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On Griess Algebras
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we prove that for any commutative (but in general non-associative) algebra $A$ with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra $V=V_0\oplus V_2\oplus V_3\oplus\cdots$, such that $\dim V_0=1$ and $V_2$ contains $A$. We can choose $V$ so that if $A$ has a unit $e$, then $2e$ is the Virasoro element of $V$, and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as well. In addition, the algebra $V$ can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.
Mots-clés : vertex algebra; Griess algebra. 
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}
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Michael Roitman. On Griess Algebras. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a56/

[1] Borcherds R. E., “Vertex algebras, Kac–Moody algebras, and the Monster”, Proc. Nat. Acad. Sci. U.S.A., 83 (1986), 3068–3071 | DOI | MR | Zbl

[2] Borcherds R. E., “Monstrous moonshine and monstrous Lie superalgebras”, Invent. Math., 109 (1992), 405–444 | DOI | MR | Zbl

[3] Dong C., “Representations of the moonshine module vertex operator algebra”, Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, MA, 1992), Contemp. Math., 175, 1994, 27–36 | MR | Zbl

[4] Dong C., Li H., Mason G., Montague P. S., “The radical of a vertex operator algebra”, The Monster and Lie algebras (Columbus, OH, 1996), Ohio State Univ. Math. Res. Inst. Publ., 7, de Gruyter, Berlin, 1998, 17–25 ; q-alg/9608022 | MR | Zbl

[5] Dong C., Lin Z., Mason G., “On vertex operator algebras as $\mathrm{sl}_2$-modules”, Groups, Difference Sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 4, de Gruyter, Berlin, 1996, 349–362 | MR | Zbl

[6] Dong C., Mason G., Zhu Y., “Discrete series of the Virasoro algebra and the moonshine module”, Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods, Part 2 (University Park, PA, 1991), Proc. Sympos. Pure Math., 56, American Mathematical Society, Providence, RI, 1994, 295–316 | MR | Zbl

[7] Frenkel I. B., Huang Y.-Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc., 104, no. 494, 1993 | MR

[8] Frenkel I. B., Lepowsky J., Meurman A., “A natural representation of the Fischer–Griess Monster with the modular function $J$ as character”, Proc. Nat. Acad. Sci. U.S.A., 81 (1984), 3256–3260 | DOI | MR | Zbl

[9] Frenkel I. B., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, 134, Academic Press, Boston, MA, 1988 | MR | Zbl

[10] Frenkel I. B., Zhu Y., “Vertex operator algebras associated to repersentations of affine and Virasoro algebras”, Duke Math. J., 66 (1992), 123–168 | DOI | MR | Zbl

[11] Griess R. L., “The friendly giant”, Invent. Math., 69 (1982), 1–102 | DOI | MR | Zbl

[12] Griess R. L., “GNAVOA. I. Studies in groups, nonassociative algebras and vertex operator algebras”, Vertex Operator Algebras in Mathematics and Physics (Toronto, ON, 2000), Fields Inst. Commun., 39, American Mathematical Society, Providence, RI, 2003, 71–88 | MR | Zbl

[13] Hubbard K., “The notion of vertex operator coalgebra and a geometric interpretation”, Comm. Algebra, 34 (2006), 1541–1589 ; math.QA/0405461 | DOI | MR | Zbl

[14] Kac V. G., Vertex algebras for beginners, 2nd ed., University Lecture Series, 10, American Mathematical Society, Providence, RI, 1998 | MR

[15] Lam C. H., “Construction of vertex operator algebras from commutative associative algebras”, Comm. Algebra, 24 (1996), 4339–4360 | DOI | MR | Zbl

[16] Lam C. H., “On VOA associated with special Jordan algebras”, Comm. Algebra, 27 (1999), 1665–1681 | DOI | MR | Zbl

[17] Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, 227, Birkhäuser Boston, Inc., Boston, MA, 2004 | MR | Zbl

[18] Li H., “Symmetric invariant bilinear forms on vertex operator algebras”, J. Pure Appl. Algebra, 96 (1994), 279–297 | DOI | MR | Zbl

[19] Li H., “Local systems of vertex operators, vertex superalgebras and modules”, J. Pure Appl. Algebra, 109 (1996), 143–195 ; hep-th/9406185 | DOI | MR | Zbl

[20] Li H., “An analogue of the Hom functor and a generalized nuclear democracy theorem”, Duke Math. J., 93 (1998), 73–114 ; q-alg/9706012 | DOI | MR | Zbl

[21] Primc M., “Vertex algebras generated by Lie algebras”, J. Pure Appl. Algebra, 135 (1999), 253–293 ; math.QA/9901095 | DOI | MR | Zbl

[22] Roitman M., “On free conformal and vertex algebras”, J. Algebra, 217 (1999), 496–527 ; math.QA/9809050 | DOI | MR | Zbl

[23] Roitman M., “Combinatorics of free vertex algebras”, J. Algebra, 255 (2002), 297–323 ; math.QA/0103173 | DOI | MR | Zbl

[24] Roitman M., “Invariant bilinear forms on a vertex algebra”, J. Pure Appl. Algebra, 194 (2004), 329–345 ; math.QA/0210432 | DOI | MR | Zbl

[25] Tits J., “On Griess' “friendly giant””, Invent. Math., 78 (1984), 491–199 | DOI | MR

[26] Wang W., “Rationality of Virasoro vertex operator algebras”, Internat. Math. Res. Notices, 1993:7 (1993), 197–211 | DOI | MR | Zbl

[27] Xu X., Introduction to vertex operator superalgebras and their modules, Mathematics and Its Applications, 456, Kluwer Academic Publishers, Dordrecht, 1998 | MR