Leibniz superalgebras with nilindex $n + m$ and characteristic sequence $(n-1, 1 | m)$ divided into four parametric classes that contain a set of non-isomorphic superalgebras. In this paper, we give a complete classification of solvable Leibniz superalgebras whose nilradical is a nilpotent Leibniz superalgebra with nilindex $n + m$ and characteristic sequence $(n-1, 1 | m)$. We obtain a condition for the value of parameters of the classes of such nilpotent superalgebras for which they have a solvable extension. Moreover, the classification of solvable Leibniz superalgebras whose nilradical is a Lie superalgebra with the maximal nilindex is given.
{"title":"Solvable Leibniz superalgebras whose nilradical has the characteristic sequence $(n-1, 1 mid m)$ and nilindex $n+m$","authors":"Khudoyberdiyev A. Kh., Muratova Kh. A","doi":"10.46298/cm.11369","DOIUrl":"https://doi.org/10.46298/cm.11369","url":null,"abstract":"Leibniz superalgebras with nilindex $n + m$ and characteristic sequence $(n-1, 1 | m)$ divided into four parametric classes that contain a set of non-isomorphic superalgebras. In this paper, we give a complete classification of solvable Leibniz superalgebras whose nilradical is a nilpotent Leibniz superalgebra with nilindex $n + m$ and characteristic sequence $(n-1, 1 | m)$. We obtain a condition for the value of parameters of the classes of such nilpotent superalgebras for which they have a solvable extension. Moreover, the classification of solvable Leibniz superalgebras whose nilradical is a Lie superalgebra with the maximal nilindex is given.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, first, we introduce the Berger-type deformed Sasaki metric on the cotangent bundle $T^{ast}M$ over a K"{a}hlerian manifold $(M^{2m}, J, g)$ and investigate the Levi-Civita connection of this metric. Secondly, we present the unit cotangent bundle equipped with Berger-type deformed Sasaki metric, and we investigate the Levi-Civita connection. Finally, we study the geodesics on the cotangent bundle and on unit cotangent bundle with respect to the Berger-type deformed Sasaki metric.
{"title":"Note on geodesics of cotangent bundle with Berger-type deformed Sasaki metric over K\"ahlerian manifold","authors":"A. Zagane","doi":"10.46298/cm.11025","DOIUrl":"https://doi.org/10.46298/cm.11025","url":null,"abstract":"In this paper, first, we introduce the Berger-type deformed Sasaki metric on\u0000the cotangent bundle $T^{ast}M$ over a K\"{a}hlerian manifold $(M^{2m}, J, g)$\u0000and investigate the Levi-Civita connection of this metric. Secondly, we present\u0000the unit cotangent bundle equipped with Berger-type deformed Sasaki metric, and\u0000we investigate the Levi-Civita connection. Finally, we study the geodesics on\u0000the cotangent bundle and on unit cotangent bundle with respect to the\u0000Berger-type deformed Sasaki metric.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47291270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an abridged version of our Habilitation thesis. In these notes, we aim to summarize our research interests and achievements as well as motivate what drives our work: symmetry, structure and invariants. The paradigmatic example which permeates and often inspires our research is the Weyl algebra $mathbb{A}_{1}$.
{"title":"Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants","authors":"S. A. Lopes","doi":"10.46298/cm.11678","DOIUrl":"https://doi.org/10.46298/cm.11678","url":null,"abstract":"This is an abridged version of our Habilitation thesis. In these notes, we aim to summarize our research interests and achievements as well as motivate what drives our work: symmetry, structure and invariants. The paradigmatic example which permeates and often inspires our research is the Weyl algebra $mathbb{A}_{1}$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139353090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The integer point transform $sigma_PP$ is an important invariant of a rational polytope $PP$, and here we show that it is a complete invariant. We prove that it is only necessary to evaluate $sigma_PP$ at one algebraic point in order to uniquely determine $PP$, by employing the Lindemann-Weierstrass theorem. Similarly, we prove that it is only necessary to evaluate the Fourier transform of a rational polytope $PP$ at a single algebraic point, in order to uniquely determine $PP$. We prove that identical uniqueness results also hold for integer cones. In addition, by relating the integer point transform to finite Fourier transforms, we show that a finite number of emph{integer point evaluations} of $sigma_PP$ suffice in order to uniquely determine $PP$. We also give an equivalent condition for central symmetry of a finite point set, in terms of the integer point transform, and prove some facts about its local maxima. Most of the results are proven for arbitrary finite sets of integer points in $R^d$.
{"title":"The integer point transform as a complete invariant","authors":"S. Robins","doi":"10.46298/cm.11218","DOIUrl":"https://doi.org/10.46298/cm.11218","url":null,"abstract":"The integer point transform $sigma_PP$ is an important invariant of a\u0000rational polytope $PP$, and here we show that it is a complete invariant. We\u0000prove that it is only necessary to evaluate $sigma_PP$ at one algebraic point\u0000in order to uniquely determine $PP$, by employing the Lindemann-Weierstrass\u0000theorem. Similarly, we prove that it is only necessary to evaluate the Fourier\u0000transform of a rational polytope $PP$ at a single algebraic point, in order to\u0000uniquely determine $PP$. We prove that identical uniqueness results also hold\u0000for integer cones.\u0000 In addition, by relating the integer point transform to finite Fourier\u0000transforms, we show that a finite number of emph{integer point evaluations} of\u0000$sigma_PP$ suffice in order to uniquely determine $PP$. We also give an\u0000equivalent condition for central symmetry of a finite point set, in terms of\u0000the integer point transform, and prove some facts about its local maxima. Most\u0000of the results are proven for arbitrary finite sets of integer points in\u0000$R^d$.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42669958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Here are reproduced slightly edited notes of my lectures on the classification of discrete groups generated by complex reflections of Hermitian affine spaces delivered in October of 1980 at the University of Utrecht.
{"title":"Discrete complex reflection groups","authors":"V. Popov","doi":"10.46298/cm.11249","DOIUrl":"https://doi.org/10.46298/cm.11249","url":null,"abstract":"Here are reproduced slightly edited notes of my lectures on the\u0000classification of discrete groups generated by complex reflections of Hermitian\u0000affine spaces delivered in October of 1980 at the University of Utrecht.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46780822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a finite group and let ψ(G) denote the sum of element orders of G; later this concept has been used to define R(G) which is the product of the element orders of G. Motivated by the recursive formula for ψ(G), we consider a finite abelian group G and obtain a similar formula for R(G).
{"title":"A recursive formula for the product of element orders of finite abelian groups","authors":"Subhrajyoti Saha","doi":"10.46298/cm.10996","DOIUrl":"https://doi.org/10.46298/cm.10996","url":null,"abstract":"Let G be a finite group and let ψ(G) denote the sum of element orders of G; later this concept has been used to define R(G) which is the product of the element orders of G. Motivated by the recursive formula for ψ(G), we consider a finite abelian group G and obtain a similar formula for R(G).","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44276178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain inductive and enumerative formulas for the multiplicities of the weights of the spin module for the Clifford algebra of a Levi subalgebra in a complex semisimple Lie algebra. Our formulas involve only matrices and tableaux, and our techniques combine linear algebra, Lie theory, and combinatorics. Moreover, this suggests a relationship with complex nilpotent orbits. The case of the special linear Lie algebra $mathfrak{sl}(n,{mathbb C})$ is emphasized.
{"title":"Matrix formulas for multiplicities in the spin module","authors":"Lucas Fresse, S. Mehdi","doi":"10.46298/cm.11156","DOIUrl":"https://doi.org/10.46298/cm.11156","url":null,"abstract":"We obtain inductive and enumerative formulas for the multiplicities of the\u0000weights of the spin module for the Clifford algebra of a Levi subalgebra in a\u0000complex semisimple Lie algebra. Our formulas involve only matrices and\u0000tableaux, and our techniques combine linear algebra, Lie theory, and\u0000combinatorics. Moreover, this suggests a relationship with complex nilpotent\u0000orbits. The case of the special linear Lie algebra $mathfrak{sl}(n,{mathbb\u0000C})$ is emphasized.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49449176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this short survey we want to present some of the impact of Minkowski's successive minima within Convex and Discrete Geometry. Originally related to the volume of an $o$-symmetric convex body, we point out relations of the successive minima to other functionals, as e.g., the lattice point enumerator or the intrinsic volumes and we present some old and new conjectures about them. Additionally, we discuss an application of successive minima to a version of Siegel's lemma.
{"title":"Minkowski's successive minima in convex and discrete geometry","authors":"I. Aliev, M. Henk","doi":"10.46298/cm.11155","DOIUrl":"https://doi.org/10.46298/cm.11155","url":null,"abstract":"In this short survey we want to present some of the impact of Minkowski's\u0000successive minima within Convex and Discrete Geometry. Originally related to\u0000the volume of an $o$-symmetric convex body, we point out relations of the\u0000successive minima to other functionals, as e.g., the lattice point enumerator\u0000or the intrinsic volumes and we present some old and new conjectures about\u0000them. Additionally, we discuss an application of successive minima to a version\u0000of Siegel's lemma.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49090055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we give a first study of perfect copositive $n times n$ matrices. They can be used to find rational certificates for completely positive matrices. We describe similarities and differences to classical perfect, positive definite matrices. Most of the differences occur only for $n geq 3$, where we find for instance lower rank and indefinite perfect matrices. Nevertheless, we find for all $n$ that for every classical perfect matrix there is an arithmetically equivalent one which is also perfect copositive. Furthermore we study the neighborhood graph and polyhedral structure of perfect copositive matrices. As an application we obtain a new characterization of the cone of completely positive matrices: It is equal to the set of nonnegative matrices having a nonnegative inner product with all perfect copositive matrices.
{"title":"Perfect Copositive Matrices","authors":"Valentin Dannenberg, Achill Schurmann","doi":"10.46298/cm.11141","DOIUrl":"https://doi.org/10.46298/cm.11141","url":null,"abstract":"In this paper we give a first study of perfect copositive $n times n$\u0000matrices. They can be used to find rational certificates for completely\u0000positive matrices. We describe similarities and differences to classical\u0000perfect, positive definite matrices. Most of the differences occur only for $n\u0000geq 3$, where we find for instance lower rank and indefinite perfect matrices.\u0000Nevertheless, we find for all $n$ that for every classical perfect matrix there\u0000is an arithmetically equivalent one which is also perfect copositive.\u0000Furthermore we study the neighborhood graph and polyhedral structure of perfect\u0000copositive matrices. As an application we obtain a new characterization of the\u0000cone of completely positive matrices: It is equal to the set of nonnegative\u0000matrices having a nonnegative inner product with all perfect copositive\u0000matrices.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48474532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we prove that uniform Diophantine exponents of lattices attain only trivial values.
本文证明了格的一致丢番图指数只得到平凡值。
{"title":"On triviality of uniform Diophantine exponents of lattices","authors":"O. German","doi":"10.46298/cm.11137","DOIUrl":"https://doi.org/10.46298/cm.11137","url":null,"abstract":"In this paper we prove that uniform Diophantine exponents of lattices attain\u0000only trivial values.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46522794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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