A vertex operator algebra related to E 8 with automorphism group O + (0,2) Robert L. Griess, Jr. Abstract. We study a particular VOA which is a subvoa of the E 8 -lattice VOA and determine its automorphism group. Some of this group may be seen within the group E 8 (C), but not all of it. The automorphism group turns out to be the 3-transposition group O + (0, 2) of order 2 2 3 5 5 2 7.7.3 and it contains the simple group Ω + (0, 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 in MI, the monster, of the form 2 0+6.Ω + (0, 2). Hypotheses We review some definitions, based on the usual definitions for the elements, products and inner products for lattice VOAs; see [FLM]. Notation.2. Φ is a root system whose components have types ADE, g is a Lie algebra with root system Φ, Q := Q Φ, the root lattice and V := V Q := S(Ĥ ) C[Q] is the lattice VOA in the usual notation. Remark.3. We display a few graded pieces of V ( is omitted, and here Q can be any even lattice). We write H m for H t m in the usual notation for lattice VOAs (2.) and Q m := {x Q (x, x) = 2m}, the set of lattice vectors of type m. V 0 = C, V = H, V 2 = [S 2 H + H 2 ] + H CQ + CQ 2, V 3 = [S 3 H + H H 2 + H 3 ] + [S 2 H + H 2 ]CQ + H CQ 2 + CQ 3, V 4 = [S 4 H + S 2 H H 2 + H H 3 + S 2 H 2 + H 4 ]+ [S 3 H + H H 2 + H 3 ]CQ + [S 2 H + H 2 ]CQ 2 + H CQ 3 + CQ 4. Remark.4. Let F be a subgroup of Aut(g), where g is the Lie algebra V = H + CQ with 0 th binary composition. The fixed points V F 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.5. For the rest of this article, we take Q to be the E 8 -lattice. Take F to be a 2B-pure elementary abelian 2-group of rank 5 in Aut(g) = E 8 (C); 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 E; we arrange for θ to interchange the standard Chevalley generators x α and x α. See [Gr9]. The Chevalley generator x α corresponds to the standard generator e α of the lattice VOA V Q. Notation.7. L := Q [E] = 2Q denotes the common kernel of the lattice characters associated to the elements of E; in the [Carter] notation, these characters are h (E); in the root lattice modulo 2, they correspond to the sixteen vectors in a maximal totally singular subspace. Then (.7.) V F = 0 and (.7.2) V F 2 = S2 H + 0 + CL θ 2, where the latter summand stands for the span of all e λ + e λ, where λ runs over all the 5 6 = 240 norm 4 lattice vectors in L. Thus, V2 F has dimension ( ) 9 2 + 240 2 = 36 + 20 = 56 and has a commutative algebra structure invariant under N(F) = 2 5+0 GL(5, 2). We note that N(F)/F = 2 0 : GL(5, 2) [Gr76][CoGr][Gr9]. We will show (6.0) that Aut(V F ) = O + (0, 2). 2. Inner Product. Definition 2.. The inner product on S n H m is x n, x n = n!m n x, x n. This is based on the adjointness requirement for h t k and h t k (see (.8.5), FLM,p.29). When k > 0, h t k acts like multiplication by h t k and, when h is a root, h t k acts like k times differentiation with respect to h. When n = 2, this means x 2, x 2 = 2m 2 x, x. In V F 2, m =. Definition 2.2. The Symmetric Bilinear Form. Source: [FLM], p.27. This form is associative with respect to the product (Section 3). We write H for H. The set of all g 2 and x + α spans V 2. (2.2.) g 2, h 2 = 2 g, h 2, whence (2.2.2) pq, rs = p, r q, s + p, s q, r, for p, q, r, s H. (2.2.3) x + α, x+ β = { 2 α = ±β 0 else (2.2.4) g 2, x + β = 0.
A VOA related to E 8 with automorphism group O + (0,2) 45 Notation 2.3. In addition, we have the distinguished Virasoro element ω and identity I := 2 ω on V 2 (see Section 3). If h i is a basis for H and h i the dual basis, then ω = 2 i h ih i. Remark 2.4. (2.4.) g 2, ω = g, g (2.4.2) g 2, I = g, g 2 (2.4.3) I, I = dim(h)/8 (2.4.4) ω, ω = dim(h)/2 If {x i i =,... l} is an ON basis, (2.4.5) I = 4 (2.4.6) ω = 2 l i=0 x 2 i l x 2 i. i=0 3. The Product on V F 2. Definition 3.. The product on V2 F comes from the vertex operations. We give it on standard basis vectors, namely xy S 2 H, for x, y H and v λ := e λ + e λ, for λ L 2. Note that (3..) give the Jordan algebra structure on S 2 H, identified with the space of symmetric 8 8 matrices, and with x, y = 8tr(xy). The function ε below is a standard part of notation for lattice VOAs. (3..) x 2 y 2 = 4 x, y xy, pq y 2 = 2 p, y qy + 2 q, y py, pq rs = p, r qs + p, s qr + q, r ps + q, s pr; (3..2) x 2 v λ = x, λ 2 v λ, xy v λ = x, λ y, λ v λ 0 λ, µ {0, ±, ±3}; (3..3) v λ v µ = ε λ, µ v λ+µ λ, µ = 2; λ 2 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 v λ. Notation 4.. For a subset M of H, there is a unique element ω M of S 2 H which satisfies () ω M S 2 (span(m)); (2) for all x, y span(m), x, y = ω M, xy. We define I M := 2 ω M. If M and N are orthogonal sets, we have ω M N = ω M + ω N. Define ω M := ω ω M and I M := I I M. This element can be written as ω M = 2 i x2 i, where the x i form an orthonormal basis of span(m). We have ω M, ω M = 2 dim span(m) and I M, I M = 8dim span(m). Also, ω M, xy = ω M, x y = ω, x y, where priming denotes orthogonal projection to span(m). Notation 4.2. e ± λ := f λ := 32 [λ2 ± 4v λ ], e λ = e + λ, f λ := e λ. If a Z or Z 2, define e λ,a to be e + λ or e λ, as a 0, (mod 2), respectively; see (4.7). Also, let e λ,a = e λ,a+. We define e λ,µ to be e λ,a, where a is 2 λ, µ in case µ is a vector in L, and a is [ˆλ, µ], where [.,.] is the nonsingular bilinear form on Hom(L, {±}) gotten from 2.,. by thinking of Hom(L, {±}) as 2 L/L and where ˆλ is the character gotten by reducing the inner product with 2λ modulo 2. Finally. let q be the quadratic form on Hom(L, {±}) gotten by reducing x x, x modulo 2, for x 2 L. Lemma 4.3. (i) The e ± λ are idempotents. (ii) e ± λ, e± µ = 6 λ = µ; 28 λ, µ = 2; 0 λ, µ = 0. 0 λ = µ; (iii) e ± λ, e µ = 28 λ, µ = 2; 0 λ, µ = 0. Proof. (i) (e ± λ )2 = 024 [4 4λ2 +6λ 2 ±8 4 2 v λ ] = e ± λ. (ii) and (iii) follow trivially from (2.2). Notation 4.4. For finite X L, define s(x) := x ±X/{±} x2. Lemma 4.5. If X L 2, ω, s(x) = 4 (±X)/{±} and so s(l 2 ) = L2 2 ω = 20ω. Proof. (2.2.5) Corollary 4.6. (i) For α L 2, ω α, s(α) = ω, s(α) = 4 and ω α, ω α = 2, whence s(α) = 8ω α = 6I α and I α = 6 α2. (ii) ω E7, s(φ E7 ) = ω, s(φ E7 ) = 63, whence s(φ E7 ) = 8ω ΦE7 ; (iii) ω, s(φ D8 ) = 56, whence s(φ D8 ) = 56ω ΦE7.
A VOA related to E 8 with automorphism group O + (0,2) 47 Notation 4.7. For ϕ Hom(L, {±}), define f(ϕ) := λ L ϕ(λ)v 2/{±} λ, u(ϕ) := λ L 2/{±} ϕ(λ)λ2 and e(ϕ) := 6 I + 64f(ϕ). These arguments may come from other domains, as in (4.2), and we allow mixing as in e(ϕλ), for a 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) := 2 {t Φ r, t 2a(mod 2), s, t 2b(mod 2)}. (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: 4 r, r 4 s, s 2 r, s 2n(rs00) 2n(rs0) 2n(rs0) 2n(rs) 0 0 0 48 64 64 64 0 0 56 56 56 72 0 0 64 48 64 64 0 56 56 72 56 0 0 64 64 48 64 0 56 72 56 56 0 64 64 64 48 72 56 56 56. (ii) If s = 0 and r, r = 4, then 2n(r, s, 0, 0) = 28 and 2n(r, s,, 0) = 2. If s = 0 and r, r = 8, then 2n(r, s, 0, 0) = 2 and 2n(r, s,, 0) = 28. Lemma 4.9. The f(ϕ), as ϕ ranges over all nonsingular characters of L of order 2, form a basis for CL θ 2. Proof. Use the action of the subgroup of the Weyl group stabilizing the maximal totally singular subspace L/2Q of Q/2Q (its shape is 2 5 2 8.2 +6 +.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 W E8 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 4.0. u(ϕ) := λ L 2/{±} ϕ(λ)λ2. This also makes sense for ϕ L by the identification in (4.2). Proposition 4.. Let α Hom(L, {±}). (i) u(α), ω = { 480 α = ; α(λ)λ 2 = 32 αsingular; 32 αnonsingular.
48 R. L. Griess, Jr. (ii) 240I u(α) = 6I 208I α + 48I α = 256I α + 48I = 6α 2 + 48I if α=; if α is singular; if α is nonsingular; (in the third case, α is taken to be a norm 4 lattice vector in L θ ; it is well defined up to its negative, and this suffices). Their respective norms are 57600, 256 and 8 2082 + 7 8 482 = 7424 = 2 8 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). This is so for u() since the linear group (isomorphic to the Weyl group of E 8 ) stabilizing L 2 is irreducible and so fixes a subspace of dimension just in the symmetric square of H. It follows that u() = 240I. Notice that in all cases u(α) = 2u (α) u(), where u (α) := λ Φ /{±} λ2 and Φ := {λ Φ α(λ) = }. Now to evaluate u := u (α) for α. If Φ has type D 8, we have an irreducible group as above and conclude that u (α) = bi, where b = u, I = 2 u, ω = 2 56 4 = 2. If Φ has type A E 7, we have a reducible group with two constituents and conclude that u = ci α + di α, where we interpret α as an element of L 2 and moreover as a root in the A -component of Φ. Since I α, I α = 8 and α 2, α 2 = 32, c = 6. Since I = I α +I α, 8 c+ 7 8 d = u, I = 28, whence d = 44. Thus, 2u 240I = 32I α +288I α 240I = 208I α +48I α = 256I α +48I. Lemma 4.2. f(ϕ) f(ψ) = ( ) +q(ϕψ) 4(f(ϕ) + f(ψ)) + ( ) + ϕ,ψ 64v ϕψ + u(ϕψ) if ϕ ψ; furthermore, this equals 4(f(ϕ) + f(ψ)) 6I if ϕψ singular; and equals 4(f(ϕ) + f(ψ)) + 48I 52e α, ϕ,ψ if ϕψ nonsingular; 56f(ϕ) + u() if ϕ = ψ. Proof. The left side is ϕ(λ)ψ(µ)v λ+µ +u(ϕψ) = (for ν = λ + µ ) ψ(ν) (ϕψ)(λ)v ν λ µ: µ,λ = 2 ν λ: ν,λ =2 = ν ψ(ν)(n(ν, ϕψ,, 0) n(ν, ϕψ,, ))v ν + u(ϕψ). We use (4.8) and (4.9). The coefficent of v ν is 0 if ϕψ(ν) (mod 2). If ϕψ(ν) 0(mod 2), then ϕ(ν) = ψ(ν); the coefficient is 56ψ(ν) if ϕψ =, ( ) +q(ϕψ) 8 if ϕψ or ν and, if ϕψ = ν, it is 56( ) ϕ,ψ.
A VOA related to E 8 with automorphism group O + (0,2) 49 Corollary 4.3. e(ϕ) 2 = e(ϕ). The 256 f(ϕ) live in CL θ, a space of dimension 20, so they are linearly dependent. There is a natural subset which forms a basis. Proposition 4.4. If ϕψ is singular, f(ϕ) f(ψ) = 4(f(ϕ) + f(ψ)) 6I and e(ϕ) e(ψ) = 0. Proof. It suffices to show that (4I + f(ϕ)) (4I + f(ψ)) = 0, or 6I + 4(f(ϕ) + f(ψ)) + 4( ) +q(ϕψ) (f(ϕ) + f(ψ)) + u(ϕψ) = 0. This follows from q(ϕψ) = 0 and u(ϕψ) = 6I; see (4.2.i). { 240 if ϕ = ψ; Lemma 4.5. (i) f(ϕ), f(ψ) = 6 if ϕψ singular; (ii) e(ϕ), e(ψ) = 6 if ϕ = ψ; 6 if ϕψ nonsingular. 0 if ϕψ singular; 28 if ϕψ nonsingular. Proof. (i) This inner product is 2 λ ϕψ(λ), so consider the cases that ϕψ is, singular or nonsingular. One can also use (4.2) and associativity of the form. We leave (ii) as an exercise, with (i) and (2.2). Theorem 4.6. The 2e(ϕ) are conformal vectors of conformal weight (=central charge) 2. Proof. By (4.0) and [Miy], Theorem 4., 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 (4.2) which sum to I (to prove this, use the orthogonality relations for this set of 6 distinct characters). We use the fact that conformal weight of 2e(ϕ) is at least 2 (see Proposition 6. of [Miy]). Since their conformal weights add to 8, the conformal weight of ω, we are done. Notation 4.7. 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 4.8. The idempotents e ± λ are calledidempotents of quoot type orquooty 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.. For an element x of a ring, ad x, 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 e λ, f λ or e(ϕ) of Section 4, its spectrum is (, 435, 0 20 ). We prove (5.2) in steps, treating the quooty and tooty cases separately. Table 5.3. The action of ad(e λ ) on a spanning set. Recall that e λ = 32 [λ2 + 4v λ ]. vector image underad(e λ ) dim. µ 2 32 [4 λ, µ λµ + 4 λ, µ 2 v λ ] = 8 [ λ, µ λµ + λ, µ 2 v λ ] 36 µν 32 [2 λ, µ λν + 2 λ, ν λµ + 4 λ, µ λ, ν v λ] = 6 [ λ, µ λν + λ, ν λµ + 2 λ, µ λ, ν v λ] 36 λ 2 32 [6λ2 + 64v λ ] = 6e λ λh, λ, h = 0 32 8λh = 4 λh 7 gh, g, λ = h, λ = 0 0 28 v λ 4 32 [4λ2 + 6v λ ] = 4e λ v µ, λ, µ = 0 0 63 v µ, λ, µ = 2 32 [4v λ+µ + 4v µ ] = 8 [v λ+µ + v µ ] 56 Table 5.4. The eigenspaces of ad(e λ ). eigenvalue basis element(s) dimension e λ 4 λh, λ, h = 0 7 4 v λ+µ + v µ, λ, µ = 2 28 0 v λ+µ v µ, λ, µ = 2 28 0 gh, g, λ = h, λ = 0 28 0 v µ, λ, µ = 0 63 0 f λ
A VOA related to E 8 with automorphism group O + (0,2) 5 Table 5.5. The action of ad(f λ ) on a spanning set. Recall that f λ = 32 [λ2 4v λ ] = e λ + 6 λ2, so the table below may be deduced from Table (5.3) and (3..). vector image underad(f λ ) dimension µ 2 4 λ, µ λµ 8 [ λ, µ λµ + λ, µ 2 v λ ] = 8 [ λ, µ λµ λ, µ 2 v λ ] 36 µν 8 [ λ, µ λν + λ, ν λµ] 6 [ λ, µ λν + λ, ν λµ + 2 λ, µ λ, ν v λ] = 6 [ λ, µ λν + λ, ν λµ 2 λ, µ λ, ν v λ] 36 λ 2 λ 2 6e λ = 6f λ λh, λ, h = 0 32 8λh = 4 λh 7 gh, g, λ = h, λ 0 28 = 0 v λ v λ 4e λ = 4f λ v µ, λ, µ = 0, ± 0 63 v µ, λ, µ = 2 4 v µ 8 [v λ+µ + v µ ] = 8 [ v λ+µ + v µ ] 56 Table 5.6. The eigenspaces of ad(f λ ). eigenvalue basis element(s) dimension f λ 4 λh, λ, h = 0 7 0 v λ+µ + v µ, λ, µ = 2 28 4 v λ+µ v µ, λ, µ = 2 28 0 gh, g, λ = h, λ = 0 28 0 v µ, λ, µ = 0 63 0 e λ
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() f(ψ), ψϕ singular 4(f(ϕ) + f(ψ)) 6I 20 f(ψ), ψϕ nonsingular 4(f(ϕ) + f(ψ)) + 48I 52e α, ϕ,ψ 20 u(α), α nonsingular µ ϕ(µ) λ α(λ) λ, µ 2 v µ = µ ϕ(µ)[48 6 µ, α 2 ]v µ 36 I f(ϕ) Proofs (5.7.). 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 v λ, so we just get its coefficent at v λ as 2 ξ, v λ. By associativity of the form, this is 2 u(α), f(ϕ) v λ = 2 u(α), ϕ(λ)λ2. By (4.2.ii), we have an expression for u(α). Since I, λ 2 = 2 and I α, λ 2 = 2 λ, α 2 α, α = 8 λ, α 2, the respective cases of (4.2.ii) lead to 2 u(α), ϕ(λ)λ2 = ϕ(λ)240, ϕ(λ)6 and ϕ(λ)[48 6 λ, α 2 ]. Only the latter case is recorded in the table since u(α) is otherwise a multiple of I.
A VOA related to E 8 with automorphism group O + (0,2) 53 Table 5.8. The action of ad(e(ϕ)) on a spanning set. Recall that e(ϕ) = 6 I + 64f(ϕ). We use the notation α := ϕψ, when ϕψ is nonsingular. Note that the set of such α 2 span S 2 (H). vector image underad(e(ϕ)) dim. f(ϕ) 5 6 f(ϕ) + 5 4 I f(ψ), ϕψ singular 6 f(ψ) 4 I 20 f(ψ), ϕψ nonsingular 4e(ϕ) + 8e(ψ) 8e α, ϕ,ψ 20 α 2 if α := ϕψ nonsingular 2[e(ϕ) e(ψ)] + 2e α,ϕ(α) 36 u(α), α nonsingular 6 u(α) + µ ϕ(µ)[3 4 4 µ, α 2 ]v µ 36 I e(ϕ) e(ϕ) e(ϕ) e(ψ) if ϕψ singular 0 20 e(ψ) if α := ϕψ nonsingular 2 3 [e(ϕ) + e(ψ) e α, ϕ,ψ ] 20 v α = 4(e + α e α ) ϕ(α) 2 [e α, ϕ,ψ + e(ϕ) e(ψ)] 20 e α,ϕ 8 [e α, ϕ,ψ + e(ϕ) e(ψ)] 35 e α,ϕ 0 20 Table 5.9. Eigenspaces of ad(e(ϕ)). In the table, we use the convention that α := ϕψ is nonsingular. Recall that e ± λ = 32 (λ2 ± 4e λ ). Recall that v α = 4(e + α + e α). eigenvalue basis elements dimension e(ϕ) 0 e α,ϕ 20 4 e α,ϕ + e(ψ) 35
54 R. L. Griess, Jr. Table 5.0. Action of Idempotents on Idempotents. Recall the definitions e ± λ = 32 [λ2 ± 4v λ ], e λ,ϕ = e λ, ϕ,λ, e(ϕ) = 6 I + 64f(ϕ). In expressions below, a and b are integers modulo 2. e λ,a e µ,b = 0 if λ, µ = 0 2 0 [ 8λµ + 6(( ) a v λ + ( ) b v µ + 6( ) a+b v λ+µ ] = 2 0 [ 4(λ + µ) 2 ( ) a+b 4v λ+µ +4((λ 2 + 4( ) a v λ )+ 4(µ 2 + 4( ) b v µ )] = 2 3 [e λ+µ,a+b+ + e λ,a + e µ,b ] if λ, µ = 2 e λ,a if ( λ, µ, ( ) a+b ) = (4,0), (-4,) 0 if ( λ, µ, ( ) ab ) = (4,), (-4,0) e(ϕ) if ϕ = ψ e(ϕ) e(ψ) = 0 if ϕψ singular 2 3 [e(ϕ) + e(ψ) e ϕψ,ϕ ] if ϕψ nonsingular { 0 [λ, ψ] = a + e λ,a e(ψ) = 2 3 [e(ψλ) e(ψ) e λ,ψ ] [λ, ψ] = a. Table 5.. Inner Products of Idempotents See the basic inner products in Section 2. We also need (f(ϕ), f(ψ)) from (4.5). 2 4 λ = µ (e λ,a, e µ,b ) = 2 9 λ, µ 2 + 2 5 ( ) a+b δ λ,µ = 0 λµ singular. 2 7 λµ nonsingular (e λ,a, e(ϕ)) = 2 8 + 2 8 ( ) a ϕ(α) = { 2 7 if ( ) a ϕ(α) =, i.e., a + [ϕ, α] = 0 0 if ( ) a ϕ(α) = i.e., a + [ϕ, α] =. { 240 ϕ = ψ (e(ϕ), e(ψ)) = 2 8 + 2 2 6 ϕψ singular 6 ϕψ nonsingular = { 2 4 0. 2 7
A VOA related to E 8 with automorphism group O + (0,2) 55 6. Idempotents and Involutions. Notation 6.. The polynomial p(t) := 32 3 t2 32 3 + takes values p(0) = p() = and p( 4 ) =. For an idempotent e such that ad(e) is semisimple with eigenvalues 0, 4 and, we define t(e) := p(ad(e)), an involution which is on the 0 and -eigenspaces and is on the 4 -eigenspace. Let E ± = E ± (e) = E ± (t(e)) denote the ± 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 follows from the theory in [Miy] and (5.2). In this section, we shall verify this directly on the algebra V2 F only, for the e ± λ and e(ϕ) and prove that these elements 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 + (0, 2). The Miyamoto theory proves that the t(e) are in Aut(V F ). It turns out that the group they generate restricts faithfully to V2 F is faithful, and there we can identify it. Theorem 6.4. The group generated by the t(e ± λ ) is isomorphic to the maximal 2-local subgroup of O + (0, 2) of shape 2 8 : O + (8, 2). The normal subgroup of order 2 8 is generated by all t(e + λ )t(e λ ) 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() ( means the trivial character) is generated by all t(e λ, ). Such a complement is isomorphic to the Weyl group of type E 8, modulo its center. To verify that the involution t(e) is an automorphism of V2 F, it suffices to check that E + E + + E E E + and E + E E. Proposition 6.5. If e is quooty, t(e) is an automorphism of V F 2. Proof. This is straightforward with (6.4) and (5.4). Proposition 6.6. If e is tooty, t(e) is an automorphism of V F 2. Proof. This is harder. We use (5.), (6.4), (5.8) and (5.). 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 orthogonal); 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(e λ,a ) on QTI: fixed are e µ,b if µ, λ = 0 or ±4; e(ϕ) if [λ, ϕ] = 0; interchanged are
56 R. L. Griess, Jr. e µ,b and e λ+µ,a+b+ if λ, µ = 2; e(ϕ) and e(ϕλ) if [λ, ϕ] =. (ii) The action of t(e(ϕ)) on QTI: fixed are all e λ,ϕ and all e(ψ) with ϕ = ψ or ϕψ singular; interchanged are all e(ϕλ) and e λ,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 + eigenspace for t = t(e) is the sum of the 0-eigenspace and the 4 -eigenspace for ad(e), a vector u is fixed by t(e) iff e u is in the 4 -eigenspace. Another useful observation is that if x y is negated, then (x y)2 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 on a suitable identification of this set of involutions with nonsingular points in F 0 2 with a maximal index nonsingular quadratic form. Notation 6.8. Let T := F 0 2 have a quadratic form q of maximal Witt index. Decompose T = U W, with dim(u) = 2, dim(w) = 8, both of plus type. Let U = {0, r, s, f}, where q(f) =, q(r) = q(s) = 0. Identify W with Hom(L, {±}). For x V nonsingular, write x = p + y, for p U, y W. If p = 0, correspond x to e y,. If p = r, correspond x to e y,0. If p {s, f}, correspond e(y) to x. This correspondence is G-equivariant; use (6.7). So, we have a map of G onto O + (0, 2) by restriction to V2 F. Its kernel fixes all of our idempotents, which span V2 F. By Corollary 6.2 of [DGH], this kernel is trivial. So, G = O + (0, 2) and (6.3) is proven. Proposition 6.9. G acts irreducibly on I (dimension 55). Proof. This follows from the character table of Ω + (0, 2), but we can give an elementary proof. () The subgroup H of (6.2) has an irreducible constituent P of dimension 20 with monomial basis v α, α L 2 ; (2) the squares of the v α generate the 36-dimensional orthogonal complement, P. The action fixes I and the action on the 35-dimensional space P I is nontrivial, hence irreducible (the subgroup O 2 (H) = 2 8 acts trivially and the quotient H/O 2 (H) = O + (8, 2) acts transitively on the spanning set of 20 elements vα 2 = α 2, so acts faithfully. Now, the subgroup 2 6 : O + (6, 2) = 2 6 : Sym 8 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 20, contradiction). (3) We now have V2 F = + 35 + 20 as a decomposition into H -irreducibles. But each t(e(ϕ)) fixes I and does not fix the 20-dimensional constitutent, whence irreduciblity of G on I.
A VOA related to E 8 with automorphism group O + (0,2) 57 Theorem 6.0. Aut(V F ) = G = O + (0, 2). Proof. Set A := Aut(V F ). We quote Theorem (6.3) of [Miy], which says that if X is the set of conformal vectors of central charge 2, then t(x)t(y) {, 2, 3}. So, if X is a conjugacy class, it is a set of 3-transpositions. If it is not a conjugacy class, we have a nontrivial central product decomposition of X, which is clearly impossible since A acts faithfully and G acts irreducibly on V2 F. Now, the classification of groups generated by a a class of 3-transpostions [Fi69][Fi7] may be invoked to identify A. It is a fairly straightforward exercise to eliminate any 3-transposition group which properly contains O + (0, 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 = 2 0+6 Ω + (0, 2), acts on the subvoa V of fixed points of O 2 (P); the degree 2 part V 2 contains V 2 F. In fact, V 2 is isomorphic to the direct sum of algebras V2 F (with ) and C. The proper subvoa V of V generated by the V2 F -part is isomorphic to V F (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 quooty and tooty ones, but these idempotents have 6 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 in MI, the monster, would induce O + (0, 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 Höhn, Geoffrey Mason, Masahiko Miyamoto and Steve Smith for discussions on this topic. This article was written with financial support from NSF grant DMS-9623038 and University of Michigan Faculty Recognition Grant (993-96). References [Carter] Roger Carter, Simple Groups of Lie Type, John Wiley, London, 989. [CoGr] Arjeh Cohen and Robert L. Griess, Jr., On finite simple subgroups of the complex Lie group of type E 8, Proc. Comm. Pure Math. 47(987), 367-405. [DGH] Chongying Dong, Gerald Höhn, 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, 988.
58 R. L. Griess, Jr. [Fi69] Fischer, Bernd, Finite Groups Generated by 3-Transpositions. Univ. of Warwick. Preprint 969. [Fi7] Fischer, Bernd, Finite Groups generated by 3-Transpositions. Inventions Math. 3 (97) 232 246. [Gr76] Robert L. Griess, Jr., A subgroup of order 2 5 GL(5, 2) in E 8 (C), the Dempwolff group and Aut(D 8 D 8 D 8 ), J. of Algebra 40, 27-279 (976). [Gr9] Robert L. Griess, Jr., Elementary abelian p-subgroups of algebraic groups, Geometriae Dedicata 39; 253-305, 99. [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 4809 09 USA email: rlg@math.lsa.umich.edu