Proc Natl. Acad. Sci. USA

Vol. 83, pp 3068-3071

Vertex algebras, Kac-Moody algebras, and the Monster.

Richard E. Borcherds,

Trinity College, Cambridge CB2 1TQ, England.

Communicated by Walter Feit, December 13, 1985

ABSTRACT It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space the restricts to the Lie algebra product on this subquotient. This product (together with an infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The “Moonshine representation of the Monster constructed by Frenkel and others also has products such as the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.

Section 1. Introduction.

Let A be any Kac-Moody algebra all of whose real roots have norm 2.(Everything here can be generalized to all Kac-Moody algebras but becomes a lot more complicated, so for simplicity I will mostly just describe this case.) A is defined by certain generators and relations depending on the Cartan matrix of A, and one of the most important problems about Kac-Moody algebras is to find a more explicit realization of A. This has been done only when A is finite dimensional or when A is affine, in which case A can be realized as a central extension of a twisted ring of Laurent series in some finite dimensional Lie algebra. Here we will construct a realization of an algebra that is usually slightly larger than A and is equal to A if A is finite dimensional or affine (in which case it is equivalent to the usual realization of A). For any even integral lattice R (for example, the root lattice of A), we will first construct a (well known) Fock space V = V (R). Physicists have defined “vertex operators” for every element of R, which map V to the space V {z, z^{−1}} of formal Laurent series in V, and the coefficients of these operators map V to V. I define a sort of generalized vertex operator for every element of V instead of just for elements of R. This operator is written as : Q(u, z) : (v) for u, v in V, and its coefficients are written as u_{n}(v) for u, v in V and integers n. These products on V are not associative, commutative or skew commutative but satisfy several more complicated identities.

The product u_{0}(v) is not a Lie algebra product on V, but it is a Lie algebra product on V/DV, where DV is the image of V under a certain derivation D. This Lie algebra V/DV contains the Kac-Moody algebra A as a subalgebra but is always far larger than A. To reduce V/DV to a smaller subalgebra, we will use the Virasoro algebra. This is spanned by the operators c_{i} and 1, where c is a certain element of V. The commutator of this algebra in V/DV also contains A and is not much larger than A; for example, we can calculate bounds on the dimensions of the root spaces of A from this that are sometimes the best possible.

If V is the infinite-dimensional representation of the monster constructed by Frenkel et al. (1) then V also has products u_{n}(v) that satisfy several identities.

Section 2. Construction of the Fock space V.

In this section, we recall the construction of a certain Fock space V from an even lattice R and put several structures on V, such as a product, a derivation, and an inner product (see ref. 2).

For any even lattice R there is a central extension

0 ⟶ Z_{2} ⟶ R̂ ⟶ R ⟶ 0

where Z_{2} is a group of order 2 generated by an element ϵ and R̂ has an element e^{r} for every element r of R, such that e^{r}e^{s} = ϵ^{(r,s)}e^{s}e^{r} and e^{r}e^{−r} = ϵ^{(r,r)}/2. R̂ is uniquely defined up to isomorphism by these conditions, and the automorphism group of R̂ is an extension Z_{2}^{dim R} · Aut(R) (usually nonsplit). If R is the root lattice of a Kac-Moody algebra A, then Aut(R) is not usually a subgroup of Aut(A) in any natural way, but Aut(R̂) is, as we can prove by constructing A from R̂

The Fock space V is a rational vector space given by the tensor product

Q(R) ⊗ S(R(1)) ⊗ S(R(2)) ….

Here Q(R) is the rational group algebra of R̂ quotiented out by ϵ + 1, so it has a basis of e^{r} for r in R and e^{r}e^{s} = (−1)^{(r,s)}e^{s}e^{r}. R(i) is a copy of the rational vector space of R, and its elements are written r(i) for r in R. S(R(i)) is the symmetric algebra on R(i). A typical element of V might be e^{r}s(1)^{3}t(4) for r, s, t in R.

V has the following structures.

(i) V is an algebra as each of the pieces of the tensor product defining V is. V would be commutative except that e^{r} and e^{s} do not always commute.

(ii) There are linear maps D and deg from V to V such that De^{r} = r(1)e^{r}, Dr(i) = ir(i + 1), and D is a derivation. deg(e^{r}) = \frac{1}{2}(r, r)e^{r}, deg r(i)v = r(i)(iv + deg v). If deg u = iu we can say that u has degree i. We write D^{(i)} for the operator D^{i}/i!.

(iii) V has a Cartan involution ω. ω acts on R by ω(e^{r}) = e^{−r}, and this becomes an automorphism of V with ω(e^{r}) = e^{−r}, ω(r(i)) = −r(i).

(iv) We define the operators r(i) on V for r in R and integers i as follows: If i > 0 then r(i) is multiplication by r(i). If i = 0 then r(i)e^{s} = (r, s)e^{s}. If i < 0 then r(i)e^{s} = 0. [r(i), s(j)] = j(r, s) if i = −j, 0 otherwise. (These properties characterize the operators r(i).)

(v) V has a unique inner product (,) such that the operator r(i) is the adjoint of r(−i), and (e^{r}, e^{s}) = 1 if r = s, 0 otherwise.

(vi) The integral form V_{Z} of V is defined to be the smallest subring of V containing all the e^{r} and closed under D^{(i)} for i ≥ 0. This integral form is compatible with all the structures above; i.e., it is preserved by the Cartan involution ω and the operators r(i), and the inner product is integral on it. It is generated as a ring by e^{r}, r(1), (r(2) + r(1)^{2})/2, (2r(3)+3r(2)r(1)+r(1)^{3})/6 …, which are Schur polynomials in r(1), r(2)/2, r(3)/3 . . .. If W is the sublattice of V_{Z} of elements of “R grading” 0 and degree i, then the determinant of W is an integral power of the determinant of R and, in particular, if R is unimodular then so is W. (V is graded by the lattice R by letting e^{r} have degree r and letting r(i) have degree 0.)

Section 3. Vertex Operators.

For each u in V we will define a map u from V to the ring of formal Laurent series V {z, z^{−1}}. If u is of the form e^{r} then these operators are just vertex operators, and if u is a product of r(i)s then these operators have been constructed by Frenkel (3).

We can define Q(r, z) to be the formal expression

∑_{i≠0} r(i)z^{i}/i + r(0) log(z) + r

and define Q(r(i), z) for i ≥ 1 to be (d/dz)^{i}Q(r, z)/(i − 1)!. If u = e^{r} ∏_{i} r_{i}(n_{i}) is an element of V then we define Q(u, z) to be the formal expression

e^{Q(r,z)} ∏_{i} Q(r_{i}(n_{i}), z).

This is not an operator from V to V {z, z^{−1}} as it does not converge, but we can make it into an operator by “normal ordering” it. This means that in each term of the formal expression Q(u, z) we rearrange all terms e^{r} and r(i) so that the “creation operators” e^{r} and r(i) (i ≥ 1) occur to the left of all “annihilation operators” r(i) (i ≤ 0). Note that all annihilation operators commute with each other, and so do all creation operators except for e^{r} and e^{s}. The normal ordering of Q(u, z) is denoted by : Q(u, z) :, and this is a well-defined operator from V to V {z, z^{−1}}. We define u_{n}(v) for u, v in V and integers n by

u_{n}(v) = the coefficient of z^{−n−1} in : Q(u, z) : (v).

If u and v are in the integral form of V then so is u_{n}(v). If u and v have degrees i, j then u_{n}(v) has degree i + j − n − 1. The operator r(i) is equal to r(1)_{−i}.

Section 4. Vertex Algebras.

We will list some identities satisfied by the operators u_{n} and show how to construct Lie algebras from them. u, v, and w denote elements of V, and 1 is the unit of V.

For any even lattice R the operators u_{n} on V satisfy the following relations.

(i) u_{n}(w) = 0 for n sufficiently large (depending on u, w). This ensures convergence of the following formulae.

(ii) 1_{n}(w) = 0 if n ≠ −1, w if n = −1.

(iii) u_{n}(1) = D^{(−n−1)}(u).
(iv) u_{n}(v) = ∑_{i≥0}(−1)^{i+n+1}D^{(i)}(v_{n+i}(u)).
(v) (u_{m}(v))_{n}(w) = ∑_{i≥0}(−1)^{i}\binom{m}{i}(u_{m−i}(v_{n+i}(w)) − (−1)^{m}v_{m+n−i}(u_{i}(w))).

(The binomial coefficient \binom{m}{i} is equal to m(m − 1) … (m − i + 1)/i! if i ≥ 0 and 0 otherwise.)

We will call any module with linear operators D^{(i)}(u) and bilinear operators u_{n}(v) satisfying relations i-v above a vertex algebra, so V is a vertex algebra. (When we work with Kac-Moody algebras that so not have all real roots of norm 2, we can also construct a space V and operators u_{n}; however n is not always integral, u lies in a subspace of V depending on n, and u_{n} acts on a space that is different from V.)

Another example of a vertex algebra is given by taking any ring with a derivation [i.e., maps D^{(i)} with D^{(i)} = 0 for i < 0, 1 for i = 0,

\begin{aligned}D^{(i)}D^{(j)} & = \binom{i + j}{i}D^{(i+j)} \\ D^{(i)}(uv) & = ∑_{j} D^{(j)}(u)D^{(i−j)}(v)] \end{aligned}

and defining u_{n}(v) to be D^{(−n−1)}(u)v. This satisfies conditions i-iii and v and satisfies condition iv if and only if the ring is commutative. It also satisfies u_{n}(v) = 0 if n ≥ 0, and conversely and vertex algebra satisfying this comes from a unique ring with derivation. Hence vertex algebras are a generalization of commutative rings with derivations.

A module over a vertex algebra V is a module W with operators u_{n} on W satisfying relations i-v above for u, v in V, w in W. In particular V is a V module. (Warning—if V comes from a ring with derivation then vertex algebra modules over V are not the same as ring modules over V.)

If V is any vertex algebra then V/DV is a Lie algebra, where DV is the sum of all the spaces D^{(i)}(V) for i ≥ 1 and where the Lie algebra product is [u, v] = u_{0}(v). Note that u_{0}(v) is not antisymmetric on V. Any V module W becomes a module for the Lie algebra V/DV by letting v in V/DV act as v_{0} on W. (If v is in DV then v_{0} is 0.) In particular V is a V/DV module and is usually a nonsplit extension of the adjoint representation of V/DV. The operators D^{(i)} and the products u_{n}(v) on V are invariant under the action of V/DV. (V/DV can be extended to a larger Lie algebra V [z, z^{−1}]/DV [z, z^{−1}] of operators on V that is spanned by all the operators u_{n}, but this algebra does not leave the products u_{n}(v) invariant; see Section 8.)

The free vertex algebra on some set of generators does not exist because of relation i. However if for each pair of generators u, v we fix an integer n(u, v) and include the relations u_{i}(v) = 0 for i ≥ n(u, v) then there is a universal vertex algebra with these generators and relations. It can be constructed as a subalgebra of the vertex algebra V (R) for a certain lattice R depending on the n(u, v)s, and in particular any relation between the operators u_{n} that holds for all the vertex algebras constructed from lattices can be deduced from relations i-v.

Section 5. The Virasoro Algebra.

We will construct a representation of the Virasoro algebra on V using some operators c_{n}, which are the Segal operators, and use this to reduce the space V/DV.

We assume that R is nonsingular, and we let c be the element \frac{1}{2} ∑_{i} r_{i}(1)r′_{i}(1) of V, where r_{i} runs over some base of R and r′_{i} is the dual base. We write L_{i} for the operator c_{i+1}, and we find that the L_{i} have the following properties:

L_{−1} = D, L_{0} = deg
[L_{i}, L_{j}] = (i − j)L_{i+j} + (i^{3} − i) dim(R)δ_{i,−j}/12
L_{−i} is the adjoint of L_{i}.

In particular the operators L_{i} and 1 span a copy of the Virasoro algebra. If R is a (possibly singular) lattice contained in a nonsingular lattice S, then the operators L_{i} for i ≥ −1 on the vertex algebra of S restrict to operators on the vertex algebra of R that do not depend on the lattice S containing R. In particular if i ≥ −1 then the operator L_{i} can be defined on the vertex algebra V of R even when R is singular. We define the physical subspace P^{i} to be the elements v of V with L_{n}(v) = iv if n = 0, 0 if n ≥ 1. If v is in P^{1} then the operator v_{0} commutes with the Virasoro algebra so it preserves all the spaces P^{i}. If v in P^{1} is equal to Du for some u in V then u is in P^{0}, so P^{1}/DP^{0} is a Lie algebra acting on V and commuting with the Virasoro algebra. More generally if u is in P^{1} then

[L_{j}, u_{k}] = ((j + 1)(i − 1) − k)u_{j+k}.

If R contains the root lattice of the Kac-Moody algebra A′ (possibly quotiented out by some null lattice) then A′ can be mapped to P^{1}/DP^{0} by

\begin{aligned}e_{i} & = e^{r_{i}} \\ f_{i} & = −e^{r_{i}} \\ h_{i} & = r_{i}(1). \end{aligned}

Here r_{i} are the simple roots of A, and e_{i}, f_{i}, and h_{i} are the usual generators for the derived algebra A′ of A. It is easy to check that these elements are in P^{1} and satisfy the relations for A′, so we obtain a representation of A′. If the root lattice of A′ is singular, we can either quotient out by the kernel of the bilinear form on it, in which case we will not obtain a faithful representation of A′, or embed it in a nonsingular lattice R, in which case some of the elements r(1) will be outer derivations of A′.

If r is any nonzero vector of R the then dimension of the r subspace of P^{1} or P^{0} is p_{d−1}(1−(r, r)/2) or p_{d−1}(−(r, r)/2), where d is the dimension of R and p_{d−1} is the number of partitions into d − 1 colors. Hence the dimension of the r subspace of P^{1}/DP^{0} is equal to p_{d−1}(1 − (r, r)/2) − p_{d−1}(−(r, r)/2), and this is an upper bound for the multiplicity of roots of A (provided A is connected and not affine so that the kernel of the map from A′ to P^{1}/DP^{0} is in the center of A′).

Example: If R is the 18-dimensional even unimodular Lorentzian lattice II_{17,1} and A is the Kac-Moody algebra whose Dynkin diagram is that of R, then A has roots of norm 2, 0, −2, and −4 whose multiplicity is equal to the upper bound given above. However there are several Kac-Moody algebras for which numerical evidence suggests the better upper bound p_{d−2}(1 − (r, r)/2) for the multiplicities of roots (3). Frenkel used the no-ghost theorem to prove this stronger upper bound when R is 26 dimensional and Lorentzian. In this case P^{1}/DP^{0} has a normal subalgebra such that the quotient by this subalgebra is a simple algebra with root spaces of dimension p_{d−2}(1 − (r, r)/2) for r ≠ 0.

If the lattice R is odd we can use it to construct a “super vertex algebra” V acted on by a super Virasoro algebra spanned by elements 1, L_{i}, and G_{i+1/2}. The space G_{1/2}P^{1/2}/DP^{0} is then a Lie algebra. (Not a proper superalgebra!) For example if R is I_{9,1} then G_{1/2}P^{1/2}/DP^{0} has a normal subalgebra such that the quotient by this subalgebra is a simple Lie algebra with root spaces of dimensions equal to the coefficient of x^{(1−(r,r))/2} in ∏_{i≥1}(1 − x^{i−1/2})^{8}(1 − x^{i})^{−8}. This simple algebra contains the Kac-Moody algebra that has a simple root r′ for every vector r of the lattice E_{8}, with (r′, s′) = 1 − \frac{1}{2}(r − s, r − s). (This is similar to the “monster Lie algebra”, which has a simple root r′ for every vector r of the Leech lattice, with (r′, s′) = 2 − \frac{1}{2}(r − s, r − s).)

The operator L_{1} can be used to describe the adjoint of u_{n}: if u has degree i then the adjoint of u_{n} is

(−1)^{i} ∑_{j} L_{1}^{j}(ω(u))_{2i−j−n−2}/j!

In particular if u is in P^{1} then the adjoint of u_{0} is −ω(u)_{0}, so the adjoint of e_{i} is f_{i} and the adjoint of h_{i} is −h_{i}.

Section 6. The Representations L(r).

If r is any element of the weight lattice R′ of R we construct an irreducible A module whose largest weight is r, and these representations generalize the highest weight and adjoint representations of A.

We first assume that r is in R. We take the space P^{i} with i = (r, r)/2. This has a maximal graded submodule not containing e^{r}, and if we quotient out by this we get an irreducible module that we denote by L(r). L(r) has the following properties.

(i) L(r) is irreducible.

(ii) The weight r has multiplicity 1, and if (s, s) > (r, r) then s has multiplicity 0. This implies that L(r) is integrable, so in particular if s and t are conjugate under the Weyl group they have the same multiplicity.

(iii) L(r) has a nonzero contravariant bilinear form, which is unique up to multiplication by a constant. (This is not necessarily positive definite unless r is a highest or lowest weight vector.)

(iv) All weights of L(r) have finite multiplicities. (I do not know of any formula for the multiplicities except in the cases below.)

(v) If r is a highest or lowest weight vector then L(r) is the corresponding highest or lowest weight module, and if r is a real root of A then L(r) is a quotient of the adjoint representation (and equal to A′ modulo its center if this is simple).

(vi) If r and s are conjugate under the Weyl group then L(r) = L(s). (The converse is not true; for example r and s could be two real roots of A in different orbits of Aut(R).)

If r is an element of R′ not in R then the construction above does not work because e^{r} is not in V, so we construct the space V_{r} by replacing Q(R) in the tensor product defining V by Q(R + r). All the operators u_{n} for u in V act on V_{r}, and we can construct L(r) as a subquotient of V_{r} as above.

Problem: Is L(r) the only A module satisfying conditions i and ii above? (It is if r is a highest or lowest weight vector, and in this case condition ii implies condition i.)

Section 7. Integral forms for Kac-Moody algebras.

We constructed an integral form V_{Z} for V in Section 3. Here we will use this to find an integral form for the universal enveloping algebra for the Kac-Moody algebra A.

For each r in the weight lattice R′ we define the integral form L_{Z}(r) of L(r) to be the elements of L(r) represented by elements in the integral form of V_{r}. (If r is not in R then the integral form of V_{r} is e^{r}V_{Z}.) Similarly the integral form A_{Z} of A is the set of elements of A represented by elements of V_{Z}. This acts on all the L_{Z}(r) because u_{n} preserves the integral form of V_{r} Finally we define the integral form U_{Z} of the universal enveloping algebra U of A to be the subalgebra of U preserving all the L_{Z}(r)s. Calculation shows that U_{Z} contains (e_{i})^{n}/n! and (f_{i})^{n}/n! for all integers n ≥ 0 where the es and fs are the generators for A. We can therefore use U_{Z} to define Kac-Moody groups over finite fields (or over any commutative ring) in the same way that Chevalley groups are defined, by using the automorphisms exp(te_{i}) and exp(tf_{i}) of L_{Z}(r) ⊗ F for t in the finite field F.

The element c of V is not usually in V_{Z}, but V_{Z} can be extended to a larger integral form containing 2c and containing c if dim(R) is even. In any case the operators L_{i} for i ≥ −1 and L_{±1}^{n}/n! preserve V_{Z}. (Warning—these operators do not preserve the integral form of V_{r} for r not in R.)

Tits has also constructed an integral form for Kac-Moody algebras (4).

Section 8. Affinization.

If A is a Kac-Moody algebra with root lattice R we can construct a sort of affinization of A, which when A is finite dimensional is just the affine algebra of A When A is finite dimensional the affinization is also a Kac-Moody algebra, but this is not true in general.

To construct the affinization we form the lattice R_{1} that is the sum of R and a onedimensional lattice generated by s with (s, s) = 0 and let V_{1}, P_{1}^{i} be the Fock space and physical spaces of R_{1}. Then we define the affinization Ā of A to be the subalgebra of P_{1}^{1}/DP_{1}^{0} generated by the elements e^{ns±r_{i}} where n runs through the integers, and the rs are the simple roots of A.

Ā is an extension N.A[z, z^{−1}] of an algebra N with an infinite descending central series by the algebra of Laurent polynomials in A. When A is finite dimensional and simple, N is one dimensional and we recover the usual affinization of A. If R_{2} is the lattice that is the sum of R and a lattice generated by s, t with (s, s) = (t, t) = 0, (s, t) = 1, then Ā has many representations that can be constructed as subquotients of the subspaces P_{2}^{i} of V_{2}.

There is a second way to construct the affinization of A. If V is any vertex algebra we make W = V [z, z^{−1}] into a vertex algebra by defining

uz_{i}^{m}(vz^{n}) := ∑_{j} \binom{m}{j}u_{i+j}(v)z^{m+n−j},

and we make V into a W module by defining

uz_{i}^{m}(v) := u_{i+m}(v).

(This is a special case of the tensor product of two vertex algebras acting on the tensor product of two of their modules; in this case the vertex algebras are V and the vertex algebra of the ring Z[z, z^{−1}] with derivation D^{(i)}z^{j} = \binom{j}{i}z^{j−i}, and their modules are V and a one-dimensional module generated by an element 1 with z_{i}^{j}(1) = 1 if i + j = −1, 0 otherwise. Note that this one-dimensional module is not a module for the ring Z[z, z^{−1}].) The affinization of A is then a subalgebra of the Lie algebra W/DW, and this Lie algebra acts on V. In particular we obtain a formula for the commutator of two operators u_{m} and v_{n} on V:

[u_{m}, v_{n}] = ∑_{j} \binom{m}{j}u_{j}(v)_{m+n−j}.

If V is constructed from a lattice R and u is in W = V [z, z^{−1}] then Du = 0 if and only if u is a multiple of 1, and the operator u_{0} on V is 0 if and only if u = Dv for some v in W. If u is in V then this holds if and only if u = Dv + a for some v ∈ V and some constant a.

V is usually irreducible under the action of Ā.

Section 9. The monster.

Frenkel et al. (1) constructed an infinite-dimensional graded representation V of the monster F. This representation can be given the structure of a vertex algebra that is invariant under F and is similar to the vertex algebras constructed from positive definite lattices (or more precisely to the subspace of the complexification of such algebras fixed by the Cartan involution ω—i.e., their “compact forms”). In particular it has an element c such that the operators L_{i} = c_{i+1} give a representation of the Virasoro algebra and it has a positive definite inner product such that the adjoint of u_{n} is given by the formula in Section 5 (with ω(u) = u). One important difference between this algebra and the ones coming from lattices is that the piece of degree 1 is 0 dimensional. We will call this vertex algebra V the Monster vertex algebra.

Any vertex algebra V with these properties (except that it does not have to have an action of the monster F on it) also has the following two properties for u and v in the degree 2 piece of V.

(i) u_{1}(v) = v_{1}(u). When V is the Monster vertex algebra this is essentially the Griess product (5). Also u_{1} is self adjoint.

(ii) Norton’s inequality (see ref. 6): x = (u_{1}(u), v_{1}(v)) − (u_{1}(v), u_{1}(v)) is nonnegative and zero if and only if the operators u_{1} and v_{1} commute. In fact x is the norm of w = u_{0}(v) − \frac{1}{2}D(u_{1}(v)), and the operator w_{2} is the commutator of u_{1} and v_{1}.

There are a large number of other vertex algebras with these properties.

In particular the Griess product can be extended to the whole of V in a natural way, and there are also an infinite number of other products u_{n}(v) on V invariant under the action of the Monster on V.

The product u_{1}(v) is not symmetric on the whole of V. If we want symmetric products we can define the product ×_{n} for any integer n by

u ×_{n} v = ∑_{i≥0} \frac{(−1)^{i}}{i + 1} D^{(i)}(u_{n+1+i}(v))

and these are symmetric or antisymmetric depending on whether n is even or odd. If n = 0 this is equal to the Griess product u_{1}(v) on the degree 2 piece of V, and D(u×_{0} v) = u_{0}(v)+v_{0}(u). These products have these properties for any vertex algebra over the rational numbers but do not seem to be as natural as the products u_{n}(v).

This research was supported by Trinity College, Cambridge, England and by the Natural Sciences and Engineering Research Council of Canada.

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6. Conway, J. H. (1985) Invent. Math. 79, 513-540.