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

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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 spanh (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 .
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