View source for Thompson-McKay series

According to the classification theorem, the simple finite groups (cf. also [[Simple finite group|Simple finite group]]) consist of the cyclic groups of prime order (cf. also [[Cyclic group|Cyclic group]]), the alternating groups of degree at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200701.png" /> (cf. also [[Alternating group|Alternating group]]), the Chevalley and twisted Chevalley groups (cf. also [[Chevalley group|Chevalley group]]), the Tits group and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200702.png" /> sporadic simple groups (cf. also [[Sporadic simple group|Sporadic simple group]]). The first five sporadic groups were described by E. Mathieu in the 19th century [[#References|[a24]]], [[#References|[a25]]]. No further sporadic groups were found until Z. Janko's discovery of the first modern sporadic group in 1964 [[#References|[a14]]], [[#References|[a15]]]. Evidence for the existence of the largest of the sporadic groups, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200703.png" />, was found independently by B. Fischer and R.L. Griess in 1973. This group is now known as the monster and has order

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200704.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200705.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200706.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200707.png" /></td> </tr></table>

Even before Griess's construction [[#References|[a12]]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200708.png" /> in 1981, intriguing connections between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t1200709.png" /> and other areas of mathematics had been noted:

Define

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007010.png" /></td> </tr></table>

and for each prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007011.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007012.png" />. Any discrete subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007014.png" /> acts on the upper half of the complex plane by fractional-linear transformations (cf. also [[Fractional-linear mapping|Fractional-linear mapping]]). The corresponding quotient space has the structure of a [[Riemann surface|Riemann surface]]. If this surface is isomorphic to a sphere (with a finite number of points removed, corresponding to the orbits of the fixed points of the parabolic elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007015.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007016.png" /> is said to have genus zero. In 1974, A.P. Ogg [[#References|[a27]]] observed that the primes dividing the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007017.png" /> are also the primes for which the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007018.png" /> has genus zero.

Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007019.png" /> has genus zero; then a generator for the function field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007020.png" /> is called a Hauptmodul. Suppose further that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007021.png" /> is commensurable with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007022.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007023.png" /> contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007024.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007025.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007026.png" /> has a Hauptmodul of the form:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007027.png" /></td> </tr></table>

This form is unique except for the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007028.png" />. For example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007029.png" /> has a Hauptmodul

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007030.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007031.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007032.png" /></td> </tr></table>

In 1978, J. McKay observed in a letter to J.G. Thompson that the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007033.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007034.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007035.png" /> is the degree of the smallest non-trivial irreducible complex representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007036.png" />. Thompson [[#References|[a31]]] extended this observation by noting that the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007042.png" /> are simple linear combinations of the seven smallest irreducible degrees of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007043.png" />. He also asked whether there exists a graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007044.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007045.png" />, now known as the moonshine module, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007046.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007047.png" /> is the identity element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007048.png" />.

In 1979, J.H. Conway and S.P. Norton [[#References|[a7]]] coined the term  "moonshine"  for the study of the links between modular functions and the sporadic simple groups (cf. also [[Modular group]]; [[Modular form]]; [[Modular function]]). They generalized the observations of McKay and Thompson by attaching to each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007049.png" /> a genus-zero group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007050.png" />, depending only on the [[conjugacy class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007051.png" />, with Hauptmodul:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007052.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007053.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007055.png" />, are now known as the Thompson–McKay series. Conway and Norton conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007056.png" />, so that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007057.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007058.png" /> defines a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007059.png" /> (cf. also [[Character of a group|Character of a group]]), which they called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007060.png" />th Head character. They also conjectured identities between the Thompson–McKay series, which they called the replication identities:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007061.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007062.png" /> is the unique polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007063.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007064.png" /></td> </tr></table>

These conjectures, and others contained in [[#References|[a7]]], constitute the moonshine conjectures.

Given the character table of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007065.png" /> (which was computed by B. Fischer, D. Livingstone and M.P. Thorne) and the Hauptmoduls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007067.png" />, Thompson observed that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007068.png" /> is a virtual character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007069.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007070.png" /> if certain congruence conditions hold for a finite number of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007071.png" />. He also showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007072.png" /> is a proper character for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007073.png" />, then it is a proper character for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007074.png" />. Thus, in principle, the proof of the existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007075.png" /> can be reduced to a finite computation. Using these methods, A.O.L. Atkin, P. Fong and S.D. Smith [[#References|[a10]]], [[#References|[a30]]] showed that the Head characters are indeed virtual characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007076.png" /> and gave very strong evidence that they are proper characters.

These calculations, however, did not give any information leading to a more conceptual understanding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007077.png" /> and the moonshine conjectures. Another approach to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007078.png" /> was suggested by the explanations by J. Lepowsky [[#References|[a20]]] and V.G. Kac [[#References|[a16]]] of another observation of McKay that the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007079.png" /> appear to be character degrees of the complex [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007080.png" />. The underlying structure in this case is that of the affine Kac–Moody Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007081.png" /> (cf. also [[Kac–Moody algebra|Kac–Moody algebra]]). The corresponding construction for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007082.png" /> was found by I. Frenkel, Lepowsky and A. Meurman [[#References|[a11]]]. The additional structure is that of a [[vertex operator algebra]]. R.E. Borcherds [[#References|[a2]]], [[#References|[a5]]], [[#References|[a17]]] defined a (real) [[vertex algebra]] to be a [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007083.png" /> with an infinite number of bilinear products <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007085.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007087.png" />, such that:

V.1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007088.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007089.png" /> sufficiently large (depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007091.png" />);

V.2) (the  "Jacobi identity" )

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007092.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007093.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007094.png" /> and for all integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007097.png" />;

V.3) there is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007098.png" /> such that, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t12007099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070100.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070102.png" />. Also, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070103.png" /> (the identity transformation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070104.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070105.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070106.png" />.

A vertex operator algebra (VOA) is a vertex algebra which incorporates the action of the Virasoro Lie algebra (cf. also [[Virasoro algebra|Virasoro algebra]]). Frenkel, Lepowsky and Meurman constructed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070107.png" /> as a vertex operator algebra of conformal dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070108.png" />. They made a crucial link between the vertex operator algebra structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070109.png" /> and Griess's construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070110.png" /> via the Griess algebra and were thus able to show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070111.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070112.png" />-module. Their construction provides explicit formulas for the graded traces of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070113.png" /> which commute with an element in the class 2B (Atlas notation, cf. [[#References|[a1]]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070114.png" />. For example, the corresponding formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070115.png" /> function is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070116.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070117.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070118.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070119.png" /> is the theta-function of the [[Leech lattice|Leech lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070121.png" />. In principle, this gives a case-by-case verification of the moonshine conjectures for these elements.

To complete the proof of the moonshine conjectures, Borcherds [[#References|[a4]]] exploited the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070122.png" /> to construct a Lie algebra, which he called the Monster Lie algebra. The construction also shows that the Monster Lie algebra is a generalized Kac–Moody Lie algebra (GKM Lie algebra), which is defined by Borcherds [[#References|[a3]]] to be a [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070123.png" /> such that:

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070124.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070125.png" />-grading <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070126.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070127.png" /> is finite dimensional if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070128.png" />;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070129.png" /> has an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070130.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070131.png" /> and acts as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070132.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070133.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070134.png" /> has an invariant bilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070135.png" /> such that:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070136.png" /> is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070137.png" />;

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070139.png" /> are orthogonal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070140.png" />;

c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070141.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070142.png" /> is a non-zero homogeneous element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070143.png" /> of non-zero degree.

Remarkably, generalized Kac–Moody Lie algebras have many of the properties of Kac–Moody Lie algebras. In particular, they have  "Weyl denominator"  formulas, which in the case of the Monster Lie algebra is the product formula for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070145.png" />-function:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070146.png" /></td> </tr></table>

The monster Lie algebra inherits an action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070147.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070148.png" />, and so there are also  "twisted"  versions of the denominator formula, which can be interpreted as the denominator formulas of certain Lie superalgebras related to the Monster Lie algebra [[#References|[a3]]]. Moreover, by the  "no-ghost"  theorem, which was first discovered in string theory [[#References|[a3]]], [[#References|[a13]]], these twisted versions of the denominator formula provide relations between the Thompson–McKay series of the moonshine module. These relations are exactly the replication identities conjectured by Conway and Norton.

One consequence of these identities is that each Thompson–McKay series satisfies certain recurrence relations. For the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070149.png" />-function these recurrences were first found by D.H. Lehmer [[#References|[a19]]] and were rediscovered by K. Mahler [[#References|[a21]]]. These relations determine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070150.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070151.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070152.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070153.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070155.png" /> are known. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070156.png" /> has odd order, then the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070157.png" /> can be calculated once <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070159.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070161.png" /> are known. The moonshine conjectures now follow from a calculation of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070162.png" /> coefficients of the Thompson–McKay series and the proof by M. Koike [[#References|[a18]]] that the Hauptmoduls in question also satisfy the replication identities.

The original moonshine conjectures have been generalized in several directions and there has also been much work on the theory and applications of vertex operator algebras and generalized Kac–Moody Lie algebras. A far from complete list is: Norton's replicable functions [[#References|[a26]]] and generalized moonshine conjectures [[#References|[a23]]], [[#References|[a8]]]; moonshine for other groups [[#References|[a22]]], [[#References|[a28]]]; A.J.E. Ryba's modular moonshine conjectures [[#References|[a29]]], [[#References|[a4]]], [[#References|[a6]]] and the development of the general theory of vertex operator algebras [[#References|[a9]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.H. Conway,   R.T. Curtis,   S.P. Norton,   R.A. Parker,   R.A. Wilson,   "Atlas of finite groups. Maximal subgroups and ordinary characters for simple groups. With computational assistance from J.G. Thackray" , Oxford Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.E. Borcherds,   "Vertex algebras, Kac–Moody algebras, and the Monster"  ''Proc. Nat. Acad. Sci. USA'' , '''83'''  (1986)  pp. 3068–3071</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.E. Borcherds,   "Monstrous Moonshine and monstrous Lie superalgebras"  ''Invent. Math.'' , '''109'''  (1992)  pp. 405–444</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.E. Borcherds,   "Modular moonshine. III"  ''Duke Math. J.'' , '''93'''  (1998)  pp. 129–154</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.E. Borcherds,   "Generalized Kac–Moody algebras"  ''J. Algebra'' , '''115'''  (1988)  pp. 501–512</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  R.E. Borcherds,   A.J.E. Ryba,   "Modular Moonshine. II"  ''Duke Math. J.'' , '''83'''  (1996)  pp. 435–459</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.H. Conway,   S.P. Norton,   "Monstrous Moonshine"  ''Bull. London Math. Soc.'' , '''11'''  (1979)  pp. 308–339</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C. Dong,   H. Li,   G. Mason,   "Modular invariance of trace functions in orbifold theory"  ''preprint''  (1997)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C. Dong,   G. Mason,   "Vertex operator algebras and Moonshine: a survey" , ''Progress in Algebraic Combinatorics (Fukuoka, 1993)'' , ''Adv. Stud. Pure Math.'' , '''24''' , Math. Soc. Japan  (1996)  pp. 101–136</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  P. Fong,   "Characters arising in the Monster-modular connection" , ''The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979'' , ''Proc. Symp. Pure Math.'' , '''37''' , Amer. Math. Soc.  (1980)  pp. 557–559</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  I.B. Frenkel,   J. Lepowsky,   A. Meurman,   "Vertex operators and the monster" , Acad. Press  (1988)</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  R.L. Griess,   "The friendly giant"  ''Invent. Math.'' , '''69'''  (1982)  pp. 1–102</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  P. Goddard,   C.B. Thorne,   "Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model"  ''Phys. Lett. B'' , '''40'''  (1972)  pp. 235–238</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  Z. Janko,   "A new finite simple group with abelian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070163.png" />-Sylow subgroups"  ''Proc. Nat. Acad. Sci. USA'' , '''53'''  (1965)  pp. 657–658</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  Z. Janko,   "A new finite simple group with abelian Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070164.png" />-subgroups and its characterization"  ''J. Algebra'' , '''3'''  (1966)  pp. 147–186</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  V.G. Kac,   "An elucidation of: Infinite-dimensional algebras, Dedekind's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070165.png" />-function, classical Möbius function and the very strange formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070166.png" /> and the cube root of the modular invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070167.png" />"  ''Adv. Math.'' , '''35''' :  3  (1980)  pp. 264–273</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  V.G. Kac,   "Vertex algebras for beginners" , ''Univ. Lecture Ser.'' , '''10''' , Amer. Math. Soc.  (1997)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  M. Koike,   "On replication formula and Hecke operators"  ''Preprint Nagoya Univ.''  (unpublished)</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  D.H. Lehmer,   "Properties of the coefficients of the modular invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070168.png" />"  ''Amer. J. Math.'' , '''64'''  (1942)  pp. 488–502</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  J. Lepowsky,   "Euclidean Lie algebras and the modular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070169.png" />" , ''The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979)'' , ''Proc. Symp. Pure Math.'' , '''37''' , Amer. Math. Soc.  (1980)  pp. 567–570</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  K. Mahler,   "On a class of non-linear functional equations connected with modular functions"  ''J. Austral. Math. Soc.'' , '''22A'''  (1976)  pp. 65–118</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  G. Mason,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070170.png" /> and certain automorphic forms" , ''Finite groups: coming of age (Montreal, Que., 1982)'' , ''Contemp. Math.'' , '''45''' , Amer. Math. Soc.  (1985)  pp. 223–244</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  G. Mason,   "Finite groups and modular functions" , ''The Arcata Conference on Representations of Finite Groups (Arcata, Calif. 1986)'' , ''Proc. Symp. Pure Math.'' , '''47:1''' , Amer. Math. Soc.  (1987)  pp. 181–210  (Appendix by S.P. Norton)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  E. Mathieu,   "Memoire sur l'étude des fonctions de plusieurs quantités"  ''J. Math. Pures Appl.'' , '''6'''  (1861)  pp. 241–323</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  E. Mathieu,   "Sur les fonctions cinq fois transitives de <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120070/t120070171.png" /> quantités"  ''J. Math. Pures Appl.'' , '''18'''  (1873)  pp. 25–46</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  S.P. Norton,   "More on Moonshine"  M.D. Atkinson (ed.) , ''Computational Group Theory'' , Acad. Press  (1984)  pp. 185–193</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  A.P. Ogg,   "Automorphismes des courbes modulaires" , ''Théorie des Nombres, Fasc. 1, Exp. 7, 8'' , ''Sém. Delange–Pisot–Poitou (16e année (1974/75)'' , Secr. Math. Paris  (1975)</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  L. Queen,   "Modular functions arising from some finite groups"  ''Math. Comp.'' , '''37''' :  156  (1981)  pp. 547–580</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top">  A.J.E. Ryba,   "Modular Moonshine?"  C. Dong (ed.) , ''Moonshine, the Monster, and related topics (South Hadley, MA, (1994)'' , ''Contemp. Math.'' , '''193''' , Amer. Math. Soc.  (1996)  pp. 307–336</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top">  S.D. Smith,   "On the head characters of the Monster simple group" , ''Finite groups: coming of age (Montreal, Que. 1982)'' , ''Contemp. Math.'' , '''45''' , Amer. Math. Soc.  (1985)  pp. 303–313</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top">  J.G. Thompson,   "Some numerology between the Fischer–Griess monster and the elliptic modular function"  ''Bull. London Math. Soc.'' , '''11''' :  3  (1979)  pp. 352–353</TD></TR></table>