Journal of Algebra and Its Applications最新文献

{"title":"Weak Hopf algebras, smash products and applications to adjoint-stable algebras","authors":"Zhimin Liu, Shenglin Zhu","doi":"10.1142/s0219498825501567","DOIUrl":"https://doi.org/10.1142/s0219498825501567","url":null,"abstract":"<p>For a semisimple quasi-triangular Hopf algebra <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>H</mi><mo>,</mo><mi>R</mi><mo stretchy=\"false\">)</mo></math></span><span></span> over a field <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>k</mi></math></span><span></span> of characteristic zero, and a strongly separable quantum commutative <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span>-module algebra <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span>, we show that <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi><mi>#</mi><mi>H</mi></math></span><span></span> is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mo>End</mo><msup><mrow><mi>A</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup><mo stretchy=\"false\">⊗</mo><mi>H</mi></math></span><span></span>. With these structures, <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow></mrow><mrow><mi>A</mi><mi>#</mi><mi>H</mi></mrow></msub><mo>Mod</mo></math></span><span></span> is the monoidal category introduced by Cohen and Westreich, and <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow></mrow><mrow><mo>End</mo><msup><mrow><mi>A</mi></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup><mo stretchy=\"false\">⊗</mo><mi>H</mi></mrow></msub><mi mathvariant=\"cal\">ℳ</mi></math></span><span></span> is tensor equivalent to <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow></mrow><mrow><mi>H</mi></mrow></msub><mi mathvariant=\"cal\">ℳ</mi></math></span><span></span>. If <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>A</mi></math></span><span></span> is in the Müger center of <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow></mrow><mrow><mi>H</mi></mrow></msub><mi mathvariant=\"cal\">ℳ</mi></math></span><span></span>, then the embedding is a quasi-triangular weak Hopf algebra morphism. This explains the presence of a subgroup inclusion in the characterization of irreducible Yetter–Drinfeld modules for a finite group algebra.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cohomology of modified Rota–Baxter Leibniz algebra of weight λ","authors":"Bibhash Mondal, Ripan Saha","doi":"10.1142/s0219498825501579","DOIUrl":"https://doi.org/10.1142/s0219498825501579","url":null,"abstract":"<p>Rota–Baxter operators have been paid much attention in the last few decades as they have many applications in mathematics and physics. In this paper, our object of study is modified Rota–Baxter operators on Leibniz algebras. We investigate modified Rota–Baxter Leibniz algebras from the cohomological point of view. We study a one-parameter formal deformation theory of modified Rota–Baxter Leibniz algebras and define the associated deformation cohomology that controls the deformation. Finally, as an application, we characterize equivalence classes of abelian extensions in terms of second cohomology groups.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kostant’s generating functions and Mckay–Slodowy correspondence","authors":"Naihuan Jing, Zhijun Li, Danxia Wang","doi":"10.1142/s0219498825501713","DOIUrl":"https://doi.org/10.1142/s0219498825501713","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi><mo>⊴</mo><mi>G</mi></math></span><span></span> be a pair of finite subgroups of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span> and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>V</mi></math></span><span></span> a finite-dimensional fundamental <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>-module. We study Kostant’s generating functions for the decomposition of the <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mstyle><mtext mathvariant=\"normal\">SL</mtext></mstyle></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">(</mo><mi>ℂ</mi><mo stretchy=\"false\">)</mo></math></span><span></span>-module <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msup><mo stretchy=\"false\">(</mo><mi>V</mi><mo stretchy=\"false\">)</mo></math></span><span></span> restricted to <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>N</mi><mo>◃</mo><mi>G</mi></math></span><span></span> in connection with the McKay–Slodowy correspondence. In particular, the classical Kostant formula was generalized to a uniform version of the Poincaré series for the symmetric invariants in which the multiplicities of any individual module in the symmetric algebra are completely determined.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Duplex Hecke algebras of type B","authors":"Yu Xie, An Zhang, Bin Shu","doi":"10.1142/s021949882550166x","DOIUrl":"https://doi.org/10.1142/s021949882550166x","url":null,"abstract":"&lt;p&gt;As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in &lt;i&gt;Algebra Colloq.&lt;/i&gt;], in this article we first introduce a so-called duplex Hecke algebras of type &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle mathvariant=\"sans-serif\"&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mstyle&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; which is a &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;ℚ&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;-algebra associated with the Weyl group &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"script\"&gt;𝒲&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mstyle mathvariant=\"sans-serif\"&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mstyle&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; of type &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle mathvariant=\"sans-serif\"&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mstyle&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, and symmetric groups &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;𝔖&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; for &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;l&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle mathvariant=\"sans-serif\"&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mstyle&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; arising from the related &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in &lt;i&gt;Algebra Colloq.&lt;/i&gt;]). A duplex Hecke algebra of type &lt;span&gt;&lt;math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle mathvariant=\"sans-serif\"&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mstyle&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; admits natural representations on certain tensor spaces. We then establish a Levi-type &lt;span&gt;&lt;math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;-Schur–Weyl duality of type &lt;span&gt;&lt;math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mstyle mathvariant=\"sans-serif\"&gt;&lt;mi&gt;B&lt;/mi&gt;&lt;/mstyle&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;, which reveals the double centralizer property between such duplex Hecke algebras and &lt;span&gt;&lt;math altimg=\"eq-00014.g","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"69 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140150688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two results on character codegrees","authors":"Yang Liu, Yong Yang","doi":"10.1142/s0219498825501580","DOIUrl":"https://doi.org/10.1142/s0219498825501580","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> be a finite group and <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Irr</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the set of irreducible characters of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span>. The codegree of an irreducible character <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>χ</mi></math></span><span></span> of the group <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi></math></span><span></span> is defined as <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">cod</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>χ</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mo>|</mo><mi>G</mi><mo>:</mo><mstyle><mtext mathvariant=\"normal\">ker</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>χ</mi><mo stretchy=\"false\">)</mo><mo>|</mo><mo stretchy=\"false\">/</mo><mi>χ</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span>. In this paper, we study two topics related to the character codegrees. The first result is related to the prime graph of character codegrees and we show that the codegree prime graphs of several classes of groups can be characterized only by graph theoretical terms. The second result is about the <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi></math></span><span></span>-parts of the codegrees and character degrees.</p>","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"135 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140156706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A type C study of braverman-gaitsgory-ginzburg'sconstruction of sln representations","authors":"Zhijie Dong, Haitao Ma","doi":"10.1142/s0219498825501610","DOIUrl":"https://doi.org/10.1142/s0219498825501610","url":null,"abstract":". In [FMX19], it is proved that the convolution algebra of top Borel-Moore homology on Steinberg variety of type B/C realizes U ( sl θn ), where sl θn is the fixed point subalgebra of involution on sl n . So top Borel-Moore homology of the partial Springer’s fibers gives the representations of U ( sl θn ). In this paper, we study these representations using the Schur-Weyl duality and Springer theory.","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138973185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-linear Jordan Triple *-Derivation on Prime *-Algebras","authors":"Ali Taghavi","doi":"10.1142/s0219498825501646","DOIUrl":"https://doi.org/10.1142/s0219498825501646","url":null,"abstract":"","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"7 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139002628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Variations of primeness of ideals in rings of continuous functions","authors":"A. R. Aliabad, M. Ghoulipour, M. Paimann","doi":"10.1142/s0219498825501683","DOIUrl":"https://doi.org/10.1142/s0219498825501683","url":null,"abstract":"","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"25 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139002941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Partial actions of sweedler hopf algebra on generalized quaternion algebra","authors":"Yong Deng, Quanguo Chen, Dingguo Wang","doi":"10.1142/s0219498825501592","DOIUrl":"https://doi.org/10.1142/s0219498825501592","url":null,"abstract":"","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"205 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139002318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tropical matrix groups and Boolean matrix groups","authors":"Lin Yang, Miao-Miao Ren, Ling-Li Zeng","doi":"10.1142/s0219498825501671","DOIUrl":"https://doi.org/10.1142/s0219498825501671","url":null,"abstract":"","PeriodicalId":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"45 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138974896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}