{"title":"Cyclic homology of Jordan superalgebras and related Lie superalgebras","authors":"Consuelo Martínez, Efim Zelmanov, Zezhou Zhang","doi":"arxiv-2409.03726","DOIUrl":"https://doi.org/arxiv-2409.03726","url":null,"abstract":"We study the relationship between cyclic homology of Jordan superalgebras and\u0000second cohomologies of their Tits-Kantor-Koecher Lie superalgebras. In particular, we focus on Jordan superalgebras that are Kantor doubles of\u0000bracket algebras. The obtained results are applied to computation of second\u0000cohomologies and universal central extensions of Hamiltonian and contact type\u0000Lie superalgebras over arbitrary rings of coefficients.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192686","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}
{"title":"Based cluster algebras of infinite ranks","authors":"Fan Qin","doi":"arxiv-2409.02881","DOIUrl":"https://doi.org/arxiv-2409.02881","url":null,"abstract":"We extend based cluster algebras to infinite ranks. By extending (quantum)\u0000cluster algebras associated with double Bott-Samelson cells, we recover\u0000infinite rank cluster algebras arising from representations of (shifted)\u0000quantum affine algebras. As the main application, we show that the fundamental\u0000variables of the cluster algebras associated with double Bott-Samelson cells\u0000could be computed via a braid group action when the Cartan matrix is of finite\u0000type. We also obtain the result A=U for the associated infinite rank (quantum)\u0000cluster algebras. Additionally, several conjectures regarding quantum virtual\u0000Grothendieck rings by Jang-Lee-Oh and Oh-Park follow as consequences. Finally,\u0000we quantize cluster algebras arising from representations of shifted quantum\u0000affine algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"74 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192691","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}
{"title":"Deformation maps of Quasi-twilled associative algebras","authors":"Shanshan Liu, Abdenacer Makhlouf, Lina Song","doi":"arxiv-2409.02651","DOIUrl":"https://doi.org/arxiv-2409.02651","url":null,"abstract":"In this paper, we introduce two types of deformation maps of quasi-twilled\u0000associative algebras. Each type of deformation maps unify various operators on\u0000associative algebras. Right deformation maps unify modified Rota-Baxter\u0000operators of weight $lambda$, derivations, homomorphisms and crossed\u0000homomorphisms. Left deformation maps unify relative Rota-Baxter operators of\u0000weight 0, twisted Rota-Baxter operators, Reynolds operators and deformation\u0000maps of matched pairs of associative algebras. Furthermore, we give the\u0000controlling algebra and the cohomology of these two types of deformation maps.\u0000On the one hand, we obtain some existing results for modified Rota-Baxter\u0000operators of weight $lambda$, derivations, homomorphisms, crossed\u0000homomorphisms, relative Rota-Baxter operators of weight 0, twisted Rota-Baxter\u0000operators and Reynolds operators. On the other hand, we also obtain some new\u0000results, such as the controlling algebra of a modified Rota-Baxter operator of\u0000weight $lambda$ on an associative algebra, the controlling algebra and the\u0000cohomology of a deformation map of a matched pair of associative algebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192693","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}
{"title":"Cardinality of groups and rings via the idempotency of infinite cardinals","authors":"Abolfazl Tarizadeh","doi":"arxiv-2409.02488","DOIUrl":"https://doi.org/arxiv-2409.02488","url":null,"abstract":"An important classical result in ZFC asserts that every infinite cardinal\u0000number is idempotent. Using this fact, we obtain several algebraic results in\u0000this article. The first result asserts that an infinite Abelian group has a\u0000proper subgroup with the same cardinality if and only if it is not a Pr\"ufer\u0000group. In the second result, the cardinality of any monoid-ring $R[M]$ (not\u0000necessarily commutative) is calculated. In particular, the cardinality of every\u0000polynomial ring with any number of variables (possibly infinite) is easily\u0000computed. Next, it is shown that every commutative ring and its total ring of\u0000fractions have the same cardinality. This set-theoretic observation leads us to\u0000a notion in ring theory that we call a balanced ring (i.e. a ring that is\u0000canonically isomorphic to its total ring of fractions). Every zero-dimensional\u0000ring is a balanced ring. Then we show that a Noetherian ring is a balanced ring\u0000if and only if its localization at every maximal ideal has zero depth. It is\u0000also proved that every self-injective ring (injective as a module over itself)\u0000is a balanced ring.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192692","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}
{"title":"Ternary Associativity and Ternary Lie Algebra at Cube Root of Unity","authors":"Viktor Abramov","doi":"arxiv-2409.02557","DOIUrl":"https://doi.org/arxiv-2409.02557","url":null,"abstract":"We propose a new approach to extending the notion of commutator and Lie\u0000algebra to algebras with ternary multiplication laws. Our approach is based on\u0000ternary associativity of the first and second kind. We propose a ternary\u0000commutator, which is a linear combination of six (all permutations of three\u0000elements) triple products. The coefficients of this linear combination are the\u0000cube roots of unity. We find an identity for the ternary commutator that holds\u0000due to ternary associativity of the first or second kind. The form of the found\u0000identity is determined by the permutations of the general affine group GA(1,5).\u0000We consider the found identity as an analogue of the Jacobi identity in the\u0000ternary case. We introduce the concept of a ternary Lie algebra at the cubic\u0000root of unity and give examples of such an algebra constructed using ternary\u0000multiplications of rectangular and three-dimensional matrices. We point out the\u0000connection between the structure constants of a ternary Lie algebra with three\u0000generators and an irreducible representation of the rotation group.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192689","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}
Ryszard R. Andruszkiewicz, Tomasz Brzeziński, Krzysztof Radziszewski
{"title":"Lie affgebras vis-à-vis Lie algebras","authors":"Ryszard R. Andruszkiewicz, Tomasz Brzeziński, Krzysztof Radziszewski","doi":"arxiv-2409.01996","DOIUrl":"https://doi.org/arxiv-2409.01996","url":null,"abstract":"It is shown that any Lie affgebra, that is an algebraic system consisting of\u0000an affine space together with a bi-affine bracket satisfying affine versions of\u0000the antisymmetry and Jacobi identity, is isomorphic to a Lie algebra together\u0000with an element and a specific generalised derivation (in the sense of Leger\u0000and Luks, [G.F. Leger & E.M. Luks, Generalized derivations of Lie algebras,\u0000{em J. Algebra} {bf 228} (2000), 165--203]). These Lie algebraic data can be\u0000taken for the construction of a Lie affgebra or, conversely, they can be\u0000uniquely derived for any Lie algebra fibre of the Lie affgebra. The close\u0000relationship between Lie affgebras and (enriched by the additional data) Lie\u0000algebras can be employed to attempt a classification of the former by the\u0000latter. In particular, up to isomorphism, a complex Lie affgebra with a simple\u0000Lie algebra fibre $mathfrak{g}$ is fully determined by a scalar and an element\u0000of $mathfrak{g}$ fixed up to an automorphism of $mathfrak{g}$, and it can be\u0000universally embedded in a trivial extension of $mathfrak{g}$ by a derivation.\u0000The study is illustrated by a number of examples that include all Lie affgebras\u0000with one-dimensional, nonabelian two-dimensional,\u0000$mathfrak{s}mathfrak{l}(2,mathbb{C})$ and $mathfrak{s}mathfrak{o}(3)$\u0000fibres. Extensions of Lie affgebras by cocycles and their relation to cocycle\u0000extensions of tangent Lie algebras is briefly discussed too.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192760","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}
{"title":"Admissible groups over number fields","authors":"Deependra Singh","doi":"arxiv-2409.02333","DOIUrl":"https://doi.org/arxiv-2409.02333","url":null,"abstract":"Given a field K, one may ask which finite groups are Galois groups of field\u0000extensions L/K such that L is a maximal subfield of a division algebra with\u0000center K. This connection between inverse Galois theory and division algebras\u0000was first explored by Schacher in the 1960s. In this manuscript we consider\u0000this problem when K is a number field. For the case when L/K is assumed to be\u0000tamely ramified, we give a complete classification of number fields for which\u0000every solvable Sylow-metacyclic group is admissible, extending J. Sonn's result\u0000over the field of rational numbers. For the case when L/K is allowed to be\u0000wildly ramified, we give a characterization of admissible groups over several\u0000classes of number fields, and partial results in other cases.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192694","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}
{"title":"On Poisson conformal bialgebras","authors":"Yanyong Hong, Chengming Bai","doi":"arxiv-2409.01619","DOIUrl":"https://doi.org/arxiv-2409.01619","url":null,"abstract":"We develop a conformal analog of the theory of Poisson bialgebras as well as\u0000a bialgebra theory of Poisson conformal algebras. We introduce the notion of\u0000Poisson conformal bialgebras, which are characterized by Manin triples of\u0000Poisson conformal algebras. A class of special Poisson conformal bialgebras\u0000called coboundary Poisson conformal bialgebras are constructed from\u0000skew-symmetric solutions of the Poisson conformal Yang-Baxter equation, whose\u0000operator forms are studied. Then we show that the semi-classical limits of\u0000conformal formal deformations of commutative and cocommutative antisymmetric\u0000infinitesimal conformal bialgebras are Poisson conformal bialgebras. Finally,\u0000we extend the correspondence between Poisson conformal algebras and\u0000Poisson-Gel'fand-Dorfman algebras to the context of bialgebras, that is, we\u0000introduce the notion of Poisson-Gel'fand-Dorfman bialgebras and show that\u0000Poisson-Gel'fand-Dorfman bialgebras correspond to a class of Poisson conformal\u0000bialgebras. Moreover, a construction of Poisson conformal bialgebras from\u0000pre-Poisson-Gel'fand-Dorfman algebras is given.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192697","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}
{"title":"Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms","authors":"Stephen Scully","doi":"arxiv-2409.02059","DOIUrl":"https://doi.org/arxiv-2409.02059","url":null,"abstract":"Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$\u0000of characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar\u0000extension to the function field of the affine quadric with equation $p=0$. In\u0000this article, we establish a strong constraint on $i$ in terms of the dimension\u0000of $q$ and two stable birational invariants of $p$, one of which is the\u0000well-known \"Izhboldin dimension\", and the other of which is a new invariant\u0000that we denote $Delta(p)$. Examining the contribution from the Izhboldin\u0000dimension, we obtain a result that unifies and extends the quasilinear\u0000analogues of two fundamental results on the isotropy of non-singular quadratic\u0000forms over function fields of quadrics in arbitrary characteristic due to\u0000Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the\u0000quasilinear case of a general conjecture previously formulated by the author,\u0000suggesting that a substantial refinement of this conjecture should hold.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192722","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}
{"title":"A complete classification of perfect unitary Cayley graphs","authors":"Ján Mináč, Tung T. Nguyen, Nguyen Duy Tân","doi":"arxiv-2409.01922","DOIUrl":"https://doi.org/arxiv-2409.01922","url":null,"abstract":"Due to their elegant and simple nature, unitary Cayley graphs have been an\u0000active research topic in the literature. These graphs are naturally connected\u0000to several branches of mathematics, including number theory, finite algebra,\u0000representation theory, and graph theory. In this article, we study the\u0000perfectness property of these graphs. More precisely, we provide a complete\u0000classification of perfect unitary Cayley graphs associated with finite rings.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142192701","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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