{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-GQ4XHUOU/93932ff0-5895-492d-b394-4e795d561dd6/PDF","dcterms:extent":"474 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-GQ4XHUOU/2aca9af2-0507-4e27-a244-afec192ae47d/TEXT","dcterms:extent":"44 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-GQ4XHUOU","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2014","dc:creator":"Decelle, Sophie","dc:format":[{"@xml:lang":"sl","#text":"številka:1"},{"@xml:lang":"sl","#text":"letnik:7"},{"@xml:lang":"sl","#text":"str. 83-103"}],"dc:identifier":["COBISSID:16793433","ISSN:1855-3966","URN:URN:NBN:SI:doc-GQ4XHUOU"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Društvo matematikov, fizikov in astronomov Slovenije"},"dc:subject":[{"@xml:lang":"sl","#text":"algebra"},{"@xml:lang":"en","#text":"Conway-Griess-Norton algebra"},{"@xml:lang":"sl","#text":"grupe"},{"@xml:lang":"en","#text":"Majorana representation"},{"@xml:lang":"en","#text":"Monster group"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"The L sub 2(11)-subalgebra of the Monster algebra|"},"dc:description":{"@xml:lang":"sl","#text":"We study a subalgebra ?$V$? of the Monster algebra, ?$V_\\mathbb{M}$?, generated by three Majorana axes ?$a_x$?, ?$a_y$? and ?$a_z$ ?indexed by the ?$2A$?-involutions ?$x$?, ?$y$? and ?$z$? of ?$\\mathbb{M}$?, the Monster simple group. We use the notation ?$V = \\langle \\langle a_x, a_y, a_z \\rangle \\rangle$?. We assume that ?$xy$? is another ?$2A$?-involution and that each of ?$xz$?, ?$yz$? and ?$xyz$? has order 5. Thus a subgroup ?$G$? of ?$\\mathbb{M}$? generated by ?$\\{x, y, z\\}$? is a non-trivial quotient of the group ?$G^{(5, 5, 5)} = \\langle x, y, z | x^2, y^2, (xy)^2, z^2, (xz)^5, (yz)^5, (xyz)^5 \\rangle$?. It is known that ?$G^{(5, 5, 5)}$? is isomorphic to the projective special linear group ?$L_2(11)$? which is simple, so that ?$G$? is isomorphic to ?$L_2(11)$?. It was proved by S. Norton that (up to conjugacy) ?$G$? is the unique ?$2A$?-generated ?$L_2(11)$?-subgroup of? $V_\\mathbb{M}$? and that? $K = C_\\mathbb{M}(G)$? is isomorphic to the Mathieu group ?$M_{12}$?. For any pair ?$\\{t, s\\}$? of ?$2A$?-involutions, the pair of Majorana axes ?$\\{a_t, a_s\\}$? generates the dihedral subalgebra ?$\\langle \\langle a_t, a_s \\rangle \\rangle$? of ?$V_\\mathbb{M}$?, whose structure has been described in S. P. Norton, The Monster algebra, some new formulae, Contemp. Math. 193 (1996), 297306. In particular, the subalgebra ?$\\langle \\langle a_t, a_s \\rangle \\rangle$? contains the Majorana axis ?$a_{tst}$? by the conjugacy property of dihedral subalgebras. Hence from the structure of its dihedral subalgebras, ?$V$? coincides with the subalgebra of? $V_\\mathbb{M}$? generated by the set of Majorana axes ?$\\{a_t | t \\in T\\}$?, indexed by the 55 elements of the unique conjugacy class ?$T$? of involutions of ?$G \\cong L_2(11)$?. We prove that ?$V$? is 101-dimensional, linearly spanned by the set? $\\{a_t \\cdot a_s | s, t \\in T\\}$?, and with ?$C_{V_\\mathbb{M}}(K) = V \\oplus \\iota_\\mathbb{M}$?, where ?$\\iota_\\mathbb{M}$? is the identity of ?$V_\\mathbb{M}$?. Lastly we present a recent result of Á. Seress proving that ?$V$? is equal to the algebra of the unique Majorana representation of ?$L_2(11)$?"},"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-GQ4XHUOU","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-GQ4XHUOU"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-GQ4XHUOU/93932ff0-5895-492d-b394-4e795d561dd6/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-GQ4XHUOU/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-GQ4XHUOU"}}}}