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In [[mathematics]], the '''Griess algebra''' is a [[commutative]] [[ |
In [[mathematics]], the '''Griess algebra''' is a [[commutative]] [[non-associative algebra]] on a [[real number|real]] [[vector space]] of [[dimension]] 196884 that has the [[Monster group]] ''M'' as its [[automorphism group]]. It is named after mathematician [[R. L. Griess]], who constructed it in 1980 and subsequently used it in 1982 to construct ''M''. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional [[orthogonal complement]] of this 1-space. |
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(The Monster preserves the standard [[inner product]] on the 196884-space.) |
(The Monster preserves the standard [[inner product]] on the 196884-space.) |
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Griess's construction was later simplified by [[Jacques Tits]] and [[John H. Conway]]. |
Griess's construction was later simplified by [[Jacques Tits]] and [[John H. Conway]]. |
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The Griess algebra is the same as the degree 2 piece of the [[monster vertex algebra]], and the Griess product is one of the vertex algebra products. |
The Griess algebra is the same as the degree 2 piece of the [[monster vertex algebra]], and the Griess product is one of the vertex algebra products. |
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==References== |
==References== |
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| ⚫ | *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=A simple construction for the Fischer-Griess monster group | doi=10.1007/BF01388521 |mr=782233 | year=1985 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=79 | issue=3 | pages=513–540| bibcode=1985InMat..79..513C }} |
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| ⚫ | *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=A simple construction for the Fischer-Griess monster group |
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[[Category:Sporadic groups]] |
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[[it:Algebra di Griess]] |
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Latest revision as of 10:56, 28 November 2024
 | This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (May 2024) |
In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct M. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space. (The Monster preserves the standard inner product on the 196884-space.)
Griess's construction was later simplified by Jacques Tits and John H. Conway.
The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products.
References
[edit]- Conway, John Horton (1985), "A simple construction for the Fischer-Griess monster group", Inventiones Mathematicae, 79 (3): 513–540, Bibcode:1985InMat..79..513C, doi:10.1007/BF01388521, ISSN 0020-9910, MR 0782233
- R. L. Griess Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102
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