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17
Associative subalgebras of the Griess algebra and related topics
 Ohio State University
, 1996
"... In [DMZ] it was shown that the moonshine module V ♮ contains a sub vertex operator algebra isomorphic to the tensor product L ( 1 2, 0)⊗48 where L ( 1 ..."
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Cited by 24 (5 self)
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In [DMZ] it was shown that the moonshine module V ♮ contains a sub vertex operator algebra isomorphic to the tensor product L ( 1 2, 0)⊗48 where L ( 1
McKay’s observation and vertex operator algebras generated by two conformal vectors of central charge 1/2, in preparation
 MR 2006h:17034 Zbl 1082.17015 KENICHIRO TANABE AND HIROMICHI
"... This paper is a continuation of the authors ’ work [33] in which several coset subalgebras of the lattice VOA V √ were constructed and the relationship between such algebras ..."
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Cited by 6 (5 self)
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This paper is a continuation of the authors ’ work [33] in which several coset subalgebras of the lattice VOA V √ were constructed and the relationship between such algebras
Class 2 Moufang loops small Frattini Moufang loops and code loops
 http://lanl.arxiv.org/pdf/math.GR/9611214 64. Vasantha.W.B and Rajadurai.R.S, Hyperloops and hypergroupoids, Ultra Sci. Vol.12
, 1996
"... Abstract. Let L be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearlyderived subloop (normal associator subloop) L ∗ of L has exponent dividing 6. It follows that Lp (the subloop of L of elements of ppower order) is associative for p> 3. Next, a loop L is s ..."
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Cited by 4 (0 self)
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Abstract. Let L be a Moufang loop which is centrally nilpotent of class 2. We first show that the nuclearlyderived subloop (normal associator subloop) L ∗ of L has exponent dividing 6. It follows that Lp (the subloop of L of elements of ppower order) is associative for p> 3. Next, a loop L is said to be a small Frattini Moufang loop, or SFML, if L has a central subgroup Z of order p such that C ∼ = L/Z is an elementary abelian pgroup. C is thus given the structure of what we call a coded vector space, or CVS. (In the associative/group case, CVS’s are either orthogonal spaces, for p = 2, or symplectic spaces with attached linear forms, for p> 2.) Our principal result is that every CVS may be obtained from an SFML in this way, and two SFML’s are isomorphic in a manner preserving the central subgroup Z if and only if their CVS’s are isomorphic up to scalar multiple. Consequently, we obtain the fact that every SFM 2loop is a code loop, in the sense of Griess, and we also obtain a relatively explicit characterization of isotopy in SFM 3loops. (This characterization of isotopy is easily extended to Moufang loops of class 2 and exponent 3.) Finally, we sketch a method for constructing any finite Moufang loop which is centrally nilpotent of class 2. 1.
VOAs generated by two conformal vectors whose τinvolutions generate S3
 J. Algebra
"... We determined the inner products of two conformal vectors with central charge 1 2 whose τinvolutions generates S3 if none of τinvolutions are trivial. We also see that a subVA generated by such conformal vectors is a VOA with central charge and has a Griess algebra of dimension three or four, resp ..."
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Cited by 2 (0 self)
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We determined the inner products of two conformal vectors with central charge 1 2 whose τinvolutions generates S3 if none of τinvolutions are trivial. We also see that a subVA generated by such conformal vectors is a VOA with central charge and has a Griess algebra of dimension three or four, respectively. 1 2
Local subgroups of the Monster and odd code loops
 Trans. Amer. Math. Soc
, 1995
"... Abstract. The main result of this work is an explicit construction of plocal subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order 32, 52, and 72 and have shapes 32+5+10(Af11 xGL(2, 3)), 52+2+4(S3xGL(2, ..."
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Abstract. The main result of this work is an explicit construction of plocal subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order 32, 52, and 72 and have shapes 32+5+10(Af11 xGL(2, 3)), 52+2+4(S3xGL(2, 5)), and 72+1+2 • GL(2, 7). These groups result from a general construction which proceeds in three steps. We start with a selforthogonal code C of length n over the field Fp, where p is an odd prime. The first step is to define a code loop L whose structure is based on C. The second step is to define a group N of permutations of functions from F2 to L. The final step is to show that N has a normal subgroup K of order p2. The result of this construction is the quotient group N/K of shape p2+m+2m(S x GL(2,p)), where m + 1 = dim(C) and S is the group of permutations of Aut(C). To show that the groups we construct are contained in the Monster, we make use of certain lattices A(C), defined in terms of the code C. One step in demonstrating this is to show that the centralizer of an element of order p in N/K is contained in the centralizer of an element of order p in the Monster. The lattices are useful in this regard since a quotient of the automorphism group of the lattice is a composition factor of the appropriate centralizer in the Monster. This work was inspired by a similar construction using code loops based on binary codes that John Conway used to construct a subgroup of the Monster of shape 22+11+22 • (M24 x GL(2, 2)). 1.
Octonions and the Leech lattice
, 2008
"... We give a new, elementary, description of the Leech lattice in terms of octonions, thereby providing the first real explanation of the fact that the number of minimal vectors, 196560, can be expressed in the form 3 × 240 × (1 + 16 + 16 × 16). We also give an easy proof that it is an even selfdual l ..."
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Cited by 2 (1 self)
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We give a new, elementary, description of the Leech lattice in terms of octonions, thereby providing the first real explanation of the fact that the number of minimal vectors, 196560, can be expressed in the form 3 × 240 × (1 + 16 + 16 × 16). We also give an easy proof that it is an even selfdual lattice. 1
3transposition groups of symplectic type and vertex operator algebras
 Journal of the Mathematical Society of Japan
"... The 3transposition groups that act on a vertex operator algebra in the way described by Miyamoto in [Mi1] are classified under the assumption that the group is centerfree and the VOA carries a positivedefinite invariant Hermitian form. This generalizes and refines the result of Kitazume and Miyamo ..."
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Cited by 1 (0 self)
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The 3transposition groups that act on a vertex operator algebra in the way described by Miyamoto in [Mi1] are classified under the assumption that the group is centerfree and the VOA carries a positivedefinite invariant Hermitian form. This generalizes and refines the result of Kitazume and Miyamoto [KM]. Application to a similar but different situation is also considered in part by a slight generalization of the argument.
Arithmetic groups and the affine E8 Dynkin diagram
, 2007
"... Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2(R) to nodes of the affine E8 Dy ..."
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Several decades ago, John McKay suggested a correspondence between nodes of the affine E8 Dynkin diagram and certain conjugacy classes in the Monster group. Thanks to Monstrous Moonshine, this correspondence can be recast as an assignment of discrete subgroups of PSL2(R) to nodes of the affine E8 Dynkin diagram. The purpose of this article is to give an explanation for this latter correspondence using elementary properties of the group PSL2(R). We also obtain a super analogue of McKay’s observation, in which conjugacy classes of the Monster are replaced by conjugacy classes of Conway’s group — the automorphism group of the Leech lattice. 1
ISING VECTORS IN THE VERTEX OPERATOR ALGEBRA V + Λ ASSOCIATED WITH THE LEECH LATTICE Λ CHING HUNG LAM AND HIROKI SHIMAKURA
, 810
"... Abstract. In this article, we study the Ising vectors in the vertex operator algebra V + Λ associated with the Leech lattice Λ. The main result is a characterization of the Ising vectors in V + Λ. We show that for any Ising vector e in V Λ, there is a sublattice E ∼ = √ 2E8 of Λ such that e ∈ V + ..."
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Abstract. In this article, we study the Ising vectors in the vertex operator algebra V + Λ associated with the Leech lattice Λ. The main result is a characterization of the Ising vectors in V + Λ. We show that for any Ising vector e in V Λ, there is a sublattice E ∼ = √ 2E8 of Λ such that e ∈ V + E. Some properties about their corresponding τinvolutions in the moonshine vertex operator algebra V ♮ are also discussed. We show that there is no Ising vector of σtype in V ♮. Moreover, we compute the centralizer C Aut V ♮(z, τe) for any Ising vector e ∈ V + Λ, where z is a 2B element in Aut V ♮ which fixes V + Λ. Based on this result, we also obtain an explanation for the 1A case of an observation by GlaubermanNorton (2001), which describes some mysterious relations between the centralizer of z and some 2A elements commuting z in the Monster and the Weyl groups of certain sublattices of the root lattice of type E8. 1.
Vertex operators and sporadic groups
, 2007
"... In the 1980’s, the work of Frenkel, Lepowsky and Meurman, along with that of Borcherds, culminated in the notion of vertex operator algebra, and an example whose full symmetry group is the largest sporadic simple group: the Monster. Thus it was shown that the vertex operators of mathematical physics ..."
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In the 1980’s, the work of Frenkel, Lepowsky and Meurman, along with that of Borcherds, culminated in the notion of vertex operator algebra, and an example whose full symmetry group is the largest sporadic simple group: the Monster. Thus it was shown that the vertex operators of mathematical physics play a role in finite group theory. In this article we describe an extension of this phenomenon by introducing the notion of enhanced vertex operator algebra, and constructing examples that realize other sporadic simple groups, including one that is not involved in the Monster. 1