CHAPTER 0 Introduction Algebras range in diversity from such fundamental classes as associative and Lie algebras to rather exotic examples such as the Chevalley and Griess algebras. The Chevalley algebra is a commutative nonassociative algebra constructed on the direct sum of the three 8-dimensional representations of the Lie algebra D4 The Griess algebra is a commutative nonassociative algebra constructed on the direct sum of the trivial representation and the minimal nontrivial representa- tion of the Monster group. Both exceptional algebras have some similarity to the exceptional Lie algebra Es, but one does not expect to obtain them as sub- structures of algebras in either general class of algebras. However, it does turn out that these exotic algebras are substructures of examples of a new class of algebras having remarkable similarities to the associative and Lie algebras. The new class of algebras, called vertex operator algebras or chiral algebras, has been introduced independently in mathematics [B,FLM] and in physics (cf. [BPZ]). Vertex operator algebras can be thought of as an algebraic foundation of conformal field theory, which has become the crossroads of numerous mathemat- ical and physical theories. The physical motivation for the study of conformal field theory and chiral algebras was the development of two-dimensional statis- tical mechanics and string theory, the modern approach to quantum gravity. In the most promising models of string theory the exceptional Lie algebra Es plays a central role. (See [GSW] for references to the physics literature.) An important mathematical motivation for the introduction of vertex operator algebras was the attempt to understand a natural realization of the Monster as a symmetry group, and to give an axiomatic approach to the Griess algebra [G]. After the axioms of the vertex operator algebras finally crystalized, it became clear that, in a deep sense, these algebraic structures are analogous to the associative and Lie algebras. Furthermore, one particular vertex operator algebra contains the Griess algebra as a (weight 2) substructure [FLM]. The purpose of this work is multifold. First we want to understand certain generalizations of vertex operator algebras. In one such algebra the Chevalley algebra can be found as a (weight ~) substructure, showing the close analogy with the Griess algebra. The generalizations of vertex operator algebras which we study are natural and useful. The simplest generalizations, vertex operator superalgebras [Go,T], are involved in superconformal field theory. Beyond that, 1
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