A class of continuous non-associative algebras arising from algebraic groups including $E_8$
We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$, the algebra A is obtained by adjoining a unit to the 38...
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2021-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509420000663/type/journal_article |
| _version_ | 1811156195878109184 |
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| author | Maurice Chayet Skip Garibaldi |
| author_facet | Maurice Chayet Skip Garibaldi |
| author_sort | Maurice Chayet |
| collection | DOAJ |
| description | We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type
$E_8$, the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the
$E_8$ case has been requested for some time, and interest has been increased by the recent proof that
$E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum.
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| first_indexed | 2024-04-10T04:47:27Z |
| format | Article |
| id | doaj.art-ba270bbf4d9c4f519b6df0a7767f99bd |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:27Z |
| publishDate | 2021-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-ba270bbf4d9c4f519b6df0a7767f99bd2023-03-09T12:34:52ZengCambridge University PressForum of Mathematics, Sigma2050-50942021-01-01910.1017/fms.2020.66A class of continuous non-associative algebras arising from algebraic groups including $E_8$Maurice Chayet0Skip Garibaldi1https://orcid.org/0000-0001-8924-5933ECAM-EPMI, 13 Boulevard de l'Hautil, 95092 Cergy Pointoise Cedex, France; E-mail:IDA Center for Communications Research-La Jolla, 4320 Westerra Ct, San Diego, CA 92121, USA; E-mail:We give a construction that takes a simple linear algebraic group G over a field and produces a commutative, unital, and simple non-associative algebra A over that field. Two attractions of this construction are that (1) when G has type $E_8$, the algebra A is obtained by adjoining a unit to the 3875-dimensional representation; and (2) it is effective, in that the product operation on A can be implemented on a computer. A description of the algebra in the $E_8$ case has been requested for some time, and interest has been increased by the recent proof that $E_8$ is the full automorphism group of that algebra. The algebras obtained by our construction have an unusual Peirce spectrum. https://www.cambridge.org/core/product/identifier/S2050509420000663/type/journal_article17B2517D9920G41 |
| spellingShingle | Maurice Chayet Skip Garibaldi A class of continuous non-associative algebras arising from algebraic groups including $E_8$ Forum of Mathematics, Sigma 17B25 17D99 20G41 |
| title | A class of continuous non-associative algebras arising from algebraic groups including $E_8$ |
| title_full | A class of continuous non-associative algebras arising from algebraic groups including $E_8$ |
| title_fullStr | A class of continuous non-associative algebras arising from algebraic groups including $E_8$ |
| title_full_unstemmed | A class of continuous non-associative algebras arising from algebraic groups including $E_8$ |
| title_short | A class of continuous non-associative algebras arising from algebraic groups including $E_8$ |
| title_sort | class of continuous non associative algebras arising from algebraic groups including e 8 |
| topic | 17B25 17D99 20G41 |
| url | https://www.cambridge.org/core/product/identifier/S2050509420000663/type/journal_article |
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