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Keywords:
nonassociative algebra; nonassociative commutative algebra; groups of Lie type; sporadic groups; vertex operator algebras; lattice type vertex operator algebras; axioms; $(B,N)$-pair; monster; $2A$-involutions; Jordan algebra; pairwise orthogonal idempotents; $E_8$; $E_6$; polynomial identity
nonassociative algebra; nonassociative commutative algebra; groups of Lie type; sporadic groups; vertex operator algebras; lattice type vertex operator algebras; axioms; $(B,N)$-pair; monster; $2A$-involutions; Jordan algebra; pairwise orthogonal idempotents; $E_8$; $E_6$; polynomial identity
Summary:
We discuss some examples of nonassociative algebras which occur in VOA (vertex operator algebra) theory and finite group theory. Methods of VOA theory and finite group theory provide a lot of nonassociative algebras to study. Ideas from nonassociative algebra theory could be useful to group theorists and VOA theorists.
References:
[1] Cohen A.M., Griess R.L., Jr.: On finite simple subgroups of the complex Lie group of type $E_8$. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 367–405, Proc. Sympos. Pure Math., 47, Part 2, Amer. Math. Soc., Providence, RI, 1987. . MR 0933426
[2] Ashihara T., Miyamoto M.: Deformation of central charges, vertex operator algebras whose Griess algebras are Jordan algebras. J. Algebra 321 (2009), no. 6, 1593–1599. MR 2498258
[3] Cohen A.M., Wales D.B.: Finite subgroups of $F\sb 4(C)$ and $E\sb 6(C)$. Proc. London Math. Soc. (3) 74 (1997), no. 1, 105–150. MR 1416728
[4] Conway J.H.: A simple construction for the Fischer-Griess monster group. Invent. Math. 79 (1985), no. 3, 513–540. MR 0782233 | Zbl 0564.20010
[5] Dong C., Nagatomo K.: Automorphism groups and twisted modules for lattice vertex operator algebras. in Recent Developments in Quantum Affine Algebras and Related Topics, Contemp. Math., 248, pp. 117–133, Amer. Math. Soc., Providence, RI, 1999. MR 1745258 | Zbl 0953.17014
[6] Dong C., Griess R.L., Jr.: Rank one lattice type vertex operator algebras and their automorphism groups. J. Algebra 208 (1998), 262–275. q-alg/9710017. MR 1644007 | Zbl 1028.17019
[7] Dong C., Griess, R.L., Jr., Ryba A.J.E.: Rank one lattice type vertex operator algebras and their automorphism groups. II: E-series, J. Algebra 217 (1999), 701–710. MR 1700522 | Zbl 1028.17019
[8] Frenkel I., Lepowsky J., Meurman A.: Vertex Operator Algebras and the Monster. Pure and Applied Math., 134, Academic Press, Boston, 1988. MR 0996026 | Zbl 0674.17001
[9] Gebert R.W.: Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebra. Internat. J. Modern Phys. A 8 (1993), no. 31, 5441–5503. MR 1248070 | Zbl 0860.17039
[10] Griess R.L., Jr.: A construction of $F_1$ as automorphisms of a $196,883$-dimensional algebra. Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 686–691. MR 0605419 | Zbl 0452.20020
[11] Griess R.L., Jr.: The friendly giant. Invent. Math. 69 (1982), 1–102. MR 0671653 | Zbl 0498.20013
[12] Griess R.L., Jr.: Sporadic groups, code loops and nonvanishing cohomology. J. Pure Appl. Algebra 44 (1987), 191–214. MR 0885104 | Zbl 0611.20009
[13] Griess R.L., Jr.: Code loops and a large finite group containing triality for $D_4$. Proc. Atti del Convegno Internazionale di Teoria Dei Gruppi e Geometria Combinatoria (Firenze, October 1986), Serie II, 19, 1988, pp. 79–98. MR 0988189
[14] Griess R.L., Jr.: A Moufang loop, the exceptional Jordan algebra and a cubic form in $27$ variables. J. Algebra 131 (1990), no. 1, 281–293. MR 1055009 | Zbl 0718.17028
[15] Griess R.L., Jr.: Codes, Loops and $p$-Locals. Groups, Difference Sets and the Monster (Columbus, OH, 1993), pp. 369–375, de Gruyter, Berlin, 1996. MR 1400428 | Zbl 0865.94030
[16] Griess R.L., Jr.: A vertex operator algebra related to $E_8$ with automorphism group $O^+(10,2)$. The Monster and Lie Algebras (Columbus, OH 1996), ed. J. Ferrar, K. Harada, de Gruyter, Berlin, 1998. MR 1650633
[17] Griess R.L., Jr.: The monster and its nonassociative algebra. in Proceedings of the Montreal Conference on Finite Groups, Contemporary Mathematics, 45, 121-157, 1985, American Mathematical Society, Providence, RI. MR 0822237 | Zbl 0582.20007
[18] Griess R.L., Jr., Ryba A.J.E.: Finite simple groups which projectively embed in an exceptional Lie group are classified!. Bull. Amer. Math. Soc. 36 (1999), no. 1, 75–93. MR 1653177 | Zbl 0916.22008
[19] Griess R.L., Jr., Ryba A.J.E.: Quasisimple finite subgroups of exceptional algebraic groups. Journal of Group Theory, 2002, 1–39. MR 1879514
[20] Griess R.L., Jr., Höhn G.: Virasoro frames and their stabilizers for the $E_8$ lattice type vertex operator algebra. J. Reine Angew. Math. 561 (2003), 1–37. MR 1998606
[21] Griess R.L., Jr.: GNAVOA, I. Studies in groups, nonassociative algebras and vertex operator algbras. Vertex Operator Algebras in Mathematics and Physics (Toronto, 2000), pp. 71–88, Fields Ins. Commun., 39, Amer. Math. Soc., Providence, 2003. MR 2029791
[22] Lam C.H.: Construction of vertex operator algebras from commutative associative algebras. Comm. Algebra 24 (1996), no. 14, 4339–4360. MR 1421193 | Zbl 0892.17020
[23] Lam C.H.: On VOA associated with special Jordan algebras. Comm. Algebra 27 (1999), no. 4, 1665–1681. MR 1679676 | Zbl 0934.17016
[24] Miyamoto M.: Griess algebras and conformal vectors in vertex operator algebras. J. Algebra 179 (1996), no. 2, 523–548. MR 1367861 | Zbl 0964.17021
[25] Meyer W., Neutsch W.: Associative subalgebras of the Griess algebra. J. Algebra 158 (1993), no. 1, 1–17. MR 1223664 | Zbl 0789.17002
[26] Norton S.: The Monster Algebra: some new formulae. appeared in AMS Contemp. Math. 193 “Moonshine, the Monster and Related Topics” (eds. Chongying Dong and Geoffrey Mason), 1996, pp. 297–306 (Conference at Mount Holyoke, 1994). MR 1372728 | Zbl 0847.11023
[27] Roitman M.: On Griess algebras. Symmetry, Integrability and Geometry, Methods and Applications, SIGMA 4 (2008); http://www.emis.de/journals/SIGMA/2008/057/ MR 2434941
[28] Smith S.D.: Nonassociative commutative algebras for triple covers of $3$-transposition groups. Michigan Math. J. 24 (1977), 273–287. MR 0491931 | Zbl 0391.20011
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