Simplicity of a Vertex Operator Algebra Whose Griess Algebra is the Jordan Algebra of Symmetric Matrices

Abstract

Let r ∈ ℂ be a complex number, and d ∈ ℤ_≥2 a positive integer greater than or equal to 2. Ashihara and Miyamoto [4] introduced a vertex operator algebra V _𝒥 of central charge dr, whose Griess algebra is isomorphic to the simple Jordan algebra of symmetric matrices of size d. In this article, we prove that the vertex operator algebra V _𝒥 is simple if and only if r is not an integer. Further, in the case that r is an integer (i.e., V _𝒥 is not simple), we give a generator system of the maximal proper ideal I _r of the VOA V _𝒥 explicitly.

Key Words:

• Jordan algebra
• Vertex operator algebra

2000 Mathematics Subject Classification:

• Primary 17B69, 17C50
• Secondary 16W10

ACKNOWLEDGMENTS

The authors would like to express their sincere gratitude to Professor Masahiko Miyamoto, Professor Toshiyuki Abe, Professor Hiroki Shimakura, Professor Hiroshi Yamauchi, and Dr. Takahiro Ashihara for valuable discussions.

Notes

Communicated by D. Nakano.