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.

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