A Vertex Operator Algebra Related to E8 with Automorphism Group O+(10, 2)

A vertex operator algebra related to E8 with automorphism group O+(10, 2)

Robert L. Griess, Jr.

Abstract. We study a particular VOA which is a subVOA of the E8-lattice VOA and determine its automorphism group. Some of this group may be seen within the group E8(C), but not all of it. The automorphism group turns out to be the 3-transposition group O+(10, 2) of order 22135527.17.31 and it contains the simple group Ω+(10, 2) with index 2. We use a recent theory of Miyamoto to get involutory automorphisms associated to conformal vectors. This VOA also embeds in the moonshine module and has stabilizer inIM, the monster, of the form 210+16.Ω+(10, 2).

Hypotheses We review some deﬁnitions, based on the usual deﬁnitions for the elements, products and inner products for lattice VOAs; see [FLM].

Notation 1.2. Φ is a root system whose components have types ADE, g is a Lie algebra with root system Φ, Q := QΦ, the root lattice and V := VQ := S(Hˆ−) C[Q] is the lattice VOA in the usual notation. ⊗

Remark 1.3. We display a few graded pieces of V ( is omitted, and here Q can −m ⊗ be any even lattice). We write Hm for H t in the usual notation for lattice VOAs (2.1) and Q := x Q (x, x)=2m⊗ , the set of lattice vectors of type m. m { ∈ | } V0 = C, V1 = H1,

2 V2 = [S H1 + H2]+ H1CQ1 + CQ2, 3 2 V3 = [S H1 + H1H2 + H3] + [S H1 + H2]CQ1 + H1CQ2 + CQ3, 4 2 2 V4 = [S H1 + S H1H2 + H1H3 + S H2 + H4]+ 3 2 [S H1 + H1H2 + H3]CQ1 + [S H1 + H2]CQ2 + H1CQ3 + CQ4.

Remark 1.4. Let F be a subgroup of Aut(g), where g is the Lie algebra V1 = th F H1 + CQ1 with 0 binary composition. The ﬁxed points V of F on V form a subVOA. We have an action of N(F )/F as automorphisms of this sub VOA. 44 R. L. Griess, Jr.

Notation 1.5. For the rest of this article, we take Q to be the E8 -lattice. Take C F to be a 2B-pure elementary abelian 2-group of rank 5 in Aut(g) ∼= E8( ); it is ﬁxed point free. Let E := F T where T is the standard torus and where F is chosen to make rank(E)=4.∩ Let θ F r E; we arrange for θ to interchange the ∈ standard Chevalley generators xα and x−α. See [Gr91]. The Chevalley generator α xα corresponds to the standard generator e of the lattice VOA VQ .

[E] Notation 1.7. L := Q ∼= √2Q denotes the common kernel of the lattice characters associated to the elements of E; in the [Carter] notation, these characters are h−1(E); in the root lattice modulo 2, they correspond to the sixteen vectors in a maximal totally singular subspace. Then F (1.7.1) V1 =0 and F 2 C θ (1.7.2) V2 = S H1 +0+ L2, where the latter summand stands for the span of all eλ + e−λ, where λ runs over all the 15 16 = 240 norm 4 lattice vectors in L. Thus, V F has dimension 9 + 240 = · 2 2 2 36+120 = 156 and has a commutative algebra structure invariant under N(F ) = ∼ 25+10 GL(5, 2). We note that N(F )/F = 210: GL(5, 2) [Gr76][CoGr][Gr91]. · ∼ F + We will show (6.10) that Aut(V ) ∼= O (10, 2).

2. Inner Product.

n n n n n Deﬁnition 2.1. The inner product on S Hm is x , x = n!m x, x . This is based on the adjointness requirement for h tk andh h t−ki (see (1.8.15),h i FLM,p.29). When k > 0, h t−k acts like multiplication⊗ by h⊗ t−k and, when h is a root, h tk acts like k⊗times differentiation with respect to⊗h. ⊗ When n =2, this means x2, x2 =2m2 x, x . In V F , m =1. h i h i 2 Deﬁnition 2.2. The Symmetric Bilinear Form. Source: [FLM], p.217. This form is associative with respect to the product (Section 3). We write H for H1. The set 2 + of all g and xα spans V2. (2.2.1) g2,h2 =2 g,h 2, h i h i whence

(2.2.2) pq,rs = p, r q,s + p,s q, r , for p,q,r,s H. h i h ih i h ih i ∈

+ + 2 α = β (2.2.3) xα , x = ± h β i 0 else n (2.2.4) g2, x+ =0. h β i + A VOA related to E8 with automorphism group O (10, 2) 45

Notation 2.3. In addition, we have the distinguished Virasoro element ω and identity I 1 ∗ := 2 ω on V2 (see Section 3). If hi is a basis for H and hi the dual basis, then 1 ∗ ω = 2 i hihi . RemarkP 2.4. (2.4.1) g2,ω = g,g h i h i

1 (2.4.2) g2, I = g,g h i 2h i

(2.4.3) I, I = dim(H)/8 h i

(2.4.4) ω,ω = dim(H)/2 h i If x i =1, ...ℓ is an ON basis, { i | }

ℓ 1 (2.4.5) I = x2 4 i i=0 X

ℓ 1 (2.4.6) ω = x2. 2 i i=0 X

F 3. The Product on V2 . F Deﬁnition 3.1. The product on V2 comes from the vertex operations. We give it 2 λ −λ on standard basis vectors, namely xy S H1, for x, y H1 and vλ := e + e , ∈ ∈ 2 for λ L2. Note that (3.1.1) give the Jordan algebra structure on S H1, identiﬁed ∈ 1 with the space of symmetric 8 8 matrices, and with x, y = 8 tr(xy). Thefunction ε below is a standard part of notation× for lattice VOAs.h i

(3.1.1) x2 y2 =4 x, y xy, pq y2 =2 p,y qy +2 q,y py, × h i × h i h i pq rs = p, r qs + p,s qr + q, r ps + q,s pr; × h i h i h i h i

(3.1.2) x2 v = x, λ 2v , xy v = x, λ y, λ v × λ h i λ × λ h ih i λ 0 λ, µ 0, 1, 3 ; h i∈{ ± ± } (3.1.3) vλ vµ = ε λ, µ vλ+µ λ, µ = 2; ×  h2 i h i −  λ λ = µ.  46 R. L. Griess, Jr.

Convention 3.2. Recall that L = Q[E]. Since (L,L) 2Z, we may and do assume that ε is trivial on L L. ≤ ×

4. Some Calculations with Linear Combinations of the vλ. 2 Notation 4.1. For a subset M of H, there is a unique element ωM of S H which 2 satisfies (1) ωM S (span(M)); (2) for all x, y span(M), x, y = ωM , xy . I ∈ 1 ∈ h i h i We define M := 2 ωM . If M and N are orthogonal sets, we have ωM∪N = ′ I′ I I ωM + ωN . Define ωM := ω ωM and M := M . This element can be 1 2 − − written as ωM = 2 i xi , where the xi form an orthonormal basis of span(M). 1 I I 1 We have ωM ,ωM = 2 dim span(M) and M , M = 8 dim span(M). Also, h i′P′ ′ ′ h i ωM , xy = ωM , x y = ω, x y , where priming denotes orthogonal projection to hspan(Mi). h i h i

Notation 4.2. e± := f ∓ := 1 [λ2 4v ], e = e+, f := e−. If a Z or Z , λ λ 32 ± λ λ λ λ λ ∈ 2 deﬁne e to be e+ or e−, as a 0, 1(mod 2), respectively; see (4.7). Also, let λ,a λ λ ≡ e′ = e . We deﬁne e to be e , where a is 1 λ, µ in case µ is a vector in λ,a λ,a+1 λ,µ λ,a 2 h i L, and a is [λ,ˆ µ], where [., .] is the nonsingular bilinear form on Hom(L, 1 ) 1 ˆ {± } gotten from 2 ., . by thinking of Hom(L, 1 ) as 2 L/L and where λ is the h i {± } 1 character gotten by reducing the inner product with 2 λ modulo 2. Finally. let q be the quadratic form on Hom(L, 1 ) gotten by reducing x x, x modulo 2, for x 1 L. {± } 7→ h i ∈ 2 ± Lemma 4.3. (i) The eλ are idempotents. 1 16 λ = µ; (ii) e±,e± = 1 λ, µ = 2; h λ µ i  128 0 hλ, µi =0−.  h i 0 λ = µ; (iii) e±,e∓ = 1 λ, µ = 2; λ µ  128 h i 0 hλ, µi =0−.  h i ± 2 1 2 2 2 ± Proof. (i) (eλ ) = 1024 [4 4λ +16λ 8 4 vλ]= eλ . (ii) and (iii) follow trivially from (2.2). · ± ·

Notation 4.4. For ﬁnite X L, deﬁne s(X) := x2. ⊆ x∈±X/{±1} P |L2| Lemma 4.5. If X L2, ω,s(X) = 4 ( X)/ 1 and so s(L2) = 2 ω = 120ω. ⊆ h i | ± {± }|

Proof. (2.2.5)

1 Corollary 4.6. (i) For α L2, ωα,s(α) = ω,s(α) = 4 and ωα,ωα = 2 , ∈I hI 1 i2 h i h i whence s(α)=8ωα = 16 α and α = 16 α . (ii) ω ,s(Φ ) = ω,s(Φ ) = 63, whence s(Φ )=18ω E ; h E7 E7 i h E7 i E7 Φ 7 (iii) ω,s(Φ ) = 56, whence s(Φ )=56ω E . h D8 i D8 Φ 7 + A VOA related to E8 with automorphism group O (10, 2) 47

Notation 4.7. For ϕ Hom(L, 1 ), deﬁne f(ϕ) := ϕ(λ)v , ∈ {± } λ∈L2/{±1} λ u(ϕ) := ϕ(λ)λ2 and e(ϕ) := 1 I + 1 f(ϕ). These arguments may λ∈L2/{±1} 16 64 P come from other domains, as in (4.2), and we allow mixing as in e(ϕλ), for a P character ϕ and lattice vector λ. We prove later that e(ϕ) is an idempotent.

Lemma 4.8. Let r, s L, a,b Z and let ∈ ∈ n(r,s,a,b) := 1 t Φ r, t 2a(mod 2), s,t 2b(mod 2) . 2 |{ ∈ |h i≡ h i≡ }| (i) Suppose that the images of r and s in L/2L are nonzero and distinct. The values of n(r,s,a,b) depend only on the isometry type of the images of the ordered pair r, s in L/2L and are listed below: h i

1 r, r 1 s,s 1 r, s 2n(rs00) 2n(rs01) 2n(rs10) 2n(rs11) 4 h i 4 h i 2 h i 0 0 0 48 64 64 64 0 0 1 56 56 56 72 0 1 0 64 48 64 64 0 1 1 56 56 72 56 1 0 0 64 64 48 64 1 0 1 56 72 56 56 1 1 0 64 64 64 48 1 1 1 72 56 56 56. (ii) If s =0 and r, r =4, then 2n(r, s, 0, 0) = 128 and 2n(r, s, 1, 0) = 112. If s =0 and r, r =8h , theni 2n(r, s, 0, 0) = 112 and 2n(r, s, 1, 0) = 128. h i Lemma 4.9. The f(ϕ), as ϕ ranges over all nonsingular characters of L of order C θ 2, form a basis for L2.

Proof. Use the action of the subgroup of the Weyl group stabilizing the maximal −5 8 1+6 totally singular subspace L/2Q of Q/2Q (its shape is 2 2 .2+ .GL(4, 2) ); it also stabilizes the maximal totally singular subspace 2Q/2L of L/2L (halve the + quadratic form on L, then reduce modulo 2). Since WE8 induces the group O (8, 2) on Q/2Q, Witt’s theorem implies that the stabilizer of a maximal isotropic subspace is transitive on the nonsingular vectors outside it. The action of this group on L/2L has the analogous property.

Notation 4.10. u(ϕ) := ϕ(λ)λ2 . This also makes sense for ϕ L by λ∈L2/{±1} ∈ the identiﬁcation in (4.2). P Proposition 4.11. Let α Hom(L, 1 ). (i) ∈ {± } 480 α = 1; u(α),ω = α(λ)λ2 = 32 αsingular; h i ( 32− αnonsingular. X 48 R. L. Griess, Jr.

(ii) 240I if α=1; u(α)= 16I if α is singular;  −208I + 48I′ = 256I + 48I = 16α2 + 48I if α is nonsingular;  − α α − α − θ (in the third case, α is taken to be a norm 4 lattice vector in L ; it is well defined up to its negative, and this suffices). Their respective norms are 57600, 256 and 1 2 7 2 8 8 208 + 8 48 = 7424= 2 29. Proof. We deal with cases, making use of inner product results (2.2) and (2.3); at once, we get (i). If u(α) were known to be a multiple of I, this inner product information would be enough to determine u(α). Thisissofor u(1) since the linear group (isomorphic to the Weyl group of E8 ) stabilizing L2 is irreducible and so fixes a subspace of dimension just 1 in the symmetric square of H. It follows that u(1) = 240I. ′ ′ 2 Notice that in all cases u(α)=2u (α) u(1), where u (α) := ′ λ − λ∈Φ /{±1} and Φ′ := λ Φ α(λ)=1 . { ∈ | ′ }′ ′ P Now to evaluate u := u (α) for α = 1. If Φ has type D8, we have an irreducible group as above and conclude6 that u′(α) = bI, where b = u′, I = 1 ′ 1 ′ h i 2 u ,ω = 2 56 4 = 112. If Φ has type A1E7, we have a reducible group with two h i · ′ constituents and conclude that u = cIα + dIα⊥ , where we interpret α as an element of L and moreover as a root in the A -component of Φ′. Since I , I = 1 and 2 1 h α αi 8 2 2 I I I ⊥ 1 7 ′ I α , α = 32, c = 16. Since = α + α , 8 c+ 8 d = u , = 128, whence d = h144. Thus,i 2u′ 240I = 32I +288I′ 240I = 208I h +48iI′ = 256I +48I. − α α − − α α − α Lemma 4.12. f(ϕ) f(ψ)= × 1+q(ϕψ) 1+hϕ,ψi ( 1) 4(f(ϕ)+ f(ψ)) + ( 1) 64vϕψ + u(ϕψ) − if ϕ = ψ; furthermore,− this equals  4(f(ϕ)+ f(ψ)) 6 16I   − − if ϕψ singular; and equals   4(f(ϕ)+ f(ψ))+48I 512e  α,hϕ,ψi  − if ϕψ nonsingular; 56f(ϕ)+ u(1)   if ϕ = ψ.   Proof. The left side is

ϕ(λ)ψ(µ)vλ+µ+u(ϕψ)= (for ν = λ + µ ) ψ(ν) (ϕψ)(λ)vν ν Xλ µ:hµ,λXi=−2 X λ:hXν,λi=2 = ψ(ν)(n(ν, ϕψ, 1, 0) n(ν, ϕψ, 1, 1))v + u(ϕψ). − ν ν X We use (4.8) and (4.9). The coefﬁcent of vν is 0 if ϕψ(ν) 1(mod 2). If ϕψ(ν) 0(mod 2), then ϕ(ν) = ψ(ν); the coefﬁcient is 56ψ≡(ν) if ϕψ = 1, ≡ ( 1)1+q(ϕψ)8 if ϕψ =1 or ν and, if ϕψ = ν, it is 56( 1)hϕ,ψi. − 6 − − + A VOA related to E8 with automorphism group O (10, 2) 49

Corollary 4.13. e(ϕ)2 = e(ϕ). The 256 f(ϕ) live in CLθ , a space of dimension 120, so they are linearly dependent. There is a natural subset which forms a basis.

Proposition 4.14. If ϕψ is singular, f(ϕ) f(ψ) = 4(f(ϕ)+ f(ψ)) 16I and e(ϕ) e(ψ)= 0. × − − × Proof. It sufﬁces to show that (4I + f(ϕ)) (4I + f(ψ))=0, or 16I + 4(f(ϕ)+ × f(ψ)) + 4( 1)1+q(ϕψ)(f(ϕ)+ f(ψ)) + u(ϕψ)=0. This follows from q(ϕψ)=0 and u(ϕψ)=− 16I; see (4.12.i). − 240 if ϕ = ψ; Lemma 4.15. (i) f(ϕ),f(ψ) = 16 if ϕψ singular; h i ( −16 if ϕψ nonsingular. 1 16 if ϕ = ψ; (ii) e(ϕ),e(ψ) = 0 if ϕψ singular; h i  1  128 if ϕψ nonsingular.

Proof. (i) This inner product is 2 λ ϕψ(λ), so consider the cases that ϕψ is 1, singular or nonsingular. One can also use (4.12) and associativity of the form. We leave (ii) as an exercise, with (i) andP (2.2). Theorem 4.16. The 2e(ϕ) are conformal vectors of conformal weight (=central 1 charge) 2 . Proof. By (4.10) and [Miy], Theorem 4.1, these are conformal vectors. Fix ϕ. Choose a maximal, totally singular subspace, J, of L modulo 2L. Let J be the set of distinct linear characters of L which contain J in their kernel. The e(ψ), for ψ ϕJ, are pairwise orthogonal idempotents (4.12) which sum to I (to prove this, use∈ the orthogonality relations for this set of 16 distinct characters). We use the fact 1 that conformal weight of 2e(ϕ) is at least 2 (see Proposition 6.1 of [Miy]). Since their conformal weights add to 8, the conformal weight of ω, we are done. Notation 4.17. In an integral lattice, an element of norm 2 is called a root and an element of norm 4 is called a quoot (suggested by the term “quartic" for degree 4).

± Notation 4.18. The idempotents eλ are called idempotents of quoot type or quooty idempotents and the e(ϕ) are called idempotents of tout type or tooty idempotents (suggested by “tout" or “tutti"). The set of all such is denoted QI, TI, respectively. Set QTI := QI TI. ∪

5. Eigenspaces.

Notation 5.1. For an element x of a ring, adx, ad(x) denotes the endomorphism: right multiplication by x. If the ring is a ﬁnite dimensional algebra over a ﬁeld, the spectrum of x means the spectrum of the endomorphism ad(x). The main result of this section is the following. 50 R. L. Griess, Jr.

Theorem 5.2. If e is one of the idempotents eλ, fλ or e(ϕ) of Section 4, its spectrum 1 1 35 120 is (1 , 4 , 0 ). We prove (5.2) in steps, treating the quooty and tooty cases separately.

——————————————————————————————————

1 2 Table 5.3. The action of ad(eλ) on a spanning set. Recall that eλ = 32 [λ +4vλ].

vector image underad(eλ) dim.

2 1 2 µ 32 [4 λ, µ λµ +4 λ, µ vλ]= 1h[ λ, µi λµ + hλ, µ i2v ] 36 8 h i h i λ 1 µν 32 [2 λ, µ λν +2 λ, ν λµ +4 λ, µ λ, ν vλ]= 1h[ λ, µi λν + hλ, νiλµ +2hλ, µihλ, νiv ] 36 16 h i h i h ih i λ 2 1 2 λ 32 [16λ + 64vλ]=16eλ 1

λh, λ, h =0 1 8λh = 1 λh 7 h i 32 4 gh, g, λ = h, λ =0 0 28 h i h i 1 2 vλ 4 32 [4λ + 16vλ]=4eλ 1

v , λ, µ =0 0 63 µ h i 1 1 vµ, λ, µ = 2 32 [4vλ+µ +4vµ]= 8 [vλ+µ + vµ] 56 —————————————————————————————————–h i −

Table 5.4. The eigenspaces of ad(eλ). eigenvalue basis element(s) dimension

1 eλ 1 1 4 λh, λ, h =0 7 1 v + vh, λ,i µ = 2 28 4 λ+µ µ h i − 0 vλ+µ vµ, λ, µ = 2 28 0 gh, −g, λ =h h, λi =0− 28 0 hv , iλ, µh =0i 63 µ h i 0 fλ 1 —————————————————————————————————– + A VOA related to E8 with automorphism group O (10, 2) 51

Table 5.5. The action of ad(f ) on a spanning set. Recall that f = 1 [λ2 4v ]= λ λ 32 − λ e + 1 λ2 , so the table below may be deduced from Table (5.3) and (3.1.1). − λ 16

vector image underad(fλ) dimension

2 1 1 2 µ 4 λ, µ λµ 8 [ λ, µ λµ + λ, µ vλ]= h i 1 [ −λ, µ hλµ i λ, µ h2v ]i 36 8 h i − h i λ 1 µν 8 [ λ, µ λν + λ, ν λµ] 1 h i h i 16 [ λ, µ λν + λ, ν λµ +2 λ, µ λ, ν vλ]= − 1 [h λ, µi λν +h λ, νi λµ 2h λ, µih λ, νi v ] 36 16 h i h i − h ih i λ λ2 λ2 16e = 16f 1 − λ λ λh, λ, h =0 1 8λh = 1 λh 7 h i 32 4 gh, g, λ = h, λ 0 28 h =0i h i

v v 4e = 4f 1 λ λ − λ − λ v , λ, µ =0, 1 0 63 µ h i ± v , λ, µ = 2 1 v 1 [v + v ]= 1 [ v + v ] 56 µ h i − 4 µ − 8 λ+µ µ 8 − λ+µ µ —————————————————————————————————–

Table 5.6. The eigenspaces of ad(fλ). eigenvalue basis element(s) dimension

1 fλ 1 1 λh, λ, h =0 7 4 h i 0 vλ+µ + vµ, λ, µ = 2 28 1 h i − 4 vλ+µ vµ, λ, µ = 2 28 0 gh, −g, λ =h h, λi =0− 28 0 hv , iλ, µh =0i 63 µ h i 0 eλ 1 —————————————————————————————————– 52 R. L. Griess, Jr.

Table 5.7. The action of ad(f(ϕ)) on a spanning set.

vector image underad(f(ϕ)) dim.

f(ϕ) 56f(ϕ)+ u(1) 1

f(ψ), ψϕ singular 4(f(ϕ)+ f(ψ)) 16I 120 − − f(ψ), ψϕ nonsingular 4(f(ϕ)+ f(ψ))+48I 512e 120 − α,hϕ,ψi u(α), α nonsingular ϕ(µ) α(λ) λ, µ 2v = µ λ h i µ ϕ(µ)[48 16 µ, α 2]v 36 P µ P − h i µ I P f(ϕ) 1 —————————————————————————————————– Proofs (5.7.1). Proofs of the above are straightforward. We give a proof only of the formula for ξ := f(ϕ) u(α). Clearly, ξ is a linear combination of the vλ, so we just get its coefﬁcent× at v as 1 ξ, v . By associativity of the form, this λ 2 h λi is 1 u(α),f(ϕ) v = 1 u(α), ϕ(λ)λ2 . By (4.12.ii), we have an expression 2 h × λi 2 h i for u(α). Since I, λ2 = 2 and I , λ2 = 1 λ, α 2 α, α −1 = 1 λ, α 2 , the h i h α i 2 h i h i 8 h i respective cases of (4.12.ii) lead to 1 u(α), ϕ(λ)λ2 = ϕ(λ)240, ϕ(λ)16 and 2 h i − ϕ(λ)[48 16 λ, α 2]. Only the latter case is recorded in the table since u(α) is otherwise− a multipleh i of I. —————————————————————————————————– + A VOA related to E8 with automorphism group O (10, 2) 53

1 I Table 5.8. The action of ad(e(ϕ)) on a spanning set. Recall that e(ϕ) = 16 + 1 64 f(ϕ). We use the notation α := ϕψ, when ϕψ is nonsingular. Note that the set of such α2 span S2(H).

vector image underad(e(ϕ)) dim.

15 15 I f(ϕ) 16 f(ϕ)+ 4 1

f(ψ), ϕψ singular 1 f(ψ) 1 I 120 − 16 − 4 f(ψ), ϕψ nonsingular 4e(ϕ)+8e(ψ) 8e 120 − α,hϕ,ψi α2 if α := ϕψ nonsingular 2[e(ϕ) e(ψ)]+2e 36 − α,ϕ(α) u(α), α nonsingular 1 u(α)+ ϕ(µ)[ 3 1 µ, α 2]v 36 16 µ 4 − 4 h i µ I P e(ϕ) 1

e(ϕ) e(ϕ) 1

e(ψ) if ϕψ singular 0 120

e(ψ) if α := ϕψ nonsingular 2−3[e(ϕ)+ e(ψ) e ] 120 − α,hϕ,ψi v = 4(e+ e−) ϕ(α) 1 [e + e(ϕ) e(ψ)] 120 α α − α 2 α,hϕ,ψi − 1 eα,ϕ 8 [eα,hϕ,ψi + e(ϕ) e(ψ)] 35 ′ − eα,ϕ 0 120 —————————————————————————————————–

Table 5.9. Eigenspaces of ad(e(ϕ)). In the table, we use the convention that α := ϕψ is nonsingular. Recall that e± = 1 (λ2 4e ). Recall that v = 4(e+ + e−). λ 32 ± λ α α α eigenvalue basis elements dimension

1 e(ϕ) 1 ′ 0 eα,ϕ 120 1 4 eα,ϕ + e(ψ) 35 —————————————————————————————————–− 54 R. L. Griess, Jr.

± Table 5.10. Action of Idempotents on Idempotents. Recall the deﬁnitions eλ = 1 2 1 I 1 32 [λ 4vλ], eλ,ϕ = eλ,hϕ,λi, e(ϕ)= 16 + 64 f(ϕ). In expressions below, a and b are integers± modulo 2.

0 if λ, µ =0 −10 a h i 2 [ 8λµ + 16(( 1) vλ+ b− −a+b  ( 1) vµ + 16( 1) vλ+µ]=  −−10 −2 a+b  2 [ 4(λ + µ) ( 1) 4vλ+µ  −2 a− −  +4((λ + 4( 1) vλ)+  2 −b eλ,a eµ,b =  4(µ + 4( 1) vµ)] = ×  −3 −  2 [eλ+µ,a+b+1 + eλ,a + eµ,b] if λ, µ = 2 e if h( λ, µi , ( −1)a+b) λ,a h i −  = (4,0), (-4,1)   0 if ( λ, µ , ( 1)ab)=  h i −  (4,1), (-4,0)    e(ϕ) if ϕ = ψ e(ϕ) e(ψ)= 0 if ϕψ singular  × 2−3[e(ϕ)+ e(ψ) e ] if ϕψ nonsingular  − ϕψ,ϕ  0 [λ, ψ]= a +1 eλ,a e(ψ)= −3 × 2 [e(ψλ) e(ψ) eλ,ψ] [λ, ψ]= a. ——————————————————————————– − − ———————

Table 5.11. Inner Products of Idempotents

See the basic inner products in Section 2. We also need (f(ϕ),f(ψ)) from (4.15).

2−4 λ = µ −9 2 −5 a+b (eλ,a,eµ,b)=2 λ, µ +2 ( 1) δλ,µ = 0 λµ singular . h i −  −7  2 λµ nonsingular  (eλ,a,e(ϕ)) = 2−7 if ( 1)aϕ(α)=1, i.e., a + [ϕ, α]=0 2−8 +2−8( 1)aϕ(α)= − . − 0 if ( 1)aϕ(α)= 1 i.e., a + [ϕ, α]=1 − −

240 ϕ = ψ 2−4 (e(ϕ),e(ψ))=2−8 +2−12 16 ϕψ singular = 0 . − ( 16 ϕψ nonsingular ( 2−7 + A VOA related to E8 with automorphism group O (10, 2) 55

6. Idempotents and Involutions.

32 2 32 Notation 6.1. The polynomial p(t) := 3 t 3 +1 takes values p(0) = p(1) = 1 1 − and p( 4 )= 1. For an idempotent e such that ad(e) is semisimple with eigenvalues 1 − 0, 4 and 1, we deﬁne t(e) := p(ad(e)), an involution which is 1 on the 0 and 1 − 1-eigenspaces and is 1 on the 4 -eigenspace. Let E± = E±(e)= E±(t(e)) denote the 1 eigenspace of− this involution. ± The main results of this section are the following.

Theorem 6.2. For a quooty or tooty idempotent, e, t(e) is an automorphism of V F . This followsfromthe theoryin [Miy] and (5.2). Inthis section, we shall verify this F ± directly on the algebra V2 only, for the eλ and e(ϕ) and prove that these elements 1 are all the idempotents whose doubles are conformal vectors of conformal weight 2 . See (6.5) and (6.6).

Theorem 6.3. The subgroup of Aut(V F ) generated by all t(e) as in (6.2) is isomorphic to O+(10, 2). The Miyamoto theory proves that the t(e) are in Aut(V F ). It turns out that the F group they generate restricts faithfully to V2 is faithful, and there we can identify it. ± Theorem 6.4. The group generated by the t(eλ ) is isomorphic to the maximal 2-local subgroup of O+(10, 2) of shape 28: O+(8, 2). The normal subgroup of order 28 is + − generated by all t(eλ )t(eλ ) and acts regularly on the set of weighty idempotents. A complement to this normal subgroup is the stabilizer of any e(ϕ), for example, the stabilizer of e(1) (1 means the trivial character) is generated by all t(eλ,1). Sucha complement is isomorphic to the Weyl group of type E8, modulo its center.

F To verify that the involution t(e) is an automorphism of V2 , it sufﬁces to check that E E + E E E and E E E . + + − − ≤ + + − ≤ − F Proposition 6.5. If e is quooty, t(e) is an automorphism of V2 . Proof. This is straightforward with (6.4) and (5.4).

F Proposition 6.6. If e is tooty, t(e) is an automorphism of V2 . Proof. This is harder. We use (5.1), (6.4), (5.8) and (5.11). It is easy to verify that E+ E+ E+. To prove E− E− E+, we verify that (E− E−, E−)=0 (this× sufﬁces≤ since the eigenspaces× are≤ nonsingular and pairwise orthogo× nal); the veriﬁcation is a straightforward checking of cases. To prove that E− E+ E−, we use the previous result, commutativity of the product and associativity× of the≤ form.

Table 6.7. (i) The action of t(eλ,a) on QTI: ﬁxed are eµ,b if µ, λ =0 or 4; e(ϕ) if [λ, ϕ]=0; interchanged are h i ± 56 R. L. Griess, Jr.

e and e if λ, µ = 2; e(ϕ) and e(ϕλ) if [λ, ϕ]=1. µ,b λ+µ,a+b+1 h i − (ii) The action of t(e(ϕ)) on QTI: fixed are ′ all eλ,ϕ and all e(ψ) with ϕ = ψ or ϕψ singular; interchanged are all e(ϕλ) and eλ,ϕ with λ a quoot. Proof. For an involution t to interchange vectors x and y in characteristic not 2, it is necessary and sufficient that t fix x+y and negate x y. The following is a useful observation: since the +1 eigenspace for t = t(e) is− the sum of the 0-eigenspace 1 and the 4 -eigenspace for ad(e), a vector u is fixed by t(e) iff e u is in the 1 × 2 4 -eigenspace. Another useful observation is that if x y is negated, then (x y) is fixed. The proof of (i) and (ii) is an exercise in checking− cases. − The identification of G, the group generated by all such t(e), e QTI, is based ∈ F10 on a suitable identification of this set of involutions with nonsingular points in 2 with a maximal index nonsingular quadratic form. F10 Notation 6.8. Let T := 2 have a quadratic form q of maximal Witt index. Decom- pose T = U W , with dim(U)=2, dim(W )=8, both of plus type. Let U = 0,r,s,f , where⊥ q(f)=1, q(r) = q(s)=0. Identify W with Hom(L, 1 ). For{ x V} nonsingular, write x = p + y, for p U,y W . If p =0, correspond{± } ∈ ∈ ∈ x to ey,1. If p = r, correspond x to ey,0. If p s,f , correspond e(y) to x. This correspondence is G-equivariant; use (6.7). ∈ { }

+ F So, we have a map of G onto O (10, 2) by restriction to V2 . Its kernel fixes F all of our idempotents, which span V2 . By Corollary 6.2 of [DGH], this kernel is + trivial. So, G ∼= O (10, 2) and (6.3) is proven. Proposition 6.9. G acts irreducibly on I⊥ (dimension 155). Proof. This follows from the character table of Ω+(10, 2), but we can give an elementary proof. (1) The subgroup H of (6.2) has an irreducible constituent P of dimension 120 with monomial basis v , α L ; α ∈ 2 (2) the squares of the vα generate the 36-dimensional orthogonal complement, P ⊥. The action fixes I and the action on the 35-dimensional space P ⊥ I⊥ is 8 ∩ nontrivial, hence irreducible (the subgroup O2(H) = 2 acts trivially and the quotient + ∼ 2 2 H/O2(H) = O (8, 2) acts transitively on the spanning set of 120 elements vα = α , ∼ 6 + 6 so acts faithfully. Now, the subgroup 2 : O (6, 2) ∼= 2 : Sym8 has smallest faithful irreducible degrees 28 and 35; if H is reducible on P ⊥ I⊥, then 28 occurs and H has an irreducible R of dimension d, 28 d 34 and∩ so P ⊥ R⊥ is a trivial module of dimension 36 d 2. This is impossible≤ ≤ since P ⊥ is an∩ H -constituent of a transitive permutation− module≥ of degree 120, contradiction). F (3) We now have V2 = 1+35+120 as a decomposition into H -irreducibles. But each t(e(ϕ)) fixes I and does not fix the 120-dimensional constitutent, whence irreduciblity of G on I⊥. + A VOA related to E8 with automorphism group O (10, 2) 57

F + Theorem 6.10. Aut(V )= G ∼= O (10, 2). Proof. Set A := Aut(V F ). We quote Theorem (6.13) of [Miy], which says that if X 1 is the set of conformal vectors of central charge 2 , then t(x)t(y) 1, 2, 3 . So, if X is a conjugacy class, it is a set of 3-transpositions. If it| is not a con|∈{jugacy class,} we have a nontrivial central product decomposition of X , which is clearly impossible h Fi since A acts faithfully and G acts irreducibly on V2 . Now, the classiﬁcation of groups generated by a a class of 3-transpostions [Fi69][Fi71] may be invoked to identify A. It is a fairly straightforward exercise to eliminate any 3-transposition group which properly contains O+(10, 2).

7. A related subVOA of V ♮. The VOA defined in [FLM], denoted V ♮, has the monster as its automorphism group. One of the parabolics, P = 210+16Ω+(10, 2), acts on the subVOA V ′ of fixed ∼ ′ F ′ points of O2(P ); the degree 2 part V2 contains V2 . In fact, V2 is isomorphic F C ′′ to the direct sum of algebras V2 (with ) and . The proper subVOA V of ′ F × F V generated by the V2 -part is isomorphic to V (this is so because we can see our L = Q[E] embedded in the Leech lattice, as the fixed point sublattice of an involution). This subVOA V ′′ contains idempotents given by formulas like ours for 1 ♮ quooty and tooty ones, but these idempotents have 16 in their spectrum on V , so the involutions associated to them by the Miyamoto theory act trivially on V ′. The involutory automorphisms of V ′ given by our forumlas in Section 6 do not extend to automorphisms of V ♮ since otherwise the stabilizer of this subVOA inIM, the monster, would induce O+(10, 2) on it, contrary to the above structure of the maximal 2-local P ; we mention that the maximal 2-locals have been classifed [Mei].

Acknowledgements. We thank Chongying Dong, Gerald Hohn,¨ Geoffrey Mason, Masahiko Miyamoto and Steve Smith for discussions on this topic. This article was written with ﬁnancial sup- port from NSF grant DMS-9623038 and University of Michigan Faculty Recognition Grant (1993-96).

References

[Carter] Roger Carter, Simple Groups of Lie Type, John Wiley, London, 1989. [CoGr] Arjeh Cohen and Robert L. Griess, Jr., On ﬁnite simple subgroups of the complex Lie group of type E8, Proc. Comm. Pure Math. 47(1987), 367-405. [DGH] Chongying Dong, Gerald Hohn,¨ Robert L. Griess, Jr., Framed Vertex Operator Algebras and the Moonshine Module, submitted. [FLM] Igor Frenkel, James Lepowsky, Arne Meurman, Vertex Operator Algebras and the Monster, Academic Press, San Diego, 1988. 58 R. L. Griess, Jr.

[Fi69] Fischer, Bernd, Finite Groups Generated by 3-Transpositions. Univ. of Warwick. Preprint 1969. [Fi71] Fischer, Bernd, Finite Groups generated by 3-Transpositions. Inventions Math. 13 (1971) 232–246. 15 [Gr76] Robert L. Griess, Jr., A subgroup of order 2 |GL(5, 2)| in E8(C), the Dempwolff group and Aut(D8 ◦ D8 ◦ D8), J. of Algebra 40, 271-279 (1976). [Gr91] Robert L. Griess, Jr., Elementary abelian p-subgroups of algebraic groups, Geometriae Dedicata 39; 253-305, 1991. [Mei] Ulrich Meierfrankenfeld, The maximal 2-locals of the monster, preprint. [Miy] Masahiko Miyamoto, Griess algebras and conformal vectors in vertex operator algebras, to appear in Journal of Algebra.

Department of Mathematics University of Michigan Ann Arbor, Michigan 48109•1109 USA email: [email protected]