient proof of previously known results on the annihilators of the bounded irreducible modules of <span><math><mrow><mi>Vec</mi><mspace></mspace><mi>R</mi></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125600"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192927","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 : 2025-03-01DOI: 10.1016/j.exmath.2024.125621
Piero Truini , Alessio Marrani , Michael Rios , Willem de Graaf
We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but ) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call algebras) on each vertex of such a projection. We then focus on the Magic Star algebra , which generalizes the non-simply laced exceptional Lie algebra , and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.
{"title":"Exceptional Periodicity and Magic Star algebras","authors":"Piero Truini , Alessio Marrani , Michael Rios , Willem de Graaf","doi":"10.1016/j.exmath.2024.125621","DOIUrl":"10.1016/j.exmath.2024.125621","url":null,"abstract":"<div><div>We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but <span><math><msub><mrow><mi>g</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call <span><math><mi>H</mi></math></span> algebras) on each vertex of such a projection. We then focus on the Magic Star algebra <span><math><msubsup><mrow><mi>f</mi></mrow><mrow><mn>4</mn></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msubsup></math></span>, which generalizes the non-simply laced exceptional Lie algebra <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the <span><math><mi>H</mi></math></span> algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125621"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-01DOI: 10.1016/j.exmath.2024.125592
Paolo Aniello , Sonia L’Innocente , Stefano Mancini , Vincenzo Parisi , Ilaria Svampa , Andreas Winter
We determine the Haar measure on the compact -adic special orthogonal groups of rotations in dimension , by exploiting the machinery of inverse limits of measure spaces, for every prime . We characterise the groups as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each . Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on . Our results pave the way towards the study of the irreducible projective unitary representations of the -adic rotation groups, with potential applications to the recently proposed -adic quantum information theory.
{"title":"Characterising the Haar measure on the p-adic rotation groups via inverse limits of measure spaces","authors":"Paolo Aniello , Sonia L’Innocente , Stefano Mancini , Vincenzo Parisi , Ilaria Svampa , Andreas Winter","doi":"10.1016/j.exmath.2024.125592","DOIUrl":"10.1016/j.exmath.2024.125592","url":null,"abstract":"<div><div>We determine the Haar measure on the compact <span><math><mi>p</mi></math></span>-adic special orthogonal groups of rotations <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> in dimension <span><math><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math></span>, by exploiting the machinery of inverse limits of measure spaces, for every prime <span><math><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow></math></span>. We characterise the groups <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span>. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on <span><math><mrow><mi>SO</mi><msub><mrow><mrow><mo>(</mo><mi>d</mi><mo>)</mo></mrow></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span>. Our results pave the way towards the study of the irreducible projective unitary representations of the <span><math><mi>p</mi></math></span>-adic rotation groups, with potential applications to the recently proposed <span><math><mi>p</mi></math></span>-adic quantum information theory.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125592"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141880872","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 : 2025-03-01DOI: 10.1016/j.exmath.2024.125631
L. Andrianopoli , R. D’Auria
In this contribution, we present the geometric approach to supergravity. In the first part, we discuss in some detail the peculiarities of the approach and apply the formalism to the case of pure supergravity in four space-time dimensions. In the second part, we extend the discussion to theories in higher dimensions, which include antisymmetric tensors of degree higher than one, focussing on the case of eleven dimensional space–time. Here, we report the formulation first introduced by R. D’Auria and P. Fré in 1981, corresponding to a generalization of a Chevalley–Eilenberg Lie algebra, together with some more recent results, pointing out the relation of the formalism with the mathematical framework of algebras.
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