# Griess algebra

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 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 of this 1-space.
(The Monster preserves the standard inner product on the 196884-space.)

Griess's construction was later simplified by Jacques Tits and John H. Conway.

The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products.

## References

- Conway, John Horton (1985), "A simple construction for the Fischer-Griess monster group",
*Inventiones Mathematicae*,**79**(3): 513–540, doi:10.1007/BF01388521, ISSN 0020-9910, MR 0782233 - R. L. Griess, Jr,
*The Friendly Giant*, Inventiones Mathematicae 69 (1982), 1-102

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