mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the monster Lie algebra is an

infinite-dimensional In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...

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, Borch ...

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 246320597611213317192329314147 ...

, which was used 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.

Structure

The monster Lie algebra ''m'' is a ''Z²''-

graded Lie algebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket oper ...

. The piece of degree (''m'', ''n'') has dimension ''c''_''mn'' if (''m'', ''n'') ≠ (0, 0) and dimension 2 if (''m'', ''n'') = (0, 0). The

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s ''c_n'' are the coefficients of ''q''^''n'' of the ''j''-invariant as

elliptic modular function In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holom ...

j(q) -744 =   + 196884 q + 21493760 q^2 + \cdots.

The

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by � ...

is the 2-dimensional subspace of degree (0, 0), so the monster Lie algebra has rank 2. The monster Lie algebra has just one real

simple root Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...

, given by the vector (1, −1), and the

Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...

has order 2, and acts by mapping (''m'', ''n'') to (''n'', ''m''). The imaginary simple roots are the vectors (1, ''n'') for ''n'' = 1, 2, 3, ..., and they have multiplicities ''c''_''n''. The

denominator formula A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...

for the monster Lie algebra is the product formula for the ''j''-invariant: ::

j(p) - j(q) = \left( - \right) \prod_^(1-p^n q^m)^.

The denominator formula (sometimes called the Koike-Norton-Zagier infinite product identity) was discovered in the 1980s. Several mathematicians, including Masao Koike, Simon P. Norton, and

Don Zagier Don Bernard Zagier (born 29 June 1951) is an American-German mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany. He was a professor at the ''Col ...

, independently made the discovery. (See p. 1077).

Construction

There are two ways to construct the monster Lie algebra. As it is a generalized Kac–Moody algebra whose simple roots are known, it can be defined by explicit generators and relations; however, this presentation does not give an action of the monster group on it. It can also be constructed from the

monster vertex algebra The monster vertex algebra (or moonshine module) is a vertex algebra acted on by the monster group that was constructed by Igor Frenkel, James Lepowsky, and Arne Meurman. R. Borcherds used it to prove the monstrous moonshine conjectures, by ap ...

by using 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 P ...

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...

. This construction is much harder, but also proves that the

acts naturally on it.

References

* * *; ** * (Introductory study text with a brief account of Borcherds algebra in Ch. 21) {{refend Lie algebras Moonshine theory