Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 057, 35 pages      math.QA/0302021      https://doi.org/10.3842/SIGMA.2008.057
Contribution to the Special Issue on Kac-Moody Algebras and Applications

On Griess Algebras

Michael Roitman
Department of Mathematics, Kansas State University, Manhattan, KS 66506 USA

Received February 29, 2008, in final form July 28, 2008; Published online August 13, 2008

Abstract
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 = V0 Å V2 Å V3 Å ¼, such that dim V0 = 1 and V2 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.

Key words: vertex algebra; Griess algebra.

pdf (526 kb)   ps (295 kb)   tex (44 kb)

References

  1. Borcherds R.E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 3068-3071.
  2. Borcherds R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405-444.
  3. Dong C., Representations of the moonshine module vertex operator algebra, in Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, MA, 1992), Contemp. Math. 175 (1994), 27-36.
  4. Dong C., Li H., Mason G., Montague P.S., The radical of a vertex operator algebra, in The Monster and Lie algebras (Columbus, OH, 1996), Ohio State Univ. Math. Res. Inst. Publ., Vol. 7, de Gruyter, Berlin, 1998, 17-25, q-alg/9608022.
  5. Dong C., Lin Z., Mason G., On vertex operator algebras as sl2-modules, in Groups, Difference Sets, and the Monster (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., Vol. 4, de Gruyter, Berlin, 1996, 349-362.
  6. Dong C., Mason G., Zhu Y., Discrete series of the Virasoro algebra and the moonshine module, in Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Part 2, American Mathematical Society, Providence, RI, 1994, 295-316.
  7. Frenkel I.B., Huang Y.-Z., Lepowsky J., On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494.
  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.
  9. Frenkel I.B., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Boston, MA, 1988.
  10. Frenkel I.B., Zhu Y., Vertex operator algebras associated to repersentations of affine and Virasoro algebras, Duke Math. J. 66 (1992), 123-168.
  11. Griess R.L., The friendly giant, Invent. Math. 69 (1982), 1-102.
  12. Griess R.L., GNAVOA. I. Studies in groups, nonassociative algebras and vertex operator algebras, in Vertex Operator Algebras in Mathematics and Physics (Toronto, ON, 2000), Fields Inst. Commun., Vol. 39, American Mathematical Society, Providence, RI, 2003, 71-88.
  13. Hubbard K., The notion of vertex operator coalgebra and a geometric interpretation, Comm. Algebra 34 (2006), 1541-1589, math.QA/0405461.
  14. Kac V.G., Vertex algebras for beginners, 2nd ed., University Lecture Series, Vol. 10, American Mathematical Society, Providence, RI, 1998.
  15. Lam C.H., Construction of vertex operator algebras from commutative associative algebras, Comm. Algebra 24 (1996), 4339-4360.
  16. Lam C.H., On VOA associated with special Jordan algebras, Comm. Algebra 27 (1999), 1665-1681.
  17. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
  18. Li H., Symmetric invariant bilinear forms on vertex operator algebras, J. Pure Appl. Algebra 96 (1994), 279-297.
  19. Li H., Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), 143-195, hep-th/9406185.
  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.
  21. Primc M., Vertex algebras generated by Lie algebras, J. Pure Appl. Algebra 135 (1999), 253-293, math.QA/9901095.
  22. Roitman M., On free conformal and vertex algebras, J. Algebra 217 (1999), 496-527, math.QA/9809050.
  23. Roitman M., Combinatorics of free vertex algebras, J. Algebra 255 (2002), 297-323, math.QA/0103173.
  24. Roitman M., Invariant bilinear forms on a vertex algebra, J. Pure Appl. Algebra 194 (2004), 329-345, math.QA/0210432.
  25. Tits J., On Griess' "friendly giant", Invent. Math. 78 (1984), 491-199.
  26. Wang W., Rationality of Virasoro vertex operator algebras, Internat. Math. Res. Notices 1993 (1993), no. 7, 197-211.
  27. Xu X., Introduction to vertex operator superalgebras and their modules, Mathematics and Its Applications, Vol. 456, Kluwer Academic Publishers, Dordrecht, 1998.

Previous article   Next article   Contents of Volume 4 (2008)