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Groups of Lie Type, Vertex Algebras, and Modular Moonshine

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Groups of Lie Type, Vertex Algebras, and Modular Moonshine 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 automorphisms. In this way, we get series of vertex algebras over fields whose automorphism 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 group which is a Chevalley or Steinberg variations over Fq as \most of" the full automorphism group of a vertex algebra over Fq. 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. 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 sporadic group 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 Lie algebra 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 Center 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 root system 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 simple group. 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-vector space 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 lattice 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 ring 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 abelian group A and a subgroup 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 2 V . A vertex algebra V is Z-graded if V = ⊕n2ZVn for subspaces Vn such that if a 2 Vn, b 2 Vm, then akb 2 Vn+m−k−1 for any integer k. For the definition of vertex operator algebra 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 s! 2 J, for each n, Jn := J \ Vn is an integral form of Vn,(J; J) 2 Q. Since J is a VA, 1 2 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) ⊗ CfLg 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.
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