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 to prove the existence of the Monster group, referring to it as "The Friendly Giant".
Matrix representation size
To visualize the complexity of the Monster group, an individual element <math>g \in M</math>, when acting on the 196883-dimensional irreducible subspace <math>W</math>, can be represented as a square matrix.
:<math display="block">
g = \begin{pmatrix}
g_{1,1} & g_{1,2} & \dots & g_{1,196883} \\
g_{2,1} & g_{2,2} & \dots & g_{2,196883} \\
\vdots & \vdots & \ddots & \vdots \\
g_{196883,1} & g_{196883,2} & \dots & g_{196883,196883}
\end{pmatrix}
</math>
History and inspiration
In 1973, Bernd Fischer and Robert Griess independently produced evidence for the existence of the Monster group. However, for several years, the group remained a conjecture because its immense order (approx. <math>8 \times 10^{53}</math>) made computer construction impossible at the time. To prove its existence, Griess undertook the task of constructing a specific mathematical object upon which this group would act naturally. This resulted in the 196884-dimensional algebra, announced in 1980 and published in 1982. The Griess algebra is also identified as the degree 2 piece of the monster vertex algebra. 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.)