Pub Date : 2021-10-14DOI: 10.1017/S0017089521000458
I. Vergara
Abstract We show that if G is an amenable group and H is a hyperbolic group, then the free product $Gast H$ is weakly amenable. A key ingredient in the proof is the fact that $Gast H$ is orbit equivalent to $mathbb{Z}ast H$ .
{"title":"Weak amenability of free products of hyperbolic and amenable groups","authors":"I. Vergara","doi":"10.1017/S0017089521000458","DOIUrl":"https://doi.org/10.1017/S0017089521000458","url":null,"abstract":"Abstract We show that if G is an amenable group and H is a hyperbolic group, then the free product \u0000$Gast H$\u0000 is weakly amenable. A key ingredient in the proof is the fact that \u0000$Gast H$\u0000 is orbit equivalent to \u0000$mathbb{Z}ast H$\u0000 .","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"698 - 701"},"PeriodicalIF":0.5,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44149633","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 : 2021-10-09DOI: 10.1017/S0017089522000106
Xin Li
Abstract We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.
{"title":"Left regular representations of Garside categories I. C*-algebras and groupoids","authors":"Xin Li","doi":"10.1017/S0017089522000106","DOIUrl":"https://doi.org/10.1017/S0017089522000106","url":null,"abstract":"Abstract We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-algebras. As another application, we give a complete analysis of the ideal structures of C*-algebras generated by left regular representations of Artin–Tits monoids.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"S53 - S86"},"PeriodicalIF":0.5,"publicationDate":"2021-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47764107","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 : 2021-09-20DOI: 10.1017/s001708952100032x
D. Schütz
{"title":"CORRIGENDUM TO: A FAST ALGORITHM FOR CALCULATING S-INVARIANTS","authors":"D. Schütz","doi":"10.1017/s001708952100032x","DOIUrl":"https://doi.org/10.1017/s001708952100032x","url":null,"abstract":"","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"526 - 526"},"PeriodicalIF":0.5,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45320381","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 : 2021-09-20DOI: 10.1017/S001708952200009X
Ben Davison
Abstract We introduce and study a fermionisation procedure for the cohomological Hall algebra $mathcal{H}_{Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.
摘要我们介绍并研究了上同调Hall代数$mathcal的一个费米子过程{H}_｛Pi_Q｝$预投影代数的表示，其选择性地将BPS李代数的上同调奇偶性从偶数切换到奇数。我们通过变形降维确定Etingof和Rains工作中研究的预投影代数的中心扩展的上同调Donaldson–Thomas不变量来实现这一点。通过相同的技术，我们确定了Crawley–Boevey和Holland引入的$unicode{x03BC}$变形预投影代数的表示堆栈的Borel–Moore同调，用于所有维度向量。这提供了Crawley Boevey和Van den Bergh关于变形预投影代数的表示的光滑模方案的上同调的结果的共同推广，以及我先前关于未变形预投影代的表示堆栈的Borel–Moore同调的结论。
{"title":"A boson-fermion correspondence in cohomological Donaldson–Thomas theory","authors":"Ben Davison","doi":"10.1017/S001708952200009X","DOIUrl":"https://doi.org/10.1017/S001708952200009X","url":null,"abstract":"Abstract We introduce and study a fermionisation procedure for the cohomological Hall algebra \u0000$mathcal{H}_{Pi_Q}$\u0000 of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the \u0000$unicode{x03BC}$\u0000 -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"S28 - S52"},"PeriodicalIF":0.5,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44380003","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 : 2021-08-25DOI: 10.1017/S0017089522000192
Agustín García Iglesias, José Ignacio Sánchez
Abstract We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type $A_2$ with $q=-1$ , a.k.a. the positive part of the small quantum group $mathfrak{u}^+_{sqrt{-text{1}}}(mathfrak{sl}_3)$ . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.
{"title":"On the computation of Hopf 2-cocycles with an example of diagonal type","authors":"Agustín García Iglesias, José Ignacio Sánchez","doi":"10.1017/S0017089522000192","DOIUrl":"https://doi.org/10.1017/S0017089522000192","url":null,"abstract":"Abstract We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type \u0000$A_2$\u0000 with \u0000$q=-1$\u0000 , a.k.a. the positive part of the small quantum group \u0000$mathfrak{u}^+_{sqrt{-text{1}}}(mathfrak{sl}_3)$\u0000 . We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"141 - 169"},"PeriodicalIF":0.5,"publicationDate":"2021-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48111784","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 : 2021-08-23DOI: 10.1017/S0017089521000240
Tamar Bar-On
Abstract We prove that the profinite completion of a profinite projective group is projective.
摘要我们证明了profinite射影群的profinite完备是射影的。
{"title":"THE PROFINITE COMPLETION OF A PROFINITE PROJECTIVE GROUP","authors":"Tamar Bar-On","doi":"10.1017/S0017089521000240","DOIUrl":"https://doi.org/10.1017/S0017089521000240","url":null,"abstract":"Abstract We prove that the profinite completion of a profinite projective group is projective.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"499 - 503"},"PeriodicalIF":0.5,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47915355","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 : 2021-08-09DOI: 10.1017/S0017089522000015
É. Fornaroli, M. Khrypchenko, E. A. Santulo
Abstract Let X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.
{"title":"Proper Lie automorphisms of incidence algebras","authors":"É. Fornaroli, M. Khrypchenko, E. A. Santulo","doi":"10.1017/S0017089522000015","DOIUrl":"https://doi.org/10.1017/S0017089522000015","url":null,"abstract":"Abstract Let X be a finite connected poset and K a field. We study the question, when all Lie automorphisms of the incidence algebra I(X, K) are proper. Without any restriction on the length of X, we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of X. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns, and ordinal sums of two anti-chains, we give a complete answer.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"702 - 715"},"PeriodicalIF":0.5,"publicationDate":"2021-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42368348","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 : 2021-07-30DOI: 10.1017/S0017089521000239
N. Dutertre, Juan Antonio Moya Pérez
Abstract Let $f,{:},(mathbb R^n,0)to (mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F,{:}, (mathbb{R}^{1+n},0) to (mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) cap B_epsilon^n$ , $0 < vert t vert ll epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.
{"title":"TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES","authors":"N. Dutertre, Juan Antonio Moya Pérez","doi":"10.1017/S0017089521000239","DOIUrl":"https://doi.org/10.1017/S0017089521000239","url":null,"abstract":"Abstract Let \u0000$f,{:},(mathbb R^n,0)to (mathbb R,0)$\u0000 be an analytic function germ with non-isolated singularities and let \u0000$F,{:}, (mathbb{R}^{1+n},0) to (mathbb{R},0)$\u0000 be a 1-parameter deformation of f. Let \u0000$ f_t ^{-1}(0) cap B_epsilon^n$\u0000 , \u0000$0 < vert t vert ll epsilon$\u0000 , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"64 1","pages":"484 - 498"},"PeriodicalIF":0.5,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/S0017089521000239","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47173321","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}
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