In mathematics, the Griess algebra is a

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

non-associative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...

on a

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...

196884 that has the

Monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 2463205976112133171923293141475 ...

''M'' as its

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is th ...

. It is named after mathematician

R. L. Griess Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michig ...

, 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 In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

of this 1-space. (The Monster preserves the standard

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

on the 196884-space.) Griess's construction was later simplified by

Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and ...

and John H. Conway. The Griess algebra is the same as the degree 2 piece of the

monster vertex algebra The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by appl ...

, and the Griess product is one of the vertex algebra products.

References

* *R. L. Griess, Jr, ''The Friendly Giant'', Inventiones Mathematicae 69 (1982), 1-102 {{algebra-stub Non-associative algebras Sporadic groups