Griess algebra
E656676
The Griess algebra is a 196,884-dimensional commutative nonassociative algebra over the real numbers whose automorphism group is the Monster, providing a concrete algebraic realization of this largest sporadic simple group.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
commutative algebra
ⓘ
finite-dimensional algebra ⓘ nonassociative algebra ⓘ real algebra ⓘ |
| alsoKnownAs | Monster algebra NERFINISHED ⓘ |
| arisesFrom | representation theory of the Monster group ⓘ |
| basisDimension | 196884 ⓘ |
| constructedBy | Robert L. Griess Jr. NERFINISHED ⓘ |
| containsSubrepresentation |
196883-dimensional irreducible representation of the Monster group
ⓘ
trivial representation of the Monster group ⓘ |
| decompositionUnderMonster | 1 ⊕ 196883 ⓘ |
| definedOver | real numbers ⓘ |
| dimension | 196884 ⓘ |
| fieldCharacteristic | 0 ⓘ |
| hasAutomorphismGroup | Monster group NERFINISHED ⓘ |
| hasAutomorphismGroupProperty | largest sporadic simple group ⓘ |
| hasIdentityElement | yes ⓘ |
| hasInvariantBilinearForm | yes ⓘ |
| hasProductDefinedBy | Monster-invariant bilinear form and projection rules ⓘ |
| hasProperty |
commutative multiplication
ⓘ
nonassociative multiplication ⓘ nonunital as originally defined (idempotent replaces identity) ⓘ |
| hasRank | 196884 as a real vector space ⓘ |
| hasZeroDivisors | yes ⓘ |
| isExampleOf | commutative nonassociative algebra with simple automorphism group ⓘ |
| isNot |
Jordan algebra
ⓘ
Lie algebra ⓘ associative algebra ⓘ |
| namedAfter | Robert L. Griess Jr. NERFINISHED ⓘ |
| providesConcreteRealizationOf | Monster group NERFINISHED ⓘ |
| realizesAsAutomorphismGroup | Monster group NERFINISHED ⓘ |
| relatedConcept |
Fischer–Griess Monster
NERFINISHED
ⓘ
Monster vertex operator algebra NERFINISHED ⓘ |
| relatedTo |
Monster group
NERFINISHED
ⓘ
Monstrous moonshine NERFINISHED ⓘ finite simple groups ⓘ moonshine module ⓘ sporadic simple groups NERFINISHED ⓘ vertex operator algebras ⓘ |
| studiedIn |
algebra
ⓘ
finite group theory ⓘ moonshine theory NERFINISHED ⓘ representation theory ⓘ |
| usedIn | construction of the Monster group ⓘ |
| usedToShow | existence of the Monster group ⓘ |
| yearOfConstruction | 1980 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.