Expositiones Mathematicae最新文献

{"title":"Commutators and products of Lie ideals of prime rings","authors":"Tsiu-Kwen Lee , Jheng-Huei Lin","doi":"10.1016/j.exmath.2025.125658","DOIUrl":"10.1016/j.exmath.2025.125658","url":null,"abstract":"<div><div>Motivated by some recent results on Lie ideals, it is proved that if <span><math><mi>L</mi></math></span> is a Lie ideal of a simple ring <span><math><mi>R</mi></math></span> with center <span><math><mrow><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, then <span><math><mrow><mi>L</mi><mo>⊆</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>L</mi><mo>=</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow><mi>a</mi><mo>+</mo><mi>Z</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> for some noncentral <span><math><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow></math></span>, or <span><math><mrow><mrow><mo>[</mo><mi>R</mi><mo>,</mo><mi>R</mi><mo>]</mo></mrow><mo>⊆</mo><mi>L</mi></mrow></math></span>, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let <span><math><mi>R</mi></math></span> be a prime ring with extended centroid <span><math><mi>C</mi></math></span>. We completely characterize Lie ideals <span><math><mi>L</mi></math></span> and elements <span><math><mi>a</mi></math></span> of <span><math><mi>R</mi></math></span> such that <span><math><mrow><mi>L</mi><mo>+</mo><mi>a</mi><mi>L</mi></mrow></math></span> contains a nonzero ideal of <span><math><mi>R</mi></math></span>. Given noncentral Lie ideals <span><math><mrow><mi>K</mi><mo>,</mo><mi>L</mi></mrow></math></span> of <span><math><mi>R</mi></math></span>, it is proved that <span><math><mrow><mrow><mo>[</mo><mi>K</mi><mo>,</mo><mi>L</mi><mo>]</mo></mrow><mo>=</mo><mn>0</mn></mrow></math></span> if and only if <span><math><mrow><mi>K</mi><mi>C</mi><mo>=</mo><mi>L</mi><mi>C</mi><mo>=</mo><mi>C</mi><mi>a</mi><mo>+</mo><mi>C</mi></mrow></math></span> for any noncentral element <span><math><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow></math></span>. As a consequence, we characterize noncentral Lie ideals <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> with <span><math><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow></math></span> such that <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math></span> contains a nonzero ideal of <span><math><mi>R</mi></math></span>. Finally, we characterize noncentral Lie ideals <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>’s and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>’s satisfying <span><math><mrow><mrow><mo>[</mo><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>K</mi></mr","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125658"},"PeriodicalIF":0.8,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349059","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}

{"title":"Dualistic structures in information geometry","authors":"Leonard Todjihounde","doi":"10.1016/j.exmath.2025.125654","DOIUrl":"10.1016/j.exmath.2025.125654","url":null,"abstract":"<div><div>Important basics on dualistic structures on Riemannian manifolds are revisited and presented as a fundamental concept connecting information geometry, affine geometry and Hessian geometry. Since several statistical manifolds can be seen as warped product spaces, we conclude this survey by some results on warped products of dualistic structures.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125654"},"PeriodicalIF":0.8,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143386889","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}

{"title":"An annotated bibliography for comparative prime number theory","authors":"Greg Martin,&nbsp;Pu Justin Scarfy Yang,&nbsp;Aram Bahrini,&nbsp;Prajeet Bajpai,&nbsp;Kübra Benli̇,&nbsp;Jenna Downey,&nbsp;Yuan Yuan Li,&nbsp;Xiaoxuan Liang,&nbsp;Amir Parvardi,&nbsp;Reginald Simpson,&nbsp;Ethan Patrick White,&nbsp;Chi Hoi Yip","doi":"10.1016/j.exmath.2024.125644","DOIUrl":"10.1016/j.exmath.2024.125644","url":null,"abstract":"<div><div>The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125644"},"PeriodicalIF":0.8,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143508438","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}

{"title":"Characters of the unitriangular group and the Mackey method","authors":"Mikhail Ignatev ,&nbsp;Mikhail Venchakov","doi":"10.1016/j.exmath.2025.125656","DOIUrl":"10.1016/j.exmath.2025.125656","url":null,"abstract":"<div><div>Let <span><math><mi>U</mi></math></span> be the unitriangular group over a finite field. We consider an interesting class of irreducible complex characters of <span><math><mi>U</mi></math></span>, so-called characters of depth 2. This is a next natural step after characters of maximal and submaximal dimension, whose description is already known. We explicitly describe the support of a character of depth 2 by a system of defining algebraic equations. After that, we calculate the value of such a character on an element from the support. The main technical tool used in the proofs is the Mackey little group method for semidirect products.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 3","pages":"Article 125656"},"PeriodicalIF":0.8,"publicationDate":"2025-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349100","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}

{"title":"An approach to annihilators in the context of vector field Lie algebras","authors":"Charles H. Conley ,&nbsp;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}

{"title":"Characterising the Haar measure on the p-adic rotation groups via inverse limits of measure spaces","authors":"Paolo Aniello ,&nbsp;Sonia L’Innocente ,&nbsp;Stefano Mancini ,&nbsp;Vincenzo Parisi ,&nbsp;Ilaria Svampa ,&nbsp;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>&gt;</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}

{"title":"Exceptional Periodicity and Magic Star algebras","authors":"Piero Truini ,&nbsp;Alessio Marrani ,&nbsp;Michael Rios ,&nbsp;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}

{"title":"Supergravity in the geometric approach and its hidden graded Lie algebra","authors":"L. Andrianopoli ,&nbsp;R. D’Auria","doi":"10.1016/j.exmath.2024.125631","DOIUrl":"10.1016/j.exmath.2024.125631","url":null,"abstract":"<div><div>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 <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msub></math></span> algebras.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125631"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621013","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}

{"title":"V.S. Varadarajan (1937–2019): In memoriam","authors":"Rita Fioresi","doi":"10.1016/j.exmath.2025.125661","DOIUrl":"10.1016/j.exmath.2025.125661","url":null,"abstract":"<div><div>This article is a personal recollection of some aspects of the life and mathematics of Professor V.S. Varadarajan, who passed away on April 25, 2019, in Santa Monica, California, USA.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125661"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621012","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}

{"title":"Harmonic analysis of compact Lie supergroups","authors":"M.-K. Chuah ,&nbsp;C.A. Cremonini ,&nbsp;R. Fioresi","doi":"10.1016/j.exmath.2024.125586","DOIUrl":"10.1016/j.exmath.2024.125586","url":null,"abstract":"<div><div>We realize the irreducible representations of a compact Lie supergroup <span><math><mi>G</mi></math></span>, with a contragredient simple Lie superalgebra, in the space of square integrable (in the sense of Berezin) holomorphic sections on <span><math><mrow><mi>X</mi><mo>=</mo><mi>G</mi><mi>A</mi></mrow></math></span>, <span><math><mi>A</mi></math></span> is the real torus in the complexification of <span><math><mi>G</mi></math></span>. We give an explicit realization of unitary representations when <span><math><mrow><mi>G</mi><mo>=</mo><mi>SU</mi><mrow><mo>(</mo><mn>1</mn><mo>|</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":50458,"journal":{"name":"Expositiones Mathematicae","volume":"43 2","pages":"Article 125586"},"PeriodicalIF":0.8,"publicationDate":"2025-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614865","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}