Griess Algebras and Conformal Vectors in Vertex Operator Algebras

✍ Scribed by Masahiko Miyamoto

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 275 KB
• Volume: 179
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.0023

✦ Synopsis

We define automorphisms of vertex operator algebra using the representations of the Virasoro algebra. In particular, we show that the existence of a special 1 element, which we will call a ''rational conformal vector with central charge ,'' 2 implies the existence of an automorphism of a vertex operator algebra. This result offers a simple construction of triality involutions of the Moonshine module V h . We also study the structures of Griess algebras and prove a conjecture given by Meyer Neutsch that the maximal dimension of associative subalgebras of the Griess Monster algebra is 48. ᮊ 1996 Academic Press, Inc. ''rational conformal vector.'' This element e defines a representation of 1 the Virasoro algebra with central charge on V. From the fusion rules 2 among the irreducible modules, we will prove that V has an automorphism of order at most 2. Furthermore, in the case dim V s 0 and V s 0, we e 0 1 1 shall also prove that e is a conformal vector with central charge if and 2 1 ² : only if er2 is an idempotent of Griess algebra V with er2, er2 s , 2 16 1

Ž .

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