{"title":"弱霍普夫布拉斯、粉碎乘积及其在邻接稳定布拉斯中的应用","authors":"Zhimin Liu, Shenglin Zhu","doi":"10.1142/s0219498825501567","DOIUrl":null,"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":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Weak Hopf algebras, smash products and applications to adjoint-stable algebras\",\"authors\":\"Zhimin Liu, Shenglin Zhu\",\"doi\":\"10.1142/s0219498825501567\",\"DOIUrl\":null,\"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\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-01-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219498825501567\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825501567","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
对于特征为零的域 k 上的半简单准三角形霍普夫代数(H,R)和强可分离量子交换 H 模块代数 A,我们证明 A#H 是弱霍普夫代数,并且它可以嵌入弱霍普夫代数 EndA∗⊗H 中。有了这些结构,A#HMod 就是科恩和韦斯特里希引入的单元范畴,而 EndA∗⊗Hℳ 与 Hℳ 是张量等价的。如果 A 位于 Hℳ 的 Müger 中心,那么嵌入就是准三角形弱霍普夫代数态射。这就解释了在有限群代数的不可还原Yetter-Drinfeld模块的表征中存在子群包含的原因。
Weak Hopf algebras, smash products and applications to adjoint-stable algebras
For a semisimple quasi-triangular Hopf algebra over a field of characteristic zero, and a strongly separable quantum commutative -module algebra , we show that is a weak Hopf algebra, and it can be embedded into a weak Hopf algebra . With these structures, is the monoidal category introduced by Cohen and Westreich, and is tensor equivalent to . If is in the Müger center of , 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.