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ELECTRONIC RESEARCH ANNOUNCEMENTS doi:10.3934/era.2014.21.167 IN MATHEMATICAL SCIENCES Volume 21, Pages 167–176 (November 18, 2014) S 1935-9179 AIMS (2014)

GROUPS OF LIE TYPE, VERTEX ALGEBRAS, AND MODULAR MOONSHINE

ROBERT L. GRIESS JR. AND CHING HUNG LAM

(Communicated by Walter Neumann)

Abstract. We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra . In this way, we get series of vertex algebras over fields whose groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power q, we realize each finite simple which is a Chevalley or Steinberg variations over Fq as “most of” the full of a vertex algebra over Fq. These finite simple groups are

An(q),Bn(q),Cn(q),Dn(q),(q),(q),(q),(q),(q) 2 2 3 2 and An(q), Dn(q), D4(q), E6(q), where q is a prime power. Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to com- plete the modular moonshine program of Borcherds and Ryba for proving an 15 10 3 2 embedding of the F3 of order 2 3 5 7 13·19·31 in E8(3).

1. Introduction This article is a preview of [13], where full details will be published. We begin by constructing classical vertex algebras for all types of root systems, and Chevalley groups acting on them. Extensions of this procedure give VAs for the Steinberg variations. Recent results on integral forms in vertex algebras [3, 8, 20] helped us promote the basic idea of Chevalley basis of a to the vertex algebra situation. We show that, given a field F , a Chevalley group or Steinberg variation over F is essentially the full automorphism group of some

Received by the editors May 24, 2014. 2010 Mathematics Subject Classification. Primary 20D05, Secondary 17B69. Key words and phrases. vertex algebra, integral form, Chevalley and Steinberg groups, mod- ular moonshine, sporadic group. The first author thanks Academia Sinica for hospitality during visits in 2012 and 2013, and the US National Security Agency and the University of Michigan for financial support. The second author thanks National Science Council (NSC 100-2628-M-001005-MY4) and National for Theoretical Sciences of Taiwan for financial support. We also thank Brian Parshall and James Humphreys for consultations.

c 2014 American Institute of Mathematical Sciences 167 168 ROBERT L. GRIESS JR. AND CHING HUNG LAM vertex algebra over F . When the field and satisfy some conditions, the exact automorphism group is an upwards extension of the latter group by outer diagonal automorphisms and possibly a group of graph automorphisms. In particular, the finite simple groups of Chevalley and Steinberg over finite fields (extended by certain outer automorphisms) are realized as the full automorphism groups of vertex algebras. These finite simple groups are

An(q),Bn(q),Cn(q),Dn(q),E6(q),E7(q),E8(q),F4(q),G2(q)

2 2 3 2 and An(q), Dn(q), D4(q), E6(q), where q is a prime power. As with modular Lie algebras (i.e., in positive characteristic), there is exceptional behavior for certain types of classical vertex algebras in characteristics 2 and 3. Such a reduced vertex algebra (the vertex algebra modulo a certain nontrivial ideal; see Definition 5.1) has automorphism group which is a larger Chevalley group. There is no analogue of this behavior for finite dimensional Lie algebras and groups over the complex numbers. The covering procedure developed by Frohardt and Griess [11], based on action of a graph automorphism, was used to demonstrate this exceptional behavior for Lie algebras in a uniform way (linear algebra study of a graph automorphism) without case-by-case work and special calculations used in earlier treatments (see [11] for details and history). Fortunately, these covering procedures can easily be promoted to the vertex algebra situation to construct the exceptional automorphism groups for corresponding reduced vertex algebras. A variant of the covering procedure [11] applies to the interesting study by Borcherds and Ryba [4] of a 3C element g in the Monster . Its cen- ∼ tralizer has the form C(g) = 3 × F3, where F3 is a sporadic simple group of order 15 10 3 2 2 3 5 7 13·19·31. The first proof that F3 embeds in the Chevalley group E8(3) was made by John Thompson and Peter Smith, using computers [23]. Years later, Borcherds and Ryba created a graded F3- which was a module for

C(g)/hgi and which they felt ought to be (up to some re-indexing) IVE8 /3IVE8 , where IVE8 denotes the standard integral form of the VOA VE8 [3,8, 20]. We proved their conjecture by use of several vertex algebras and a subspace of one which plays the role of a covering transversal (Definition 1.2). We now define several kinds of coverings, then describe an example of the cov- ering method. Notation 1.1. A root system is denoted by usual capitals A, B, . . . , etc. A capital symbol X or Xn stands for a root system of type X and of rank n. When the system is simply laced (types A, D, E), X can also stand for the root lattice. The Lie algebra of type Xn over the R, defined by a Chevalley basis, is denoted xn(R). A common notation for a general Lie algebra is g. The Chevalley group of type Xn over R is denoted by Xn(R). We may write the latter as X(R) if the subscript n is relatively unimportant. Definition 1.2. Given an A and a B, the subgroup C is called a covering subgroup with respect to B if C maps onto A/B, i.e., A = C + B. A covering transversal is a covering subgroup C so that A = B ⊕ C. If A is an algebra, respectively, a vertex algebra, and B is an ideal, a subalgebra, respectively, a subVA, C is said to cover A/B if C + B = A. Such a C is called a covering algebra or a covering VA with respect to B. GROUPS OF LIE TYPE, VERTEX ALGEBRAS, AND MODULAR MOONSHINE 169

Example 1.3. Here is an example for Lie algebras over a field F of characteristic 3. The algebra a2(F ) is 8-dimensional. The quotient a2(F ) modulo its 1-dimension- al center (reduced algebra) is simple. Because of embeddings a2(F ) < g2(F ) < d4(F ) defined by a graph automorphism of order 3, we can deduce that the auto- morphism group of the reduced a2(F ) is G2(F ). The subalgebra a2(F ) in g2(F ) is referred to as a covering algebra in [11]. For the analogue in vertex algebra theory, we consider a certain containment of integral forms IVA2 < IVG2 < IVD4 (see Nota- tion 4.5) defined by a graph automorphism, then argue that a quotient of F ⊗ IVA2 by an ideal has automorphism group G2(F ). We mention other integral forms of interest (besides the standard ones considered in the present article). In [8] there are several, including a Monster-invariant integral form in the Moonshine VOA and a rank 8 example associated to the VOA V + EE8 and the finite group O+(10, 2).

2. VAs and VOAs over commutative rings We use the definition of a vertex algebra over any commutative ring R as in [3] (see also [4] and [9]). Denote the vaccum vector by 1 and the translation operator by T , i.e, T a = a−21 for any a ∈ V .

A vertex algebra V is Z-graded if V = ⊕n∈ZVn for subspaces Vn such that if a ∈ Vn, b ∈ Vm, then akb ∈ Vn+m−k−1 for any k. For the definition of over C, we refer to [9]. Definition 2.1. [8] Suppose that V is a VOA (over the complex numbers) with a nondegenerate symmetric invariant bilinear form. An integral VOA form (abbreviated IVOA) for V is an abelian subgroup J of (V, +) such that J is a VA over Z, there exists a positive integer s so that sω ∈ J, for each n, Jn := J ∩ Vn is an integral form of Vn,(J, J) ∈ Q. Since J is a VA, 1 ∈ J. For each degree n, Jn has finite rank, whence there is an integer d(n) > 0 so that d(n) · (Jn,Jn) ≤ Z. Remark 2.2. An integral form J of a VOA over C will be a VA over Z. If R is a commutative ring, then J ⊗Z R is a VA over R.

3. The Chevalley basis and Chevalley group construction for vertex algebras of types ADE

We use the standard notation for the lattice vertex operator algebra VL = M(1) ⊗ C{L} associated with a positive definite even lattice L [9].

Notation 3.1. We use a common symbol, X or Xn, to indicate a rank n type ADE root lattice, the root system or the name of the root system. For a root system of type BCFG, we shall use a symbol Y or Yn for the root system, but it shall not refer to a lattice.

In [3,8, 20], an integral form of the lattice VOA VL has been given. We call this integral form the standard integral form for VL and denote it by IVL. Let X be a root lattice. The standard integral form IVX admits the group of graph automorphism of the root system, lifted in obvious way from X to VX (see [12], Appendix A). 170 ROBERT L. GRIESS JR. AND CHING HUNG LAM

th If (IVX )n := IVX ∩ Vn, then (IVX )1 is a lattice in the Lie algebra ((VX )1, 0 ) which is spanned by a Chevalley basis, C. Recall that (VX )1 has a Lie algebra structure with the Lie bracket defined by [a, b] = a0b for any a, b ∈ (VX )1. In the notation of [6], elements of C in root spaces for root α is eα. If t ∈ C, then xα(t) := exp(t ad(eα)) takes Z[t] ⊗ (IVX )1 to itself. See [6]. Since (IVX )1 generates IVX [8], xα(t) takes Z[t] ⊗ IVX to itself. The result is that for a commutative ring R and t ∈ R, xα(t) gives an automorphism of the VA R ⊗ IVX . The group which these automorphisms xα(t) generate is isomorphic to the Chevalley group of type X over the ring R. We summarize: Theorem 3.2. Let X be an indecomposable root system of type ADE and let R be a commutative ring. Then the natural action of the Chevalley group X(R) on the Lie algebra with a Chevalley basis of type X over the scalars R extends uniquely to an action as automorphisms on the vertex algebra R ⊗ IVX .

4. Definition of VA of type BCFG Definition 4.1. Let S be a subset of a group acting on a group A. Define AS to be the elements of A fixed by S. In case G := hSi is a finite group and (A, +) is a G-module, we write ν for the P G norm map defined by ν(a) = g∈G g(a). The quotient A /ν(A) is often called the 0th Tate cohomology group, Hˆ 0(G, A).

A VOA of type ADE (one root length) is just the lattice VOA VX , where X is the root lattice. For indecomposable root systems with more than one length, more care is needed. We use the definition below for a VOA of type Bn,Cn,G2,F4. Definition 4.2. Let X be the root lattice of type ADE and let γ be the standard lift of the graph automorphism of X whose fixed points on the Lie algebra at degree γ γ 1 is a Lie algebra of type Y . The vertex subalgebra VA((VX )1) < VX generated by γ (VX )1 is defined to be the classical VA of type Y . The cases are listed here: X Y central charge of central charge of γ γ VX over C VA((VX )1) over C Dn+1 Bn n + 1 n + 1/2 A2n−1 Cn 2n − 1 2n − 3n/(n + 2) D4 G2 4 14/5 E6 F4 6 26/5 γ γ Note the distinction between VA((VX )1) and VX . Notation 4.3. Let X be a root lattice of type ADE and let γ be a graph auto- morphism so that the fixed points on the Lie algebra has type Y . Denote the fixed point sublattice of γ in X by Xγ . Then, Xγ is a root sublattice of X. X Y Xγ order of γ Dn+1 Bn Dn 2 A2n−1 Cn A1 ⊕ · · · ⊕ A1 2 D4 G2 A2 3 E6 F4 D4 2 GROUPS OF LIE TYPE, VERTEX ALGEBRAS, AND MODULAR MOONSHINE 171

In the integral form IVX , we have two subobjects: γ (i) IVX , the fixed points; γ γ (ii) VA((IVX )1), the vertex subalgebra of IVX generated by the degree 1 com- γ γ ponent (IVX )1 of the standard integral form of (VX )1 , the Lie algebra at γ γ degree 1. Note that VA((IVX )1) is an integral form of VA((VX )1). γ γ Definition 4.4. We define VA((IVX )1) to be the standard integral form of VA((VX )1).

Notation 4.5. We use VY to denote a VOA of type Y over the complex numbers, i.e., VY is the lattice VOA associated to the root lattice Y if Y is of type ADE γ γ and VY = VA((VX )1) if Y is of type BCFG and the degree 1 space (VX )1 is a Lie algebra of type Y (see Definition 4.2). The standard integral form in VY is denoted by IVY . Definition 4.6. We define a classical vertex algebra of type Y over the commuta- γ γ tive ring R to be R ⊗ IVY . Note that IVY = VA((IVX )1) IVX when Y is of type BCFG. See Definition 4.2 and Notation 4.3. 4.1. The Chevalley basis and Chevalley group construction for vertex algebras of types BCFG.

Notation 4.7. If X is a root lattice of type ADE, the group associated to R⊗IVX is denoted G(IVX ,R).

The restriction of G(IVX ,R) to R ⊗ (IVX )1 gives an isomorphism of groups G(IVX ,R) → X(R). We shall usually identify these groups. If γ is a graph au- tomorphism lifted in the obvious way to the integral form IVX and to the group X(R), then the fixed point subgroup X(R)γ leaves invariant the fixed point subIF γ γ (R ⊗ IVX ) , whence also R ⊗ VA((IVX )1). From the above, we deduce the analogue of Theorem 3.2 for Chevalley groups associated to root systems of types BCFG. Theorem 4.8. Let X be a root system of type ADE and let R be a commutative ring. Let γ be a graph automorphism. Then the natural action of the Chevalley group X(R)γ on the Lie algebra with Chevalley basis of type Y (i.e., BCFG) over the scalars R extends to an action as automorphisms on the vertex algebra R ⊗ γ VA((IVX )1) (cf. Definition 4.2). We summarize: Theorem 4.9. A Chevalley group of type A, B, C, D, E, F, or G over a commu- tative ring R acts as automorphisms of a VA over R. Its degree 1 Lie algebra is the usual classical Lie algebra over R defined by use of a Chevalley basis. In Theorem 6.4, we prove more, that such a group is essentially the full automor- phism group of the relevant VA. We also show that the related families of reduced VAs has in some cases a larger Chevalley group as most of its automorphism group.

5. The covering algebra procedure lifted to VAs This procedure was devised in [11] to give a relatively uniform and computation free determination of the automorphism groups of certain classical Lie algebras in characteristic 2 or 3 modulo a nontrivial idea. First we give a few definitions. 172 ROBERT L. GRIESS JR. AND CHING HUNG LAM

Definition 5.1. Suppose that X is an indecomposable root system of type ADE and p is a prime number so that X has a graph automorphism, say γ, of order p. Let X0 denote the set of roots fixed by γ. Let F be a field of characteristic p, g be a Lie algebra over F of classical type X and let g0 be the subalgebra corresponding to X0. Then the pair (X0, p) and subalgebra of type X0 corresponding to γ are each called exceptional. Given an exceptional pair, (X0, p), there is an essentially unique pair (X, γ) which gives rise to (X0, p) as above. We call such (X, γ) the ancestor of (X0, p). The quotient of the Lie algebra, resp. VA of type X0, by its norm ideal is called the reduced Lie algebra, resp. reduced VA of type X0.

Lemma 5.2. Let X, γ be as in Notation 4.7. Let IVX and IVXγ be the standard integral forms for the respective lattice VOA VX and VXγ . Let ν be the norm map defined in Definition 4.1 and N := ν(IVX ). Then γ (IVX ) = IVXγ + N. γ That is, IVXγ is a covering subIF of (IVX ) with respect to N (cf. Definition 1.2). n Proof. For any α ∈ X, let sα,n be the coefficient of z in ! X α(−n) E−(−α, z) = exp zn , n n>0 that is, ! X α(−n) X exp zn = s zn. n α,n n>0 n≥0 By [8, Theorem 3.3], the standard integral form IVX is spanned over Z by α {sα1,n1 sα2,n2 ··· sαk,nk ⊗ e | αi, α ∈ X, ni > 0}. Therefore, γ α γ (IVX ) = SpanZ{sα1,n1 sα2,n2 ··· sαk,nk ⊗ e | all αi, α ∈ X , ni > 0} p−1  X j α γ  + SpanZ γ (sα1,n1 sα2,n2 ··· sαk,nk ⊗ e ) | some αi or α∈ / X j=0 

= IVXγ + ν(IVX ) as desired.  Remark 5.3. Let R be a commutative ring. Then the automorphism γ also acts on R ⊗ IVX . By the same proof as in Lemma 5.2, we have γ (R ⊗ IVX ) = R ⊗ IVXγ + ν(R ⊗ IVX ). γ Note also that N = ν(R ⊗ IVX ) = (R ⊗ IVX ) if (char(R), |γ|) = 1. γ Since X is also a root lattice, the integral form IVXγ is generated by (IVXγ )1 γ γ (see [8]), which is contained in (IVX )1. Hence, IVXγ < VA((IVX )1). By Lemma 5.2 and Remark 5.3, we have the following lemma. Lemma 5.4. Let R be a commutative ring. Then γ γ (R ⊗ IVX ) = R ⊗ VA((IVX )1) + ν(R ⊗ IVX ). γ γ In particular, R ⊗ VA((IVX )1) is a covering algebra in (R ⊗ IVX ) with respect to the norm ideal ν(R ⊗ IVX ). GROUPS OF LIE TYPE, VERTEX ALGEBRAS, AND MODULAR MOONSHINE 173

6. Automorphism group of a classical VA over a field For classical VAs, the automorphism group is controlled by the Lie algebra at degree 1, so we begin with some concepts for Lie algebras. This section concludes with our main results about groups of Lie type as automorphism groups of VAs. They include occurrences of all finite Chevalley groups and Steinberg variations.

Definition 6.1. We say that a classical Lie algebra of any type ABCDEFG over the field F has the idfg property if its automorphism group has the property that every automorphism is a product of inner, diagonal, field and graph. We also say that its automorphism group has the idfg property.

Proposition 6.2. Let X be an indecomposable root system of type ABCDEFG and let F be a field which satisfies one of

(a) char(F ) 6= 2 if X has type Dn or two root lengths; (b)( X, char(F )) 6= (A2, 3); (c) F is algebraically closed; (d) F is finite. Then,

(1) Let g be a classical Lie algebra of type X over the field F. Then AutF (g) has idfg-type. (2) Suppose that g is the reduced algebra for an exceptional pair (X, p) over the field F . Assume that the classical algebra of type X over F has the idfg property. Then (in the notation of Notation 5.1) the reduced Lie algebra g and Aut(g) occur in the list below: ∼ (i) Aut(a2(F )/Z(a2(F )) = G2(F ), for char(F ) = 3. ∼ (ii) Aut(g2(F )) = B3(F ), for char(F ) = 2. Note that Z(g2(F )) = 0 in this case. ∼ (iii) Aut(d4(F )/Z(d4(F )) = F4(F ), for char(F ) = 2. ∼ (iv) Aut(dn(F )/Z(dn(F )) = Bn(F ), for char(F ) = 2, n = 3 or n > 4.

Proof. This was proved by Steinberg [22] when (X, p) satisfies (a) or (b). For (c), we quote Hogeweij [14], who extended the result to F algebraically closed for all types of algebras. For (d), we use an argument in an appendix to [13] which shows that the idfg property can be passed to fixed points under certain automorphisms. This applies here by using (c) and the Frobenius automorphism on an algebraic closure of a finite field.  Theorem 6.3. Let F be a field. Let V be either a VA of classical type over F of non-exceptional type or a reduced VA of classical type over F in the exceptional case. Assume that the Lie algebra V1 has a the idfg property. Then V has the idfg property and the restriction of Aut(V ) to V1 gives an isomor- phism onto Aut(V1), the automorphism group of the reduced classical Lie algebra at degree 1 (the possibilities are listed in Proposition 6.2).

An appendix in [13] treating classical type untwisted and twisted Lie algebras and their groups may be adapted in a straightforward manner with integral forms to the situation where the Lie algebra is replaced by a classical vertex algebra. The result is the following. 174 ROBERT L. GRIESS JR. AND CHING HUNG LAM

Theorem 6.4. We assume that X is a type of root system ABCDEFG and that E is a field so that the classical Lie algebra of type X over E has the idfg property 6.1. (i) For each of the Chevalley groups

An(E),Bn(E),Cn(E),Dn(E),E6(E),E7(E),E8(E),F4(E),G2(E), there exists a classical vertex algebra over E whose automorphism group is an upwards extensions of the group by its diagonal or (when X has type An,Dn or E6), an upwards extension of the group by its diagonal outer automorphism group followed by an upwards extension by its group of graph automorphisms. (ii) Suppose that σ is a field automorphism of E and F is the fixed field. Then the conclusion of (i) holds for F in place of E. (iii) Assume that X(E) has a graph automorphism γ of order m > 1 and a field automorphism σ of order m. Define F to be the fixed field. We then define η := γσ. Let W be the classical vertex algebra over E which is associated to X(E). Then the fixed point subVA W η has group of F -automorphisms isomorphic to the corresponding Steinberg variation, one of 2 2 3 2 An(F ), Dn(F ), D4(F ), E6(F ). 6 One can also realize a group D4(F ) as automorphisms of a VA (this notation refers to the Steinberg variation associated to a Galois extension F ≤ E whose automorphism group is Sym3). We deduce a significant application to finite simple groups. Theorem 6.5. Let q be a prime power. For each of the Chevalley groups

An(q),Bn(q),Cn(q),Dn(q),E6(q),E7(q),E8(q),F4(q),G2(q) and the Steinberg variations 2 2 3 2 An(q), Dn(q), D4(q), E6(q), there exists a vertex algebra over Fq whose automorphism group is an upwards extensions of the group by its diagonal outer automorphism group or (in the case of untwisted group of type An,Dn or E6), an upwards extension of the group by its diagonal outer automorphism group followed by an upwards extension by its group of graph automorphisms. At the moment, we say nothing about the remaining groups of Lie type, the 2 2 Suzuki series B2(q) and the Ree series F4(q) (where q is an odd power of 2); and 2 finally the Ree series G2(q) (where q is an odd power of 3).

7. Modular moonshine and an embedding of the sporadic group F3 into E8(3) 15 10 3 2 The simple group F3 of order 2 3 5 7 13 · 19 · 31 embeds in E8(3), a result of Peter Smith and John Thompson proved by computer work on a 248-dimensional lattice [23]. Richard Borcherds and Alex Ryba [4] suggested a vertex algebra style proof, but their program was not fully verified. In this section, we show how a combination of the covering algebra (Section 4) and VOA integral form viewpoints allow us to complete the Borcherds-Ryba pro- gram. It is possible that some other embeddings of finite groups into groups of Lie type could be proved with similar VOA methods. GROUPS OF LIE TYPE, VERTEX ALGEBRAS, AND MODULAR MOONSHINE 175

7.1. Modular moonshine of Borcherds and Ryba. We start with one result of [4]. Theorem 7.1 ([4, Corollary 4.8]). Let g be a 3C element of the . g L Then there is a vertex algebra V = Vn defined over 3 such that n∈Z F g ∼ (1) V is acted on by the group CM(g)/hgi = F3; and P g n−1 (2) the Brauer trace tr(h| Vn)q of any 3-regular element h of C (g)/hgi n∈Z M on gV is a Hauptmodul, and is equal to the Hauptmodul of the element gh of the Monster. In particular, the character of gV is equal to X g n−1 3 −1 2 5 chg (q) = dim( V )q = ch (q ) = q + 248q + 4124q + ··· . V n VE8 n∈Z Borcherds and Ryba also conjectured that gV (up to a certain reindexing of degrees) is isomorphic to the vertex algebra IVE8 /3IVE8 over F3. (As discussed earlier in this article, IVE8 means the standard integral form for VE8 .) In [13], we give a proof of their conjecture. For reasons of space, we give here a brief sketch of our proof. 1 \ 1 \ In [4], Borcherds and Ryba used a Z[ 2 ]-form V [ 2 ] of the Moonshine VOA V , g which is invariant under the action of the Monster. The VA V over F3 is defined to be the quotient (V \[1/2])g gV := . ν(V \[1/2]) The covering object strategy would need a subobject of (V \[1/2])g which in a sense looks like IVE8 . \ + T,+ The Moonshine VOA V = VΛ ⊕ VΛ is constructed as a Z2- of the Leech lattice VOA VΛ [9]. The Leech√ lattice contains many sublattices of the \ form EE8 ⊥ EE8 ⊥ EE8, where EE8 = 2E8. Therefore, V contains a subVOA isomorphic to (V + )⊗3. In fact, one can construct the Moonshine VOA V \ directly EE8 by gluing some irreducible modules of (V + )⊗3 [19]. Miyamoto [19] also showed EE8 that the cyclic permutation of the 3 tensor factors of (V + )⊗3 can be lifted to an EE8 automorphism of V \ and the lift is in the conjugacy class 3C of Aut(V \). It is well-known [1, 21] that V + has exactly 210 inequivalent irreducible mod- EE8 ules and all of them are simple current modules. Moreover, the set of all inequiva- lent irreducible modules R(V + ) forms a 10-dimensional non-degenerate quadratic EE8 + space of plus type over F2. Let Φ be a maximal singular subspace of R(VEE ). Then L 8 V = A∈Φ A has a unique VOA structure, which is isomorphic to the lattice VOA V [21]. Moreover, one can find a subVOA (V + )⊗3 in V \ such that the space E8 EE8 L \ A∈Φ A ⊗ A ⊗ A is contained in V by the analysis in [21]. 1 ˜ 1 1 The technical part of [13] is to define Z[ 2 ] forms VE8 [ 2 ] and A[ 2 ] for VE8 1 and A ∈ Φ so they are compatible with the Z[ 2 ]-form given in [4]. A map ˜ 1 \ 1 g 1 η : VE8 [ 2 ] → (V [ 2 ]) with x 7→ x ⊗ x ⊗ x for x ∈ A[ 2 ], A ∈ Φ, is also defined. The map η is not linear, but its reduction modulo 3 becomes a homomorphism of vertex algebras. We thus give an imprecise suggestion of a covering mod 3 of \ g (V ) by VE8 . The resulting main theorem is as follows: Theorem 7.2 ([13, Theorem 7.29]). The map η induces an isomorphism of vertex ˜ ˜ g g algebras between VE8 [1/2]/3VE8 [1/2] and V over F3. In particular, V3 is a Lie algebra of type E8 over F3 and the simple group F3 embeds in E8(3). 176 ROBERT L. GRIESS JR. AND CHING HUNG LAM

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Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA. E-mail address: [email protected]

Institute of Mathematics, Academia Sinica, Taipei, 10617, Taiwan. E-mail address: [email protected]