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  "body": "\n## Description\n\nThe *Griess algebra* is a real [[vector space|000AYPA9AJ]] $B$ of [[dimension|03GSZ8RTZJ]] $196884$ with a bilinear product $B \\times B \\to B$.\nThe product is commutative but not associative.\nThe algebra also carries a positive definite symmetric bilinear form $(\\cdot, \\cdot)$ that is associative: $(ab, c) = (a, bc)$ for all $a, b, c \\in B$.\nIts automorphism group is the [[Monster group|C9VY20TAPT]], the largest [[sporadic group|PZ49Z4CBFZ]].\n\n## Remarks\n\nRobert Griess built this algebra to construct the Monster group, which acts on it by algebra automorphisms.\nAs a representation of the Monster, $B$ splits as a trivial summand of dimension $1$ and an irreducible summand of dimension $196883$.\nThe number $196884$ is also the coefficient of $q$ in the expansion of the $j$-function, a coincidence that is part of [[monstrous moonshine|E0Y7Y75T6C]].\nIn the moonshine module of Frenkel, Lepowsky and Meurman, a vertex operator algebra, the Griess algebra appears as the weight $2$ subspace.\n\n## Explanations\n\n### L1 human\n\nThe Griess algebra is a huge space of dimension $196884$ in which two vectors can be multiplied.\nIt was built to give the Monster group a concrete home: the Monster is exactly the set of symmetries of this algebra.\nThe multiplication commutes, $ab = ba$, but the order of brackets matters, $(ab)c \\ne a(bc)$ in general.\n",
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