{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}