Journal of Algebra最新文献

{"title":"Homogeneous quandles with abelian inner automorphism groups","authors":"","doi":"10.1016/j.jalgebra.2024.09.004","DOIUrl":"10.1016/j.jalgebra.2024.09.004","url":null,"abstract":"<div><div>In this paper, we give a characterization of homogeneous quandles with abelian inner automorphism groups. In particular, we show that such a quandle is expressed as an abelian extension of a trivial quandle. Our construction is a generalization of the recent work by Furuki and Tamaru, which gives a construction of disconnected flat quandles.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319859","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":"On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn","authors":"","doi":"10.1016/j.jalgebra.2024.08.033","DOIUrl":"10.1016/j.jalgebra.2024.08.033","url":null,"abstract":"<div><div>In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a <em>p</em>-adic version of this theory started to emerge: there are <em>p</em>-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for <em>p</em>-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic <span><math><mi>ℓ</mi><mo>≠</mo><mi>p</mi></math></span> has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain <em>p</em>-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for <span><math><msub><mrow><mi>GL</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319746","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":"Ore localisation for differential graded rings; towards Goldie's theorem for differential graded algebras","authors":"","doi":"10.1016/j.jalgebra.2024.08.032","DOIUrl":"10.1016/j.jalgebra.2024.08.032","url":null,"abstract":"<div><div>We study Ore localisation of differential graded algebras. Further we define dg-prime rings, dg-semiprime rings, and study the dg-nil radical of dg-rings. Then, we define dg-essential submodules, dg-uniform dimension, and apply all this to a dg-version of Goldie's theorem on prime dg-rings.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142316206","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":"Metric ultraproducts of groups — Simplicity, perfectness and torsion","authors":"","doi":"10.1016/j.jalgebra.2024.09.005","DOIUrl":"10.1016/j.jalgebra.2024.09.005","url":null,"abstract":"<div><div>We characterise the simplicity of metric ultraproducts of a family of metric groups. We also present several new examples of simple groups, such as metric ultraproducts of finite and infinite symmetric groups, linear groups, and interval exchange transformation groups. Using similar methods, we also examine concepts such as genericity, perfectness, and torsion.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326804","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":"On the polynomiality conjecture of cluster realization of quantum groups","authors":"","doi":"10.1016/j.jalgebra.2024.08.031","DOIUrl":"10.1016/j.jalgebra.2024.08.031","url":null,"abstract":"<div><div>In this paper, we give a sufficient and necessary condition for a regular element of a quantum cluster algebra <span><math><msub><mrow><mi>O</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> to be universally polynomial. This resolves several conjectures by the first author on the polynomiality of the cluster realization of quantum group generators in different families of positive representations.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142319747","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":"Seminormal forms for the Temperley-Lieb algebra","authors":"","doi":"10.1016/j.jalgebra.2024.09.003","DOIUrl":"10.1016/j.jalgebra.2024.09.003","url":null,"abstract":"<div><p>Let <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>Q</mi></mrow></msubsup></math></span> be the rational Temperley-Lieb algebra, with loop parameter 2. In the first part of the paper we study the seminormal idempotents <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> for <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>Q</mi></mrow></msubsup></math></span> for <span><math><mi>t</mi></math></span> running over two-column standard tableaux. Our main result is here a concrete combinatorial construction of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> using Jones-Wenzl idempotents <span><math><msub><mrow><mi>JW</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> for <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>Q</mi></mrow></msubsup></math></span> where <span><math><mi>k</mi><mo>≤</mo><mi>n</mi></math></span>.</p><p>In the second part of the paper we consider the Temperley-Lieb algebra <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span>. The KLR-approach to <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup></math></span> gives rise to an action of a symmetric group <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> on <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup></math></span>, for some <span><math><mi>m</mi><mo>&lt;</mo><mi>n</mi></math></span>. We show that the <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>'s from the first part of the paper are simultaneous eigenvectors for the associated Jucys-Murphy elements for <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span>. This leads to a KLR-interpretation of the <em>p</em>-Jones-Wenzl idempotent <span><math><mmultiscripts><mrow><mi>JW</mi></mrow><mrow><mi>n</mi></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mi>p</mi></mrow></mmultiscripts></math></span> for <span><math><msubsup><mrow><mi>TL</mi></mrow><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></msubsup></math></span>, that was introduced recently by Burull, Libedinsky and Sentinelli.</p></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142243690","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":"Characteristic subgroups and the R∞-property for virtual braid groups","authors":"","doi":"10.1016/j.jalgebra.2024.09.002","DOIUrl":"10.1016/j.jalgebra.2024.09.002","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;) denote the virtual braid group (resp. virtual pure braid group), let &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;) denote the welded braid group (resp. welded pure braid group) and let &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (resp. &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the group &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; the groups &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are characteristic subgroups of &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, respectively. In the second part of the paper we show that, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the virtual braid group &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, the unrestricted virtual pure braid group &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;P&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;, and the unrestricted virtual braid group &lt;span&gt;&lt;math&gt;&lt;mi&gt;U&lt;/mi&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; have the R&lt;sub&gt;∞&lt;/sub&gt;-property. As a consequence of the technique used for few strings we also prove that, for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, the welded braid group &lt;span&gt;&lt;math&gt;&lt;mi&gt;W&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; has the R&lt;sub&gt;∞&lt;/sub&gt;-property and that for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; the corresponding pure braid groups have the R&lt;sub&gt;∞&lt;/sub&gt;-property. On the other hand for &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; it is unknown if the R&lt;sub&gt;∞&lt;/sub&gt;-property holds or not for the virtual pure braid group &lt;span&gt;&lt;math&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;m","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142230718","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":"Central extensions of axial algebras","authors":"","doi":"10.1016/j.jalgebra.2024.09.001","DOIUrl":"10.1016/j.jalgebra.2024.09.001","url":null,"abstract":"<div><p>In this article, we develop a further adaptation of the method of Skjelbred-Sund to construct central extensions of axial algebras. We use our method to prove that all axial central extensions (with respect to a maximal set of axes) of complex simple finite-dimensional Jordan algebras are split, and that all non-split axial central extensions of dimension <span><math><mi>n</mi><mo>≤</mo><mn>4</mn></math></span> over an algebraically closed field of characteristic not 2 are Jordan. Also, we give a classification of 2-dimensional axial algebras and describe some important properties of these algebras.</p></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173773","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":"Groups with finitely many isomorphism classes of non-modular subgroups","authors":"","doi":"10.1016/j.jalgebra.2024.08.030","DOIUrl":"10.1016/j.jalgebra.2024.08.030","url":null,"abstract":"<div><p>Groups in which the non-moduar subgroups fall into finitely many isomorphism classes are considered, and it is proved that a (generalized) soluble group with this property either has modular subgroup lattice or is a minimax group. The corresponding result for (generalized) soluble groups with finitely many isomorphism classes of non-permutable subgroups is also obtained.</p></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021869324004873/pdfft?md5=3e23a7f7929a6d68ab2031a2b3a63eb9&pid=1-s2.0-S0021869324004873-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142162681","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":"Modular tensor categories arising from central extensions and related applications","authors":"","doi":"10.1016/j.jalgebra.2024.08.028","DOIUrl":"10.1016/j.jalgebra.2024.08.028","url":null,"abstract":"<div><p>A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to <em>n</em>-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section <span><span>2</span></span> for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of <span><math><mi>Rep</mi><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></math></span> (the representation category of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>). By further studying the simplicity of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> (whether it is a simple Hopf algebra or not), we conclude that</p><ul><li><span>(1)</span><span><p>there exists a twist <em>J</em> of <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>s</mi><msubsup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>⊕</mo><mn>3</mn></mrow></msubsup><mo>)</mo></math></span> such that <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>q</mi></mrow></msub><msup><mrow><mo>(</mo><mi>s</mi><msubsup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>⊕</mo><mn>3</mn></mrow></msubsup><mo>)</mo></mrow><mrow><mi>J</mi></mrow></msup></math></span> is a simple Hopf algebra,</p></span></li><li><span>(2)</span><span><p>there is no relation between the simplicity of a Hopf algebra <em>H</em> and the primality of <span><math><mi>Rep</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>,</p></span></li><li><span>(3)</span><span><p>there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.</p></span></li></ul></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173781","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}