---
tag: ZMPYZ8KD9N
kind: object
name: Griess algebra
msc: [20D08, 17B69]
sources:
  - url: https://en.wikipedia.org/wiki/Griess_algebra
same_as:
  wikidata: Q3514494
  wikipedia: Griess algebra
---

## Description

The *Griess algebra* is a real [[vector space|000AYPA9AJ]] $B$ of [[dimension|03GSZ8RTZJ]] $196884$ with a bilinear product $B \times B \to B$.
The product is commutative but not associative.
The algebra also carries a positive definite symmetric bilinear form $(\cdot, \cdot)$ that is associative: $(ab, c) = (a, bc)$ for all $a, b, c \in B$.
Its automorphism group is the [[Monster group|C9VY20TAPT]], the largest [[sporadic group|PZ49Z4CBFZ]].

## Remarks

Robert Griess built this algebra to construct the Monster group, which acts on it by algebra automorphisms.
As a representation of the Monster, $B$ splits as a trivial summand of dimension $1$ and an irreducible summand of dimension $196883$.
The number $196884$ is also the coefficient of $q$ in the expansion of the $j$-function, a coincidence that is part of [[monstrous moonshine|E0Y7Y75T6C]].
In the moonshine module of Frenkel, Lepowsky and Meurman, a vertex operator algebra, the Griess algebra appears as the weight $2$ subspace.

## Explanations

### L1 human

The Griess algebra is a huge space of dimension $196884$ in which two vectors can be multiplied.
It was built to give the Monster group a concrete home: the Monster is exactly the set of symmetries of this algebra.
The multiplication commutes, $ab = ba$, but the order of brackets matters, $(ab)c \ne a(bc)$ in general.