{"title":"Moufang quadrangles and affine twin buildings of type C2","authors":"Bernhard Mühlherr, Hendrik Van Maldeghem","doi":"10.2140/iig.2023.20.431","DOIUrl":"https://doi.org/10.2140/iig.2023.20.431","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"154 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Synthetic and projective properties of embeddable polar spaces","authors":"Antonio Pasini","doi":"10.2140/iig.2023.20.519","DOIUrl":"https://doi.org/10.2140/iig.2023.20.519","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134990251","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2000, Marc Burger and Shahar Mozes introduced universal groups acting on trees. Such groups provide interesting examples of totally disconnected locally compact groups. Intuitively, these are the largest groups for which all local actions satisfy a prescribed behavior. Since then, their study has evolved in various directions. In particular, Adrien Le Boudec has studied restricted universal groups, where the prescribed behavior is allowed to be violated in a finite number of vertices. On the other hand, we have been studying universal groups acting on right-angled buildings, a class of geometric objects with a much more general structure than trees. The aim of the current paper is to combine both ideas: we will study restricted universal groups acting on right-angled buildings. We show several permutational and topological properties of those groups, with as main result a precise criterion for when these groups are simple.
{"title":"Restricted universal groups for right-angled buildings","authors":"Jens Bossaert, Tom De Medts","doi":"10.2140/iig.2023.20.177","DOIUrl":"https://doi.org/10.2140/iig.2023.20.177","url":null,"abstract":"In 2000, Marc Burger and Shahar Mozes introduced universal groups acting on trees. Such groups provide interesting examples of totally disconnected locally compact groups. Intuitively, these are the largest groups for which all local actions satisfy a prescribed behavior. Since then, their study has evolved in various directions. In particular, Adrien Le Boudec has studied restricted universal groups, where the prescribed behavior is allowed to be violated in a finite number of vertices. On the other hand, we have been studying universal groups acting on right-angled buildings, a class of geometric objects with a much more general structure than trees. The aim of the current paper is to combine both ideas: we will study restricted universal groups acting on right-angled buildings. We show several permutational and topological properties of those groups, with as main result a precise criterion for when these groups are simple.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given two elements $A,B$ in a compact semisimple Lie algebra, we show that there is a regular element $X$ and elements $Y,Z$ with $A=[X,Y]$ and $B=[X,Z]$. In the course of the proof we show also that every linear subspace $V$ of codimension at most 2 in the Lie algebra contains a CSA.
{"title":"A note on commutators in compact semisimple Lie algebras","authors":"Linus Kramer","doi":"10.2140/iig.2023.20.317","DOIUrl":"https://doi.org/10.2140/iig.2023.20.317","url":null,"abstract":"Given two elements $A,B$ in a compact semisimple Lie algebra, we show that there is a regular element $X$ and elements $Y,Z$ with $A=[X,Y]$ and $B=[X,Z]$. In the course of the proof we show also that every linear subspace $V$ of codimension at most 2 in the Lie algebra contains a CSA.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On inclusions of exceptional long root geometries of type E","authors":"Anneleen De Schepper, Hendrik Van Maldeghem","doi":"10.2140/iig.2023.20.247","DOIUrl":"https://doi.org/10.2140/iig.2023.20.247","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the Kneser graph of objects of a fixed type in a building of spherical type has the unique coclique extension property when the corresponding representation has minuscule weight and also when the diagram is simply laced and the representation is adjoint.
{"title":"The unique coclique extension property for apartments of buildings","authors":"Andries E. Brouwer, Jan Draisma, Çiçek Güven","doi":"10.2140/iig.2023.20.209","DOIUrl":"https://doi.org/10.2140/iig.2023.20.209","url":null,"abstract":"We show that the Kneser graph of objects of a fixed type in a building of spherical type has the unique coclique extension property when the corresponding representation has minuscule weight and also when the diagram is simply laced and the representation is adjoint.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135690113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sur un théorème de V. V. Deodhar et de M. J. Dyer sur les groupes de Coxeter","authors":"Jean-Yves Hée","doi":"10.2140/iig.2023.20.295","DOIUrl":"https://doi.org/10.2140/iig.2023.20.295","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135733986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Short biography of Jacques Tits","authors":"Franz Bingen","doi":"10.2140/iig.2023.20.65","DOIUrl":"https://doi.org/10.2140/iig.2023.20.65","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new construction of the Lie algebra of type $E_8$, in terms of $3times3$ matrices, such that the Lie bracket has a natural description as the matrix commutator. This leads to a new interpretation of the Freudenthal-Tits magic square of Lie algebras, acting on themselves by commutation.
{"title":"An octonionic construction of E8 and the Lie algebra magic square","authors":"Robert A. Wilson, Tevian Dray, Corinne A. Manogue","doi":"10.2140/iig.2023.20.611","DOIUrl":"https://doi.org/10.2140/iig.2023.20.611","url":null,"abstract":"We give a new construction of the Lie algebra of type $E_8$, in terms of $3times3$ matrices, such that the Lie bracket has a natural description as the matrix commutator. This leads to a new interpretation of the Freudenthal-Tits magic square of Lie algebras, acting on themselves by commutation.","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134989788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mixed relations for buildings of type F4","authors":"Johannes Roth, Hendrik Van Maldeghem","doi":"10.2140/iig.2023.20.543","DOIUrl":"https://doi.org/10.2140/iig.2023.20.543","url":null,"abstract":"","PeriodicalId":36589,"journal":{"name":"Innovations in Incidence Geometry","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135734371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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