Journal of Pure and Applied Algebra最新文献

{"title":"The stable Picard group of finite Adams Hopf algebroids with an application to the R-motivic Steenrod subalgebra AR(1)","authors":"Xu Gao ,&nbsp;Ang Li","doi":"10.1016/j.jpaa.2024.107732","DOIUrl":"10.1016/j.jpaa.2024.107732","url":null,"abstract":"<div><p>In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as <em>finite Adams</em>, shedding light on its Picard group. Then, we establish a reduction process through base changes, enabling us to effectively compute the Picard group of the <figure><img></figure><em>-motivic mod</em> 2 <em>Steenrod subalgebra</em> <figure><img></figure>. Our computation shows that <figure><img></figure> is isomorphic to <figure><img></figure>, where two ranks come from the motivic grading, one from the algebraic loop functor, and the last is generated by the <figure><img></figure><em>-motivic joker J</em>.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141131150","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}

{"title":"Retraction notice to “Monogenic bialgebras over finite fields and rings of Witt vectors” [J. Pure Appl. Algebra 163 (2) (2001) 193–207]","authors":"Alan Koch","doi":"10.1016/j.jpaa.2024.107703","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107703","url":null,"abstract":"<div><p>This article has been retracted: please see Elsevier Policy on Article Withdrawal (<span>https://www.elsevier.com/about/policies/article-withdrawal</span><svg><path></path></svg>).</p><p>This article has been retracted at the request of the Author.</p><p>There is an error in [1, Prop. 2.2] and, as a result, the classification of monogenic bialgebras as provided in Theorem 1 is incomplete. Therefore, the Author requested retraction and the Managing Editors agreed with this request. The Author regrets this error.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001002/pdfft?md5=4654844eabea351f7a33b48f4a0a9c67&pid=1-s2.0-S0022404924001002-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141078500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tensor algebras over the Steenrod algebra","authors":"H.E.A. Campbell,&nbsp;Paul Selick,&nbsp;Jie Wu","doi":"10.1016/j.jpaa.2024.107730","DOIUrl":"10.1016/j.jpaa.2024.107730","url":null,"abstract":"<div><p>It is known that unstable Steenrod module structure on the polynomial algebra <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>]</mo><mo>≅</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msup><mrow><mo>(</mo><mi>R</mi><msup><mrow><mi>P</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>)</mo></mrow><mrow><mi>N</mi></mrow></msup><mo>;</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> obtained by forgetting the multiplication is isomorphic to that arising from a twisted action of <span><math><msup><mrow><mi>Sq</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>. We show that the same theorem holds for tensor algebras. As in the abelian case, the result is applied to produce a decomposition of the tensor algebra into “weight spaces”.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141145460","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}

{"title":"Quantum diagrammatics for F4","authors":"Alistair Savage,&nbsp;Bruce W. Westbury","doi":"10.1016/j.jpaa.2024.107731","DOIUrl":"10.1016/j.jpaa.2024.107731","url":null,"abstract":"<div><p>We introduce a graphical calculus for the representation theory of the quantized enveloping algebra of type <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. We do this by giving a diagrammatic description of the category of invariant tensors on the 26-dimensional fundamental representation.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001282/pdfft?md5=abc8f701428ec8e477739a08eee6bde0&pid=1-s2.0-S0022404924001282-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141140210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homogeneous ACM bundles on Grassmannians of exceptional types","authors":"Xinyi Fang ,&nbsp;Yusuke Nakayama ,&nbsp;Peng Ren","doi":"10.1016/j.jpaa.2024.107729","DOIUrl":"10.1016/j.jpaa.2024.107729","url":null,"abstract":"<div><p>In this paper, we characterize homogeneous arithmetically Cohen-Macaulay (ACM) bundles over homogeneous varieties of Picard rank one in terms of their associated data. This is a generalization of the result on homogeneous varieties of Picard rank one of classical types presented by Costa and Miró-Roig for type <em>A</em>, and Du, Fang, and Ren for types <span><math><mi>B</mi><mo>,</mo><mi>C</mi></math></span> and <em>D</em>. We show that there are only finitely many irreducible homogeneous ACM bundles by twisting line bundles over Grassmannians of exceptional types. As a consequence, we prove that some Grassmannians of exceptional types are of wild representation type.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141136709","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}

{"title":"Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups","authors":"Pravin Kumar ,&nbsp;Tushar Kanta Naik ,&nbsp;Neha Nanda ,&nbsp;Mahender Singh","doi":"10.1016/j.jpaa.2024.107713","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107713","url":null,"abstract":"<div><p>The virtual braid group <span><math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the virtual twin group <span><math><mi>V</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the virtual triplet group <span><math><mi>V</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are extensions of the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are the pure virtual braid group <span><math><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the pure virtual twin group <span><math><mi>P</mi><mi>V</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the pure virtual triplet group <span><math><mi>P</mi><mi>V</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group <span><math><mi>P</mi><mi>V</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the commutator subgroup <span><math><mi>V</mi><msubsup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> of <span><math><mi>V</mi><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and the commutator subgroup <span><math><mi>V</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> of <span><math><mi>V</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Our results complete the understanding of these groups, except that of <span><math><mi>V</mi><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span>, for which the existence of a finite presentation is not known for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>. We also prove that <span><math><mi>V</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>P</mi><mi>V</mi><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup></math></span> is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141090660","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}

{"title":"Core of an ideal in Prüfer domains","authors":"Salah Kabbaj ,&nbsp;Abdeslam Mimouni ,&nbsp;Bruce Olberding","doi":"10.1016/j.jpaa.2024.107716","DOIUrl":"10.1016/j.jpaa.2024.107716","url":null,"abstract":"<div><p>This paper contributes to the study of the core of an ideal in integral domains. Our aim is to develop explicit formulas for the core in various classes of Prüfer domains. We pay particular attention to relevant ideal-theoretic notions such as stability, invertibility, and <em>h</em>-local property. We also provide decomposition results for the core of an ideal in integral domains with effectual ramifications to Prüfer domains. All main results are illustrated with original examples, where we explicitly compute the core. We also provide counter-examples to test the limits of the assumptions used in the main results.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141037791","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}

{"title":"Symmetry classes of quantum quasigroups","authors":"Bokhee Im ,&nbsp;Alex W. Nowak ,&nbsp;Jonathan D.H. Smith","doi":"10.1016/j.jpaa.2024.107722","DOIUrl":"10.1016/j.jpaa.2024.107722","url":null,"abstract":"<div><p>The theory of groups has a twofold symmetry, sending a group to its opposite. Groups invariant under the symmetry are abelian. The theory of quasigroups has a richer, sixfold symmetry, obtained by permuting the multiplication with its two divisions. The Sixfold Way identifies the various classes of quasigroups which are invariant under the respective subgroups of the symmetry group of the theory.</p><p>Quantum quasigroups provide a self-dual framework to unify the study of quasigroups and Hopf algebras. The goal of this paper is to classify the symmetry classes of quantum quasigroups. Corresponding to the Sixfold Way for classical quasigroups, we are able to identify a Sevenfold Way for general classes exhibiting a symmetry, and initiate a study of a fuller symmetry which holds for linear quantum quasigroups.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141042448","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}

{"title":"Inner autoequivalences in general and those of monoidal categories in particular","authors":"Pieter Hofstra ,&nbsp;Martti Karvonen","doi":"10.1016/j.jpaa.2024.107717","DOIUrl":"https://doi.org/10.1016/j.jpaa.2024.107717","url":null,"abstract":"<div><p>We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence of binary coproducts, unifying various known one-dimensional results and providing tractable computational tools in the two-dimensional setting. In particular, we show that the isotropy 2-group of a monoidal category coincides with its <em>Picard</em> 2<em>-group</em>, i.e., the 2-group on its weakly invertible objects.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001142/pdfft?md5=fefd005f074cdc8059ba284878a85a38&pid=1-s2.0-S0022404924001142-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141096025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Seshadri constants and AG codes of vector bundles","authors":"Tohru Nakashima","doi":"10.1016/j.jpaa.2024.107720","DOIUrl":"10.1016/j.jpaa.2024.107720","url":null,"abstract":"<div><p>We give estimates for the minimum distances of the algebraic geometric codes on certain fibered varieties. By means of the Seshadri constants, we also obtain similar results for the codes defined from vector bundles of higher rank.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141039712","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}