Abstract
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The “Moonshine” representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
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Selected References
These references are in PubMed. This may not be the complete list of references from this article.
- Frenkel I. B., Lepowsky J., Meurman A. A natural representation of the Fischer-Griess Monster with the modular function J as character. Proc Natl Acad Sci U S A. 1984 May;81(10):3256–3260. doi: 10.1073/pnas.81.10.3256. [DOI] [PMC free article] [PubMed] [Google Scholar]