Pub Date : 2023-05-01DOI: 10.1016/s1878-6480(23)00206-9
{"title":"INDEX","authors":"","doi":"10.1016/s1878-6480(23)00206-9","DOIUrl":"https://doi.org/10.1016/s1878-6480(23)00206-9","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"431 - 431"},"PeriodicalIF":0.7,"publicationDate":"2023-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88200640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-11DOI: 10.1017/s1446788723000022
Bob Oliver
Abstract For a finite abelian p -group A and a subgroup $Gamma le operatorname {mathrm {Aut}}(A)$ , we say that the pair $(Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${mathcal {F}}$ over a finite p -group $Sge A$ such that $C_S(A)=A$ , $operatorname {mathrm {Aut}}_{{mathcal {F}}}(A)=Gamma $ as subgroups of $operatorname {mathrm {Aut}}(A)$ , and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $Gamma $ one of the Mathieu groups, that the only ${mathbb {F}}_pGamma $ -modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.
{"title":"NONREALIZABILITY OF CERTAIN REPRESENTATIONS IN FUSION SYSTEMS","authors":"Bob Oliver","doi":"10.1017/s1446788723000022","DOIUrl":"https://doi.org/10.1017/s1446788723000022","url":null,"abstract":"Abstract For a finite abelian p -group A and a subgroup $Gamma le operatorname {mathrm {Aut}}(A)$ , we say that the pair $(Gamma ,A)$ is fusion realizable if there is a saturated fusion system ${mathcal {F}}$ over a finite p -group $Sge A$ such that $C_S(A)=A$ , $operatorname {mathrm {Aut}}_{{mathcal {F}}}(A)=Gamma $ as subgroups of $operatorname {mathrm {Aut}}(A)$ , and . In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $Gamma $ one of the Mathieu groups, that the only ${mathbb {F}}_pGamma $ -modules that are fusion realizable (up to extensions by trivial modules) are the Todd modules and in some cases their duals.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"208 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134955096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-06DOI: 10.1017/s144678872200026x
{"title":"JAZ volume 114 issue 2 Cover and Back matter","authors":"","doi":"10.1017/s144678872200026x","DOIUrl":"https://doi.org/10.1017/s144678872200026x","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"51 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83406120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-03-06DOI: 10.1017/s1446788722000258
{"title":"JAZ volume 114 issue 2 Cover and Front matter","authors":"","doi":"10.1017/s1446788722000258","DOIUrl":"https://doi.org/10.1017/s1446788722000258","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"4 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74892298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-20DOI: 10.1017/s1446788722000386
Aleš Drápal, Ian M. Wanless
Abstract Let q be an odd prime power and suppose that $a,bin mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z Leftrightarrow x=y=z$ . Denote by $sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $alpha approx 0.029,08$ and $beta approx 0.012,59$ such that if $qequiv 1 bmod 4$ , then $lim sigma (q)/q^2 = alpha $ , and if $q equiv 3 bmod 4$ , then $lim sigma (q)/q^2 = beta $ .
{"title":"ON THE NUMBER OF QUADRATIC ORTHOMORPHISMS THAT PRODUCE MAXIMALLY NONASSOCIATIVE QUASIGROUPS","authors":"Aleš Drápal, Ian M. Wanless","doi":"10.1017/s1446788722000386","DOIUrl":"https://doi.org/10.1017/s1446788722000386","url":null,"abstract":"Abstract Let q be an odd prime power and suppose that $a,bin mathbb {F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (mathbb {F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ if $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z Leftrightarrow x=y=z$ . Denote by $sigma (q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $alpha approx 0.029,08$ and $beta approx 0.012,59$ such that if $qequiv 1 bmod 4$ , then $lim sigma (q)/q^2 = alpha $ , and if $q equiv 3 bmod 4$ , then $lim sigma (q)/q^2 = beta $ .","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135081074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-13DOI: 10.1017/s1446788722000234
{"title":"JAZ volume 114 issue 1 Cover and Front matter","authors":"","doi":"10.1017/s1446788722000234","DOIUrl":"https://doi.org/10.1017/s1446788722000234","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"86 1","pages":"f1 - f2"},"PeriodicalIF":0.7,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77056398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-13DOI: 10.1017/s1446788722000246
{"title":"JAZ volume 114 issue 1 Cover and Back matter","authors":"","doi":"10.1017/s1446788722000246","DOIUrl":"https://doi.org/10.1017/s1446788722000246","url":null,"abstract":"","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"1 1","pages":"b1 - b2"},"PeriodicalIF":0.7,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88798013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-13DOI: 10.1017/s1446788722000222
Xiaoyan Yang
The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport, and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Additionally, we include some computations of cosupport and provide a comparison of support and cosupport for cohomologically finite objects. Finally, we assign to any object of the category a subset of $mathrm {Spec}R$ , called the big cosupport, and study some of its properties.
{"title":"COSUPPORT FOR COMPACTLY GENERATED TRIANGULATED CATEGORIES","authors":"Xiaoyan Yang","doi":"10.1017/s1446788722000222","DOIUrl":"https://doi.org/10.1017/s1446788722000222","url":null,"abstract":"\u0000 The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport, and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Additionally, we include some computations of cosupport and provide a comparison of support and cosupport for cohomologically finite objects. Finally, we assign to any object of the category a subset of \u0000 \u0000 \u0000 \u0000$mathrm {Spec}R$\u0000\u0000 \u0000 , called the big cosupport, and study some of its properties.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"42 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80670884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-12DOI: 10.1017/s1446788722000192
REZA ESMAILVANDI, MEHDI NEMATI, NAGESWARAN SHRAVAN KUMAR
Let H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$-algebra. We also prove that the double dual $A(H)^{ast ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{ast ast }.$
{"title":"ON THE ALGEBRAS","authors":"REZA ESMAILVANDI, MEHDI NEMATI, NAGESWARAN SHRAVAN KUMAR","doi":"10.1017/s1446788722000192","DOIUrl":"https://doi.org/10.1017/s1446788722000192","url":null,"abstract":"<p>Let <span>H</span> be an ultraspherical hypergroup and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)$</span></span></img></span></span> be the Fourier algebra associated with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$H.$</span></span></img></span></span> In this paper, we study the dual and the double dual of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$A(H).$</span></span></img></span></span> We prove among other things that the subspace of all uniformly continuous functionals on <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)$</span></span></img></span></span> forms a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$C^*$</span></span></img></span></span>-algebra. We also prove that the double dual <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)^{ast ast }$</span></span></img></span></span> is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup <span>H</span> is finite. Finally, we study the unit elements of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231109001332944-0291:S1446788722000192:S1446788722000192_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$A(H)^{ast ast }.$</span></span></img></span></span></p>","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"35 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-14DOI: 10.1017/s1446788723000058
V. Gupta, Deepika Sharma
We show that all values in the interval $[0,{pi }/{2}]$ can be attained as interior angles between intermediate subalgebras (as introduced by Bakshi and the first named author [‘Lattice of intermediate subalgebras’, J. Lond. Math. Soc. (2)104(2) (2021), 2082–2127]) of a certain inclusion of simple unital $C^*$ -algebras. We also calculate the interior angles between intermediate crossed product subalgebras of any inclusion of crossed product algebras corresponding to any action of a countable discrete group and its subgroups on a unital $C^*$ -algebra.
我们证明了区间$[0，{pi}/{2}]$中的所有值都可以作为中间子代数(由Bakshi和第一作者[' Lattice of intermediate subalgebras '， J. Lond引入)之间的内角来获得。数学。Soc。(2)104(2)(2021)， 2082-2127])的简单一元$C^*$ -代数的一定包含。我们还计算了可数离散群及其子群在一元C^*$ -代数上的任意作用所对应的任意交叉积代数包含的中间交叉积子代数之间的内角。
{"title":"ON POSSIBLE VALUES OF THE INTERIOR ANGLE BETWEEN INTERMEDIATE SUBALGEBRAS","authors":"V. Gupta, Deepika Sharma","doi":"10.1017/s1446788723000058","DOIUrl":"https://doi.org/10.1017/s1446788723000058","url":null,"abstract":"\u0000 We show that all values in the interval \u0000 \u0000 \u0000 \u0000$[0,{pi }/{2}]$\u0000\u0000 \u0000 can be attained as interior angles between intermediate subalgebras (as introduced by Bakshi and the first named author [‘Lattice of intermediate subalgebras’, J. Lond. Math. Soc. (2)104(2) (2021), 2082–2127]) of a certain inclusion of simple unital \u0000 \u0000 \u0000 \u0000$C^*$\u0000\u0000 \u0000 -algebras. We also calculate the interior angles between intermediate crossed product subalgebras of any inclusion of crossed product algebras corresponding to any action of a countable discrete group and its subgroups on a unital \u0000 \u0000 \u0000 \u0000$C^*$\u0000\u0000 \u0000 -algebra.","PeriodicalId":50007,"journal":{"name":"Journal of the Australian Mathematical Society","volume":"89 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85952665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$A(H)$ be the Fourier algebra associated with 
$H.$ In this paper, we study the dual and the double dual of 
$A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on 
$A(H)$ forms a 
$C^*$-algebra. We also prove that the double dual 
$A(H)^{ast ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of 
$A(H)^{ast ast }.$
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