The monster vertex algebra (or moonshine module) is a

vertex algebra In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven usefu ...

acted on by 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 ...

that was constructed by

Igor Frenkel Igor Borisovich Frenkel (russian: Игорь Борисович Френкель; born April 22, 1952) is a Russian-American mathematician at Yale University working in representation theory and mathematical physics. Frenkel emigrated to the Uni ...

, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the

monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979 ...

conjectures, by applying the

Goddard–Thorn theorem In mathematics, and in particular in the mathematical background of string theory, the Goddard–Thorn theorem (also called the no-ghost theorem) is a theorem describing properties of a functor that quantizes bosonic strings. It is named after ...

of string theory to construct the

monster Lie algebra In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. Structure The monster Lie algebra ''m'' is a ''Z2''- ...

, an infinite-dimensional

generalized Kac–Moody algebra In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borche ...

acted on by the monster. 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. It can be constructed as

conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...

describing 24 free bosons compactified on the torus induced by the

Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...

and

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

ed by the two-element reflection group.

References

* * {{algebra-stub Non-associative algebra