Generalized vertex algebras and relative vertex operators.

*(English)*Zbl 0803.17009
Progress in Mathematics (Boston, Mass.). 112. Basel: Birkhäuser. ix, 202 p. sFr. 86.00; DM 98.00; öS 764.40; £38.00 /hc (1993).

In recent years, vertex operator algebra theory has developed in a rapid pace and several interesting connections with other areas of mathematics and physics – such as finite groups, conformal field theory and braid group theory, have been established. Vertex operator algebras are mathematical counterparts of chiral algebras in conformal field theory.

In this monograph, the authors present a systematic study of several natural generalizations of the concept of vertex operator algebra. In particular, the generalized vertex operator algebra is discussed in chapter 6 and its generalization, the generalized vertex algebra is studied in chapter 9. Finally, the notion of abelian intertwining algebra, which is a generalization of the generalized vertex algebra is presented in chapter 12. An abelian intertwining algebra can be thought of roughly as being formed by adjoining to a vertex algebra the direct sum of a family of modules (representations) of a special type. This algebra is graded by an abelian group \(G\). The subspace corresponding to the identity element of \(G\) is essentially the vertex algebra and the subspaces corresponding to the various group elements are the modules.

In chapter 13, the authors use their general theory to construct a family of intertwining operators relating higher-level standard (integrable highest weight) modules for affine Lie algebras. Finally, in the last chapter, the essential equivalence between \(Z\)-algebras and parafermion algebras has been established by realizing the parafermion algebras as canonically modified \(Z\)-algebras acting on certain quotient spaces of the vacuum spaces of standard \(A_ 1^{(1)}\)-modules defined by the action of an infinite cyclic group.

This monograph is well written, accessible and self-contained. It would be suitable for an upper level graduate course in vertex operator algebras and their generalizations.

In this monograph, the authors present a systematic study of several natural generalizations of the concept of vertex operator algebra. In particular, the generalized vertex operator algebra is discussed in chapter 6 and its generalization, the generalized vertex algebra is studied in chapter 9. Finally, the notion of abelian intertwining algebra, which is a generalization of the generalized vertex algebra is presented in chapter 12. An abelian intertwining algebra can be thought of roughly as being formed by adjoining to a vertex algebra the direct sum of a family of modules (representations) of a special type. This algebra is graded by an abelian group \(G\). The subspace corresponding to the identity element of \(G\) is essentially the vertex algebra and the subspaces corresponding to the various group elements are the modules.

In chapter 13, the authors use their general theory to construct a family of intertwining operators relating higher-level standard (integrable highest weight) modules for affine Lie algebras. Finally, in the last chapter, the essential equivalence between \(Z\)-algebras and parafermion algebras has been established by realizing the parafermion algebras as canonically modified \(Z\)-algebras acting on certain quotient spaces of the vacuum spaces of standard \(A_ 1^{(1)}\)-modules defined by the action of an infinite cyclic group.

This monograph is well written, accessible and self-contained. It would be suitable for an upper level graduate course in vertex operator algebras and their generalizations.

Reviewer: Kailash C. Misra (Raleigh)

##### MSC:

17B69 | Vertex operators; vertex operator algebras and related structures |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

17B68 | Virasoro and related algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |