Binary Codes and Vertex Operator (Super)Algebras

✍ Scribed by Masahiko Miyamoto

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 193 KB
• Volume: 181
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0116

✦ Synopsis

We study a vertex operator algebra whose Virasoro element is a sum of pairwise 1 orthogonal rational conformal vectors with central charge . The most important 2 example is the moonshine module V h . In particular, we construct a series of vertex operator algebras whose full automorphism groups are finite. Namely, we construct ϱ Ž . n vertex operator algebras M s Ý M from even linear binary code D : ‫ޚ‬ D i s 0 D i 2 Ž . and prove that if the minimum weight of D is greater than 2 then M s 0 and D 1 the full automorphism group of M is finite. From the viewpoint of finite group D theory, the construction of a vertex operator algebra has one advantage. We can expect not only automorphisms of D, but also another one. Indeed, if D contains a w x 8, 4, 4 Hamming subcode C, then C defines a nontrivial involutive automorphism of M , which is not induced from the automorphism group of D. In particular, we D Ž .Ž n H . have a finite group extension of Aut D ‫ޚ‬ rD .