Journal of AlgebraPub Date : 2024-09-24DOI: 10.1016/j.jalgebra.2024.09.006
Yuanchang Lin , Peng Zhou , Chengming Bai
{"title":"Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras","authors":"Yuanchang Lin , Peng Zhou , Chengming Bai","doi":"10.1016/j.jalgebra.2024.09.006","DOIUrl":"10.1016/j.jalgebra.2024.09.006","url":null,"abstract":"<div><div>It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given.</div></div>","PeriodicalId":14888,"journal":{"name":"Journal of Algebra","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142357553","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}
Journal of AlgebraPub Date : 2024-09-17DOI: 10.1016/j.jalgebra.2024.09.004
Takuya Saito , Sakumi Sugawara
{"title":"Homogeneous quandles with abelian inner automorphism groups","authors":"Takuya Saito , Sakumi Sugawara","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}
Journal of AlgebraPub Date : 2024-09-17DOI: 10.1016/j.jalgebra.2024.08.033
Jakub Löwit
{"title":"On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn","authors":"Jakub Löwit","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}
Journal of AlgebraPub Date : 2024-09-17DOI: 10.1016/j.jalgebra.2024.08.034
Yuta Kimura , Hiroyuki Minamoto , Kota Yamaura
{"title":"Tilting theory for finite dimensional 1-Iwanaga-Gorenstein algebras","authors":"Yuta Kimura , Hiroyuki Minamoto , Kota Yamaura","doi":"10.1016/j.jalgebra.2024.08.034","DOIUrl":"10.1016/j.jalgebra.2024.08.034","url":null,"abstract":"<div><div>We study tilting objects of the stable category <span><math><msup><mrow><munder><mrow><mrow><mi>CM</mi></mrow></mrow><mo>_</mo></munder></mrow><mrow><mi>Z</mi></mrow></msup><mspace></mspace><mi>A</mi></math></span> of graded Cohen-Macaulay modules over a finite dimensional graded Iwanaga-Gorenstein algebra <em>A</em>. We first show that if there exists a tilting object in <span><math><msup><mrow><munder><mrow><mrow><mi>CM</mi></mrow></mrow><mo>_</mo></munder></mrow><mrow><mi>Z</mi></mrow></msup><mspace></mspace><mi>A</mi></math></span>, then its endomorphism algebra always has finite global dimension. Next, to study the existence of a tilting object, we introduce a numerical invariant <span><math><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. In the case where <em>A</em> is 1-Iwanaga-Gorenstein, we give a sufficient condition on <span><math><mi>g</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> for the existence of a tilting object. As an application, for a truncated preprojective algebra <span><math><mi>Π</mi><msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mi>w</mi></mrow></msub></math></span> of a tree quiver <em>Q</em>, we prove that <span><math><msup><mrow><munder><mrow><mrow><mi>CM</mi></mrow></mrow><mo>_</mo></munder></mrow><mrow><mi>Z</mi></mrow></msup><mspace></mspace><mi>Π</mi><msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mi>w</mi></mrow></msub></math></span> always admits a tilting object.</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":"142427258","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}
Journal of AlgebraPub Date : 2024-09-17DOI: 10.1016/j.jalgebra.2024.08.032
Alexander Zimmermann
{"title":"Ore localisation for differential graded rings; towards Goldie's theorem for differential graded algebras","authors":"Alexander Zimmermann","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}
Journal of AlgebraPub Date : 2024-09-17DOI: 10.1016/j.jalgebra.2024.09.005
Jakub Gismatullin , Krzysztof Majcher , Martin Ziegler
{"title":"Metric ultraproducts of groups — Simplicity, perfectness and torsion","authors":"Jakub Gismatullin , Krzysztof Majcher , Martin Ziegler","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}
Journal of AlgebraPub Date : 2024-09-17DOI: 10.1016/j.jalgebra.2024.08.031
Ivan Chi-Ho Ip , Jeff York Ye
{"title":"On the polynomiality conjecture of cluster realization of quantum groups","authors":"Ivan Chi-Ho Ip , Jeff York Ye","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}
Journal of AlgebraPub Date : 2024-09-10DOI: 10.1016/j.jalgebra.2024.09.003
Katherine Ormeño Bastías , Steen Ryom-Hansen
{"title":"Seminormal forms for the Temperley-Lieb algebra","authors":"Katherine Ormeño Bastías , Steen Ryom-Hansen","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>></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><</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}
Journal of AlgebraPub Date : 2024-09-06DOI: 10.1016/j.jalgebra.2024.09.002
Karel Dekimpe , Daciberg Lima Gonçalves , Oscar Ocampo
{"title":"Characteristic subgroups and the R∞-property for virtual braid groups","authors":"Karel Dekimpe , Daciberg Lima Gonçalves , Oscar Ocampo","doi":"10.1016/j.jalgebra.2024.09.002","DOIUrl":"10.1016/j.jalgebra.2024.09.002","url":null,"abstract":"<div><p>Let <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>. Let <span><math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. <span><math><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) denote the virtual braid group (resp. virtual pure braid group), let <span><math><mi>W</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. <span><math><mi>W</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) denote the welded braid group (resp. welded pure braid group) and let <span><math><mi>U</mi><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. <span><math><mi>U</mi><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, the group <span><math><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and for <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> the groups <span><math><mi>W</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><mi>U</mi><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are characteristic subgroups of <span><math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, <span><math><mi>W</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and <span><math><mi>U</mi><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, respectively. In the second part of the paper we show that, for <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, the virtual braid group <span><math><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the unrestricted virtual pure braid group <span><math><mi>U</mi><mi>V</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and the unrestricted virtual braid group <span><math><mi>U</mi><mi>V</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> have the R<sub>∞</sub>-property. As a consequence of the technique used for few strings we also prove that, for <span><math><mi>n</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn></math></span>, the welded braid group <span><math><mi>W</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> has the R<sub>∞</sub>-property and that for <span><math><mi>n</mi><mo>=</mo><mn>2</mn></math></span> the corresponding pure braid groups have the R<sub>∞</sub>-property. On the other hand for <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> it is unknown if the R<sub>∞</sub>-property holds or not for the virtual pure braid group <span><math><mi>V</mi><msub><mrow><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}
Journal of AlgebraPub Date : 2024-09-05DOI: 10.1016/j.jalgebra.2024.09.001
Ivan Kaygorodov , Cándido Martín González , Pilar Páez-Guillán
{"title":"Central extensions of axial algebras","authors":"Ivan Kaygorodov , Cándido Martín González , Pilar Páez-Guillán","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}