What makes the Monster Lie Algebra special?

This is a post I’d been meaning to write for several years, but I was finally prompted to action after talking to some confused physicists. The Monster Lie Algebra, as a Lie algebra, has very little structure – it (or rather, its positive subalgebra) is quite close to being free on countably infinitely many generators. In addition to its Lie algebra structure, it has a faithful action of the monster simple group by Lie algebra automorphisms. However, the bare fact that the monster acts faithfully on the Lie algebra by diagram automorphisms is not very interesting: the almost-freeness means that the diagram automorphism group is more or less the direct product of a sequence of general linear groups of unbounded rank, and the monster embeds in any such group very easily.

The first interesting property of the Monster Lie Algebra has nothing to do with the monster simple group. Instead, the particular arrangement of generators illustrates a remarkable property of the modular J-function.

The more impressive property is a *particular* action of the monster that arises functorially from a string-theoretic construction of the Lie algebra. This action is useful in Borcherds’s proof of the Monstrous Moonshine conjecture, as I mentioned near the end of a previous post, and this usefulness is because the action satisfies a strong compatibility condition that relates the module structures of different root spaces.

What does the Monster Lie Algebra look like?

The Monster Lie Algebra is a rank 2 Borcherds-Kac-Moody Lie algebra, and this implies it has a self-centralizing subalgebra H of dimension 2, and decomposes under the adjoint H action into a collection of eigenspaces, called root spaces. Here is the decomposition as a \mathbb{Z} \oplus \mathbb{Z}-graded vector space:

decomposition of Monster Lie Algebra

A brief examination reveals a bit of a mess in the upper left and lower right. Ignoring the mess for the moment, we have a \mathbb{C}^2 in the middle, and two copies of the complex numbers in the upper left and lower right. This four dimensional vector space is a Lie subalgebra isomorphic to \mathfrak{gl}_2, i.e., we can view it as the Lie algebra of 2-by-2 matrices under conjugation, where the diagonal matrices form the 2-dimensional space in the middle.

The vector spaces in the mess have a symmetry between the degree (m,n) piece and the degree (n,m) piece, imposed by the action of the subalgebra \mathfrak{gl}_2. However, we can say more: the root multiplicities are determined by the coefficients of the normalized modular j-invariant J(\tau) = \sum c(n) q^n = q^{-1} + 196884q + 21493760q^2 + \cdots. More precisely, when either m or n is nonzero, the degree (m,n) subspace has dimension c(mn), the coefficient of the q^{mn}-coefficient of J.

The simple roots of the Monster Lie Algebra span the degree (1,n) spaces, i.e., those in the x=1 column, and they generate the “positive” subalgebra \mathfrak{n}_+. That is, one has one simple root of degree (1,-1), 198664 simple roots of degree (1,1), 21493760 simple roots of degree (1,2), etc. In the diagram below, the red line divides the Lie algebra into positive and negative subalgebras, and the semi-infinite red box contains the simple roots.

simple roots of the Monster Lie Algebra

The degree (1,-1) simple root is called “real”, while the rest are “imaginary”, because the root space has an inner product where the vector (m,n) has norm -2mn. To generate the full Monster Lie algebra from the simple roots, we follow the standard recipe for Borcherds-Kac-Moody Lie algebras, using quadratic \mathfrak{sl}_2 relations, together with Serre’s relations for real roots. This makes the positive subalgebra \mathfrak{n}_+ into a Lie algebra “freely generated over \mathfrak{gl}_2” by the imaginary simple roots.

The first miraculous fact about the Monster Lie algebra is that we started with simple roots whose multiplicities are coefficients of J, and ended up with all roots having multiplicities given by coefficients of J. This fact implies an infinite collection of identities relating the coefficients of J. For example, an examination of the degree (2,2) part reveals that \mathbb{C}^{c(4)} \cong \mathbb{C}^{c(3)} \oplus \bigwedge^2 \mathbb{C}^{c(1)}, so c(4) = c(3) + \binom{c(1)}{2}.

We can do a more systematic version of this examination using the Weyl (-Kac-Borcherds) denominator formula:

\sum_{w \in W} \det(w) w(e^{\rho} S) = e^{\rho} \prod_{\alpha \in \Delta^+} (1-e^{\alpha})^{\mathrm{mult}(\alpha)}

which arises from the Chevalley-Eilenberg resolution of the trivial representation of the positive subalgebra. Here, \rho is the Weyl vector (-1,0), and S is an alternating sum of e^\alpha over finite orthogonal subsets of simple imaginary roots that add up to \alpha. The i-th homology of the resulting complex is the subspace of the i-th exterior power of \mathfrak{n}_+ supported in degree r satisfying (r,r+2\rho) = 0. For the Monster Lie algebra, the Weyl group has order 2 (as it is isomorphic to W(\mathfrak{gl}_2)), so the contributions to homology are easy to enumerate. If we set p^m q^n to be a basis vector in degree (m,n), we have H_0(\mathfrak{n}_+) = \mathbb{C}, H_1(\mathfrak{n}_+) = \bigoplus_{n \geq -1} \mathbb{C}^{c(n)} pq^n, H_2(\mathfrak{n}_+) = \bigoplus_{m \geq 1} \mathbb{C}^{c(m)} p^m, and H_i(\mathfrak{n}_+) = 0 for i>2. We decategorify by evaluating the Hilbert-Poincaré series, and obtain the Koike-Norton-Zagier identity:

J(\sigma) - J(\tau) = p^{-1} \prod_{m>0,n \in \mathbb{Z}} (1-p^m q^n)^{c(mn)}

When considering characters, this identity is naturally an identity of formal power series, although we may turn this into a complex analytic identity near infinity by setting p = e^{2\pi i \sigma} and q = e^{2\pi i \tau}. The product actually converges in the region where the product of the imaginary parts of \sigma and \tau is greater than 1, since that is where \frac{J(\sigma) - J(\tau)}{\sigma - \tau} is nonvanishing.

This identity has one quite remarkable property, namely the power series on the left is a sum of power series that are pure in p and q, while the power series on the right appears to be full of mixed terms containing both p and q. The vanishing of mixed terms on the left is what yields the identities between coefficients of J that I mentioned before. In particular, the vanishing of the pq^2 coefficient on the right is equivalent to c(4) = c(3) + \binom{c(1)}{2}.

The Koike-Norton-Zagier identity was proved independently during the 1980s by the three people named, but it seems that none of them bothered to write up a proof. An elementary argument can be given by multiplying both sides by p and taking logs – the right side becomes a sum of p^m T_m J where T_m is the m-th Hecke operator, while the left side is the sum of p^m \Phi_m(J), where \Phi_m(J) is the unique polynomial in J of the form \frac{q^{-m}}{m} + O(q).

There is a higher-level argument using the theory of Borcherds products. Basically, the Borcherds-Harvey-Moore multiplicative theta lift sends J to a function \Psi on \mathbb{H} \times \mathbb{H} that is invariant under SL_2(\mathbb{Z}) \times SL_2(\mathbb{Z}), with zeroes of multiplicity one along the divisors \sigma = \gamma \cdot \tau for \gamma \in SL_2(\mathbb{Z}), and a cusp expansion as an infinite product whose exponents are coefficients of J. It therefore suffices to examine the polar part at infinity to identify \Psi with J(\sigma) - J(\tau), and the product formula yields the term p^{-1}(1-pq^{-1})^{c(-1)} = p^{-1} - q^{-1}.

How about that action?

I’d like to recapitulate what I said in the beginning about monster actions on the Monster Lie Algebra. Any linear action of a group on the simple root spaces extends naturally to an action by homogeneous Lie algebra automorphisms, so the bare fact that the monster acts is not so special.

However, Borcherds gave an alternative construction of the Monster Lie Algebra that produced a very well-behaved action. Instead of using generators and relations as above, he used a stringy quantization functor, which also goes by the name H^1_{BRST}. This functor takes in a representation of the Virasoro algebra at central charge 26, and produces a vector space. If the representation has a product structure, in particular from a vertex algebra, then the output has a Lie algebra structure. The “cancellation of oscillators” theorem asserts that if the input has the form U \otimes \pi_\lambda, where U is a unitarizable representation of Virasoro with central charge 24, and \pi_\lambda is a Fock space for 2 free bosons with momentum \lambda, then the output is the weight 1-\lambda^2 part of U (when \lambda \neq 0). This was first conjectured by Lovelace in 1971, and proved by Goddard and Thorn in 1972. We typically attach the name “no-ghost theorem” to this result, although the name refers to a somewhat different aspect of their theorem. In particular, the fact that the output space has no negative-norm states (known as ghosts) was a big deal in the early development of string theory.

Borcherds chose to input the tensor product of the Monster Vertex Algebra V^\natural with the Lorentzian lattice vertex algebra V_{I\!I_{1,1}}. The lattice vertex algebra is \mathbb{Z} \times \mathbb{Z}-graded, and each graded piece is a rank 2 free boson. By cancellation of oscillators, we get an identification between graded pieces of the Lie algebra in degree away from (0,0) and graded pieces of V^\natural as monster modules. If we write V^\natural = \bigoplus_{n \geq -1} V_n, then the degree (m,n) piece of the Monster Lie Algebra is identified with V_{mn}.

Here then is the distinguishing property of the monster action on the Monster Lie Algebra: for any (m,n) \neq (0,0), the root space of degree (m,n) is a monster module whose isomorphism type depends only on the product mn. That is, the monster representation is constant along hyperbolas xy=k.

This property has the following impact: The identities we found relating the coefficients of J, or equivalently the dimensions of root spaces, are promoted to relations between monster representations. For example, the identity c(4) = c(3) + \binom{c(1)}{2} of coefficients is promoted to a monster module isomorphism V_4 \cong V_3 \oplus \bigwedge^2 V_1. This requires the identification of the monster action on the degree (2,2) vector space with the action on the degree (1,4) space, which does not hold for general monster actions. This additional information means that in addition to the ordinary Weyl denominator identity, we have a twisted denominator identity for each element of the monster, and one can use equivariant Hecke operators to organize the terms.

The application to moonshine is the following: if we want to understand the character of an element in the monster acting on V^\natural, we may use the twisted denominator identities to obtain recursion relations between the traces on graded pieces. For example, the previous monster module isomorphism yields \mathrm{Tr}(g|V_4) = \mathrm{Tr}(g|V_3) + \frac{\mathrm{Tr}(g|V_1)^2-\mathrm{Tr}(g^2|V_1)}{2} for any element g in the monster simple group. The general form of these recursion relations is known as complete replicability, and Koike showed in unpublished work that the candidate moonshine functions listed by Conway and Norton are completely replicable. Once these recursions were in place for characters of the monster action on V^\natural, Borcherds proved the Monstrous Moonshine conjecture by comparing the graded pieces V_i of the monster vertex algebra to Conway and Norton’s candidate representations for i \leq 5.